Free Discontinuity Problems in Image and Signal Segmentationvittia/phd/vitti_phd.pdf · supervisor...

Free Discontinuity Problems in

Image and Signal Segmentation

Alfonso Vitti

2008

Doctoral thesis in Environmental Engineering

Free Discontinuity Problems in

Image and Signal Segmentation

Alfonso Vitti

Trento - Italy 2008

Doctoral thesis in Environmental Engineering (XVIII cycle)

Faculty of Engineering, University of Trento

Year: 2008

Supervisor: Prof. Battista Benciolini

Universita degli Studi di Trento

Trento, Italy

2008

A Barbara,

Acknowledgements

The first and biggest thanks go to Battista, being formal Prof. Giovanni Battista Benciolini, for his

scientific curiosity, always accompanied by rigour and precision, that generates and spreads around interests

and new curiosity, and for always being prompt to answer my questions, to open new ones and to explain

how things get linked each other. I really enjoy working with him and learning from such a person.

Special thanks go to Paolo Zatelli, for having, now some years ago, introduced me to GRASS, for having

shared with me many projects and discussions and for having been always ready to support me.

Remaining more strictly related to the topic of the thesis, it must be said that the first interest of my

supervisor on free discontinuity problems and on variational image segmentation comes from the lecture of

Marco Biroli on Free Boundary Problems during the school ”Geodetic Boundary Value problems in View

on the One Centimeter Geoid” organized by Fernando Sanso and Reiner Rummel in Como in 1996. A

second contact with the topic was a seminar of Riccardo March held in Trento in 2002 on invitation of

Italo Tamanini.

More recently, during the time spent in working on the thesis, the contribution of the following people

has been of particular relevance.

The author want to gratefully acknowledge:

Prof. Anneliese Defranceschi1 for the clear and useful explanations on some aspects of the Calculus of

Variations and on the Euler equation arising in variational problems and for the kind and deep review of

this work;

Prof. Riccardo March2 for the quick and useful answers and comments provided on numerical aspects;

Prof. Italo Tamanini3 for having provided me two very useful master theses on one dimensional variational

problems with free discontinuity and on their numerical implementation.

Prof. Athanasios Dermanis4 is warmly acknowledged for the kind willingness on reviewing my work and

fort the very friendly and useful discussions during our meeting in Milano and in Munich.

Alfonso Vitti

1Dipartimento di Matematica - Universita di Trento2Istituto per le Applicazioni del Calcolo ”Mauro Picone” - CNR3Dipartimento di Matematica - Universita di Trento4Department of Geodesy and Surveying - The Aristotle University of Thessaloniki

Contents

Introduction 1

I A Review of the Mathematical Theory of Variational Segmenta-

tion 5

1 Mathematical Framework 7

1.1 Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 Classical and Direct Methods in the Calculus of Variations . . . . . . . . . 8

1.2 Free Discontinuity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 A General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.2 The Space of Special Functions of Bounded Variation . . . . . . . . . . . . 12

1.2.3 Some Essential Features About Γ-convergence . . . . . . . . . . . . . . . . . 14

1.3 Variational Segmentation: The Mumford and Shah Model . . . . . . . . . . . . . . 15

2 The Mumford and Shah Model 19

2.1 The Mumford and Shah Model for Image Segmentation . . . . . . . . . . . . . . . 19

2.1.1 The Mumford and Shah Functional . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Weak Formulation in SBV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.3 Approximations of the Weak Formulation . . . . . . . . . . . . . . . . . . . 21

2.1.4 Drawbacks of the Mumford and Shah Model . . . . . . . . . . . . . . . . . 23

2.2 Controlling the Curvature in the Mumford and Shah Model . . . . . . . . . . . . . 24

2.3 Euler Equation and Gradient Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 The Euler Equation Associated to the Elliptic Approximation of the Mum-

ford and Shah Model without and with the Curvature Term . . . . . . . . . 26

2.4 Finite Differences Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 The Gradient Effect in Dimension One . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 The Blake and Zisserman Model 35

3.1 The Blake and Zisserman Model for One Dimensional Signal Segmentation . . . . 36

3.2 The Ambrosio and Tortorelli Approximation . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Finite Elements Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Contents

II Numerical Applications 49

4 Software Implementations 51

4.1 The seglib Library for Images Segmentation . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 The GRASS GIS module r.seg . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 The seglib1d Library for One Dimensional Signals Segmentation . . . . . . . . . . . 56

4.2.1 The sigseg Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Segmentation of Images 67

5.1 Segmentation of Synthetic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.1 Test Image A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1.2 Test Image B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.3 Test Image C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.4 Test Image D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Segmentation of Real Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.1 Lenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.2 Aeroplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Segmentation of Real ”Environmental” Images . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Pebbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.2 Braided River-bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3.3 High Resolution Ortho-Photo . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6 Segmentation of One Dimensional Signals 105

6.1 Segmentation of Synthetic One Dimensional Signals . . . . . . . . . . . . . . . . . 106

6.1.1 Occurrence and overcoming of the Gradient Effect . . . . . . . . . . . . . . 106

6.1.2 A Complex Signal and a Complex Noisy Signal . . . . . . . . . . . . . . . . 110

6.2 Segmentation of GNSS Coordinates Time Series . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Detection of a Singular Jump . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.2 Different Segmentations of the Same Time Series . . . . . . . . . . . . . . . 116

References 119

x

Introduction

This thesis aims to inspect and to understand, from an engineering point of view, the mathematical

framework of variational problems and in particular of ”Free Discontinuity Problems” where the

concept of image and signal segmentation can be rigorously formulated. The second objective is

to develop end-user software libraries and programs implementing two variational segmentation

models to perform practically images and one dimensional signals segmentation. In the thesis

there are no original mathematical contributions, that would be outside the competences of the

author. The mathematical problems have been approached and studied with the aim of making a

bridge between the theory and the applications. Part (I) of the thesis is the outcome of this study.

The main thesis results are the development of original software libraries and programs to perform

segmentation of images and of one dimensional signals and the application of such programs to real

data sets. The developed numerical programs allowed also to study and to reproduce in practice

the well known features of the theoretical formulations of segmentation approached as a ”Free

Discontinuity Problem”.

This work deals with the Mumford and Shah functional and with the Blake and Zisserman

functional.

The Mumford and Shah functional in dimension two is introduced, studied, implemented numer-

ically and applied to the segmentation of images. An extension of the Mumford and Shah is treated

analogously. The Blake and Zisserman functional is introduced and studied in dimension two and

then in dimension one, while the numerical implementation and the applications are performed

in dimension one, i.e to the segmentation of one dimensional signals. Specifically, the thesis faces

the problem of segmenting environmental images, such as aerophotogrammetry images and one

dimensional signal time series arising from Geodesy, such as GNSS (Global Navigation Satellite

System) permanent stations coordinates.

Segmentation can be intuitively defined as the process of partitioning a domain, according to

some criteria, into disjoint and homogeneous regions while detecting the regions boundaries. The

homogeneous regions, which are in general less noisy than the input data, are supposed to cor-

respond to meaningful objects or objects parts, regions boundaries can be used to distinguish or

recognise objects and shapes within the domain.

From the mathematical and engineering point of view a signal, either in one or in two dimensions,

can be considered as a function g(x), where x is a point of the domain Ω and the value g(x) is

the signal value at the point x. Handling the data by mean of an unstructured model, i.e. the

function g(x), and data analysis are of great relevance in Computer Vision and in Signal Analysis,

where the segmentation process, once mathematically formulated and implemented numerically,

Introduction

can be carried out automatically by computers. In image analysis, segmentation is useful as a

pre-classification process producing a smooth image, with preserved regions boundaries, which is

somehow easier to classify than the original image. More generally, in signal analysis the process

of segmentation does allow to identify the main signal features by smoothing the input data g(x),

i.e. reducing its noise level, and preserving, i.e. not smoothing, the data discontinuities so that

the meaningful structure of the signal is preserved and more easily observable.

Back to mathematics, the segmentation process can be formalised, among others, in a variational

framework. Within the framework of the Calculus of Variations it is possible to state the segmen-

tation concept as a minimum problem, that is to find a solution minimising a defined quantity. The

techniques of the Calculus of Variations allow to define appropriate quantities which minimisation

produces a solution directly depending on the properties of the defined quantities. The quantity

to be minimised in the segmentation model is a summation of penalty terms associated to the

required solution features.

The variational nature of the Mumford and Shah model and of the Blake and Zisserman model

can be basically understood considering that the penalty terms involved in both the models take

care of different requirements. In fact the segmentation criteria require: a) the solution to be as

close as possible to the input data; b) the solution to be as smooth as possible within each homo-

geneous region, that is the solution has to be less noisy than the original data and the smoothing

has not to act on the regions boundaries which are a relevant information that must not be lost; c)

the length of the regions boundaries to be as short as possible, that is the regions boundaries have

to be as smooth as possible as well. The concept of as · as possible is strictly related to the choice

of approach the segmentation problem in a variational framework, i.e. as a minimum problem,

where the concepts of close, smooth and short have to find a proper mathematical formalisation. In

particular, the Mumford and Shah model is defined by a functional where three penalty terms are

considered: the first term penalise the distance between the approximating solution and the data,

the second term penalise the wiggle of the approximating solution and the third term penalise the

length of the regions boundaries. The mathematical forms implementing the concepts of distance,

wiggle and length are briefly introduced in the following and detailed in the Chapter 2. The Blake

and Zisserman model is defined by a functional where a term still penalises the distance between

the approximating solution and the data, a term still penalises the length of the regions boundaries

and the smoothness of the solution is controlled by an higher order term with respect to the one in

the Mumford and Shah model. Moreover, the wiggle of the regions boundaries is also penalised by

an ad hoc term. The segmentation is directly derived from the minimisation of the two functionals,

that is the solutions are the minimisers of the functionals. To each penalty term it is associated

a weight, represented by a real parameter, controlling the relative influence of the penalty terms

and their overall effects on the solution.

It is important to notice that the dimension of the term controlling the regions boundaries is of

one order less than the order of the penalty term controlling the distance between the approximating

solution and the data, and of the penalty term controlling the smoothness of the approximating

solution. Moreover, since both the Mumford and Shah functional and the Blake and Zisserman

2

Introduction

functional segment the data in to homogeneous regions and detect the regions boundaries explicitly

and directly, the solution is made of two elements: one is properly the approximating solution, u,

and the other is the set K of closed curves representing the regions boundaries. This kind of

problems, where the unknown is a pair (u,K), with K varying in a class of closed subset of a fixed

open set Ω ⊂ Rn, and u : Ω \ K → Rn is a function in some function space, are generally denoted

by ”Free Discontinuity Problems”. Usually they are of the form:

min Ev(u,K) + Es(u,K) ,

with Ev(u,K), Es(u,K) being interpreted as volume and surface energies, where, in any case,

Ev(u,K) is defined in a dimension one order greater that the dimension of Es(u,K).

The Mumford and Shah model, in dimension 2, requires the minimisation of a functional of the

form:

MS(u,K) =

∫

Ω

(u − g)2dx +

∫

Ω\K

|∇u|2dx + Hn−1(K ∩ Ω).

The first term penalises the distance, in L2, between the approximating solution u and the data g,

the second term penalises the gradient of the approximating solution strictly within the homoge-

neous regions, that is the solution is required to be smooth within each homogeneous region and it

is allowed to present strong transitions along the regions boundaries. The third term penalises the

length of the regions boundaries being the length measured by the (n − 1)-Hausdorff measure of

the set K. The variational nature of the model directly derives from the definition of the penalty

quantities and from the minimisation procedure required to obtain the segmentation, that is to

find the pair (u,K) that minimises the value of MS(u,K). Moreover, MS(u,K) belongs to the

class of the so called ”Free Discontinuity Problems”, being the unknown the pair (u,K), the first

two terms two volume energies and the third term a surface energy. The same characteristics are

present in the Blake and Zisserman model which is detailed thereon.

The variational approach to the problem of segmentation should implies that ”region” terms

compete with ”boundary” terms in a way that should intrinsically impose the regularity of both

the solution inside the regions and the boundaries. Mumford and Shah conjectured such a property.

Later it has been proved that the competition of one-dimensional and two-dimensional energy terms

in a variational formulation does imply such a regularity to both regions and boundaries and, in

particular, that such a competition does not allow the existence of fractal boundaries.

The thesis is divided in two parts dedicated respectively to the Mathematical framework of the

Mumford and Shah and of the Blake and Zisserman models and to practical applications of the two

models to real images and one dimensional signals. The first part includes three Chapters. Chapter

1 is devoted to present a review of the mathematical framework where variational problems, and

in particular the so called ”Free Discontinuity Problems”, can be formulated. The Mumford and

Shah problem is already mentioned in Chapter (1), because it is a quite typical example of a free

discontinuity problem. Chapter 2 and 3 respectively present the Mumford and Shah functional

and the Blake and Zisserman functional. This part is mainly theoretical and reviews the more

3

Introduction

relevant contributions on those topics. In the second part of the thesis we first present and detail

the software implementations of the two models, Chapter 4. Then, in Chapter 5 and 6, applications

of the segmentation models to 2D images and to one dimensional signals are respectively reported.

Chapters 5 and 6 contain both simple simulated examples and applications to real data. The

purpose of the applications to simulated data is to show some peculiarity of the applied methods

and to test and validate the computer programs. The applications to some real-world data show

the full-functionality of the software and its suitability for really practical purposes.

4

Part I

A Review of the Mathematical

Theory of Variational

Segmentation

1 Mathematical Framework

This Chapter gives the principal mathematical notions necessary to cover the mathematical aspects

dealt in the thesis.

The Chapter presents obviously only a review of the main concepts and tools that are used to

study the Mumford and Shah and the Blake and Zisserman problems both from the theoretical point

of view, i.e. to prove that the functional admit a minimum, and from the practical point of view,

i.e. to build a convenient numerical approximation of the functionals. Most of the mathematics

used to treat these problems is related to the contributions of Italian mathematicians, including

Leonida Tonelli and Ennio de Giorgi1.

It must be stressed that the purpose of this chapter is to show the mathematical framework and

the main results of the study of our problems and specially to show the connection between the

originally stated problems and the approximation that are eventually computed. Moreover, the

contents of this Chapter are by no means a complete nor formal mathematical treatment of the

subject. Specially the use of the space of Special functions of Bounded Variation, (SBV ), and the

use of the Γ-convergence technique are just sketched.

The classical and the direct method in the Calculus of Variations are briefly reviewed as they

are the main mathematical tools that allow, respectively, to explicitly compute the solution and to

study the existence of minima in variational problems. The existence of the solution is guaranteed

by a generalisation of the Weierstrass Theorem. There are, however, variational problems for

which the existence conditions are not satisfied and for which the direct methods of the Calculus

of Variations can not be applied. In these cases a so called relaxation is needed to associate to the

original problem a waek formulation, which is analytically tractable.

Some of the properties of the class of Free Discontinuity Problems are introduced to underline

their inherent difficulties and to give reason of the development of mathematical tools that allow

the definition of approximating formulations that are numerically more tractable. It is important

to notice that different and many types of approximations can be defined and that, for all of them,

the asymptotical equivalence to the original problem is guaranteed and proved by the use of the

Γ-convergence theory.

The last section of the Chapter introduce the variational segmentation model proposed by Mum-

1Tonelli generalised the modern techniques used in the Calculus of Variations. De Giorgi defined the class of theFree Discontinuity Problems and the concept of Γ-convergence. The Free Discontinuity Problems are a generalclass of variational problems, that includes the Mumford and Shah and the Blake and Zisserman problems, wherethe discontinuities of a function are an essential part of the set of the unknowns. The Γ-convergence is the toolthat make it possible to go through the various steps in the long road from the theoretical formulation of theproblems to their practical solutions.

1. Mathematical Framework

ford and Shah (Mumford and Shah, 1989).

1.1 Variational Problems

The problem of finding, among all functions with prescribed boundary conditions, those which

minimise a given integral functional is one of the main goals in the Calculus of Variations2. In

general, the problem can be formalised as follows:

min F(u) : u ∈ X , (P)

where X is a suitable Banach space and

F(u) =

∫

Ω

f (x, u(x),∇u(x)) dx, (1.1)

where Ω ⊂ Rn is a bounded open set with boundary ∂Ω, u : Ω ⊂ R

n → Rm, ∇u ∈ R

nm,

f : Ω × Rn × R

nm → R is a continuous function.

1.1.1 Classical and Direct Methods in the Calculus of Variations

In general, classical methods allow to explicitly compute the solution of a minimum problem

(P) by means of the derivation of the Euler equation associated to F . Direct methods allow to

formally prove the existence of the minimum and are generalisation of the Weierstrass Theorem.

It guarantees the existence of the solution under the conditions that the functional F is lower

semicontinuous and that the function space X is compact.

Regarding the classical methods in the Calculus of Variations, the solution of the minimisation

problem can be obtained by finding the zeros of the ”derivative” of F , F ′(u) = 0, known as the

2

The Calculus of Variations, as a autonomous and well defined mathematical field, finds its origin in thepioneering works of Bernoulli and Euler. In the 1696 Giovanni Bernoulli solved the brachistochrome problemwhich is the first problem completely formulated and solved in the framework of the Calculus of Variations.Some problems studied within the Calculus of Variations were however known by at the ancient Greeks, e.g. theisoperimetry problem. During the 19th century Lagrange, Riemann, Weierstrass, Jacobi, Hamilton and othersdeveloped new methods and gave important contributions to the field of the Calculus of Variations. These newmethods are called classical. With the beginning of the 20th century, Hilbert and Lebesgue introduced new anddifferent techniques. Tonelli then generalised these methods, known as the direct methods in the Calculus ofVariations and particular importance and attention were dedicated, since then, to the study of the minimisationproblem (P).

Some of the classic problems studied in the Calculus of Variations are: the brachistochrone problem, the Fermatprinciple in geometrical optics, the Dirichlet integral, the iso-perimetry and the minimum surface problems. Mostof the classical problems of mechanics can be formulated equivalently as variational problems or by means ofdifferential equations. It must be noted that the treatment of a problem with differential equations and withvariational principles yields to different solution methods by finite dimensional approximation, the approximationof the variational formulation is generally more convenient from the computational point of view.

8


Euler equation associated to the functional F . If u ∈ C2 the Euler equations are:

F ′(u) =

(

−

n∑

i=1

∂

∂xi

(

∂f

∂ξij(x, u,∇u)

)

+∂f

∂uj(x, u,∇u)

)

1≤j≤m

, (1.2)

where f = f(x, u, ξ).

As examples, the Euler equations associated to the problem of minimising the Dirichlet integral

and to the minimal surface problem are here reported.

The Dirichlet problem is:

min

F(u) =

∫

Ω

|∇u|2dx : u = u0 on ∂Ω

,

where m = 1, n ≥ 1 and the associated Euler equation is:

−∆u = 0 in Ω

u = u0 on ∂Ω .

The minimum surface problem is:

min

F(u) =

∫

Ω

√

1 + |∇u|2dx : u = u0 on ∂Ω

,

where m = 1, n ≥ 1 and the associated Euler equation is:

−

n∑

i=1

∂

∂xi

[

(

1 + |∇u|2)−1/2 ∂u

∂xi

]

= 0 in Ω

u = u0 on ∂Ω .

A very strong link exists between Partial Differential Equations and Calculus of Variations.

In fact, variational problems produce, via their associated Euler equation, differential equations

and, on the other side, many differential equations can be studied by variational methods. For a

treatment on this topic, specifically oriented to image processing problems, we refer to (Aubert

and Kornprobst, 2002) and to (Ambrosio and Dancer, 1999) where a wider class of problems is

presented.

The direct methods in the Calculus of Variations deal directly with the functional F to prove

the existence of a minimum. The existence can be proved by defining a minimising sequences

from which, under some conditions on the function space, it is possible to extract a convergent

subsequence which, under some other conditions on the continuity of F , converges to a minimum

of the functional F . In practice, the functional F has to be lower semicontinuous with respect to

the topology defined on the function space X, the characteristics of the topology have to ensure

the compactness of the minimising sequences.

In terms of functions, let X be a metric space, and let R = R∪−∞,+∞, a function F : X → R

9


is said to be lower semicontinuous if:

F(x) ≤ limh→∞

inf F(xh)

for every sequence (xh) converging to x and for all x ∈ X.

A subset K of X is compact if every sequence in K has a subsequence which converges to a

point of K, i.e.

∀(xh) ⊂ K ∃x ∈ X, ∃(xhk) : xhk

→ x.

A function F : X → R is coercive if the closure of F ≤ t is compact in X for every t ∈ R.

The Weiestrass theorem guarantees that if F is coercive and lower semicontinuous then:

(i) F has a minimum point in X;

(ii) if xh is a minimising sequence of F in X, and x is the limit of a subsequence of xh , then x

is a minimum point of F in X;

(iii) if F is not identically +∞, then every minimising sequence for F has a converging subse-

quence.

Dacorogna (1989), presents a formal and depth treatment on the direct methods in the Calculus

of Variation.

The direct methods of the Calculus of Variations cannot always be applied directly to approach a

minimum problem. This may occur when the functional F is not lower semicontinuous or the space

X is not compact. When F is coercive but not lower semicontinuous a widely adopted strategy

to solve the minimum problem is to associate to the functional F another functional, called RF

(relaxed functional). This new functional defines a new problem, (RP ), called relaxed problem. In

particular RF admits a minimum and has the following two properties:

minRF = infF;

the minimum points for RF are the limits of minimising sequences of F , and every minimising

sequence of F has a subsequence converging to a minimum point of RF .

The relaxed functional RF is defined as the greatest lower semicontinuous functional less or equal

to F . A detailed formalisation and some analytical and practical applications of the relaxation

technique can be found in (Dacorogna, 1989), (Aubert and Kornprobst, 2002) and (Braides and

Chiado Piat, 2006).

1.2 Free Discontinuity Problems

1.2.1 A General Introduction

The terminology ”Free Discontinuity Problems” has been introduced by De Giorgi to indicate a

class of variational problems characterised by a competition between volume energies, concentrated

10


on a n-dimensional set, and surface energies, concentrated on a (n−1)-dimensional set. A relevant

feature of these problems is that the support K of the surface energies is not fixed a priori and it is

in many cases a relevant unknown of the problem. Free discontinuity problems involve functionals

whose natural domains are sets of functions which admit a finite number of discontinuities, being

K the set of the discontinuities. Since the discontinuity set K is not necessary made of closed

curves, i.e. boundaries, the class of free discontinuity problems presents only some analogies with

the class of free boundary problems and requires a specific mathematical theory.

Some of the main examples where free discontinuity problems arise are:

fracture mechanics;

theory of plasticity;

optimal partitions;

drops of liquid crystals grow;

prescribed curvature problems;

signal and image reconstruction.

In particular the segmentation model proposed by Mumford and Shah is one of the best known

free discontinuity problems.

Approaching a free discontinuity problem, and in particular the Mumford and Shah problem,

following the direct methods in the Calculus of Variations presents many difficulties mainly related

to the dependence of the involved energies on the surface K. The definition of a relaxed problem

associated to the original free discontinuity problem is hence necessary to allow the proof of the

existence of the solution. Again, even when the existence of the solution is guaranteed, the exact

computation of the solution can be very rarely performed. This motivates the need to approximate

the relaxed problem with a functional which solution can be computed practically. The effectiveness

of the approximation is formally proved through the use of the Γ-convergence techniques. The value

of the solution can be computed, for example, applying the classical methods in the Calculus of

Variations, i.e. deriving the Euler equation associated to the approximating functional. Moreover

the approximation is in general numerically tractable, i.e. solution can be computed automatically.

Numerical computations can also be useful as an heuristic guide in the mathematical analysis of still

open problems and to practically observe the theoretically foreseen proprieties of the approximated

model.

The approximation of free discontinuity problems can be achieved following many and different

techniques; some of the more relevant are:

non-local approximation;

finite-difference approximation;

finite-elements approximation;

11


slicing method;

second order singular perturbation approximation;

elliptic approximation.

For a review of these methods we refer to (Braides, 1998; Ambrosio et al., 2000; Aubert and

Kornprobst, 2002). Approximation techniques include both discrete methods, (such as the finite

elements approximation) and methods in the realm of functional problems that must be further

approximated by a discretisation to obtain a numerical solution.

In the following some notions about the space of Special Functions of Bounded Variation (SBV )

and about the Γ-convergence theory are briefly sketched.

The SBV space provides a unitary framework for the study of free discontinuity problems and

it allows a waek formulation of otherwise untractable problems.

The Γ-convergence theory allows to prove the approximation of a given functional by a new

functional, defined on different function spaces, which is more tractable from both the analytical

and the numerical viewpoint.

1.2.2 The Space of Special Functions of Bounded Variation

One of the innovative contribution given by De Giorgi to the study of free discontinuity problems

indicates to interpret K as the set of discontinuity points of the function u and to define u on

a particular space of discontinuous functions. The definition of such a function space fulfils the

following requirements:

it has to be possible to define K as a smooth set of discontinuity points of the function u;

u has to be ”differentiable” almost everywhere outside K, so that the bulk energy depending

on ∇u can be defined;

the possibility to apply the direct methods in the Calculus of Variations has to be guaranteed.

The rigorous definition and the formalisation of such a space is due to the efforts of De Giorgi and

of Ambrosio who defined the space of Special Functions of Bounded Variation (SBV ). Working in

this space has proved to be particularly fruitful allowing the definition of the so called ”weak forms”

of original problems and providing the necessary tools to study the regularity of the solutions of

relaxed problems. Various regularity results shown that a waek formulation, defined in the SBV

space, does provide a solution to a wide class of free discontinuity problems.

The SBV space is the function space that is commonly used in image analysis, the main reason

is that, as opposed to classical Sobolev spaces, functions in SBV can be discontinuous across

hypersurfaces. As for images this means that images are discontinuous across edges. We report

here a rough definition of the space of Functions of Bounded Variation (BV ) and of the space of

12


Special Functions of Bounded Variation (SBV ), we refer to (Ambrosio et al., 2000) and (Braides,

1998) for the formal definitions and details on such spaces.

Let Ω ⊆ Rn be an open set, let u : Ω → R be a measurable function and x ∈ Ω. Let u+(x) and

u−(x) denote, respectively, the approximate upper and the lower limit of u at x, defined by:

u+(x) = inf

t ∈ R : limρ→o+

|y ∈ Ω : |x − y| < ρ, u(y) > t|

ρn= 0

,

u−(x) = sup

t ∈ R : limρ→o+

|y ∈ Ω : |x − y| < ρ, u(y) < t|

ρn= 0

. (1.3)

(1.4)

If u+(x) = u−(x) ∈ R, then x is said to be a Lebesgue point of u and the common value of u+(x)

and u−(x) is called approximate limit of u at x. Let Su denote the discontinuity set of u, i.e. the

set of all x ∈ Ω which are not Lebesgue points of u. By definition u is a function of bounded

variation in Ω, if u ∈ L1(Ω) and its distributional derivative is a vector-valued measure Du with

finite total variation |Du(Ω)| defined by:

|Du(Ω)| = sup

∫

Ω

u divΦ : Φ ∈ C10 (Ω, Rn), |Φ| ≤ 1

.

The space of all functions of bounded variation on Ω will be denoted by BV (Ω). If u ∈ BV (Ω),

then Su is countably(

Hn−1, n − 1)

rectifiable, i.e.

Su = N ∪⋃

i∈N

Ki,

where Hn−1(N) = 0, and each Ki is a compact set contained in a C1 hypersurface.

The distributional derivative Du of a function u ∈ BV (Ω) can be decomposed as Du = Dau +

Dsu, where Dau is absolutely continuous and Dsu is singular with respect to the Lebesgue measure.

The density of Dau with respect to the Lebesgue measure is denoted by ∇u. Moreover, Dju denotes

the restriction of Dsu to Su and Dcu the restriction of Dsu to Ω \ Su. With these notations the

distributional derivative of u can be decomposed as

Du = Dau + Dju + Dcu,

where Dju and Dcu are respectively called the jump part of Du and the Cantor part of Du. Here

it is important to note that the measure Dju is concentrated on Su.

A function u belongs to SBV when the Cantor part Dcu of its distributional derivative Du is

null3.

3

A function u ∈ SBV (Ω) can be represented as ua + uj , where ua ∈ W 1,1(Ω) and uj ∈ X(Ω), being X(Ω) thespace of SBV function whose derivative reduces to the jump part, i.e. in X(Ω), ∇u = 0 and Dj(u) is a purelyatomic measure.

13


A generalisation of SBV spaces, namely the space of Generalised Special Functions of Bounded

Variation (GSBV ), is necessary when dealing with functions u /∈ L1(Ω). The space GSBV is

defined as

GSBV = u : Ω → R : Borel function,−k ∨ u ∧ k ∈ SBVloc(Ω) ∀k ∈ N .

where SBVloc(Ω) denotes the class of functions v ∈ SBV (Ω′) for every Ω′ ⊂⊂ Ω. This is the case,

for example, of the space where a solution of the Blake and Zisserman problem has to be searched.

In particular, the existence of minimisers of the Blake and Zisserman functional has been proved

by Carriero et al. (1996) in the space GSBV 2 ∩ L2(Ω), where

GSBV 2(Ω) = u : Ω → R : u ∈ GSBV (Ω), ∇u ∈ [GSBV ]n .

1.2.3 Some Essential Features About Γ-convergence

The features of Γ-convergence is of particular interest in solving minimum problems since it can

be used to describe the asymptotic behaviour of this class of problems and it formalises a notion

of variational convergence.

The theory of Γ-convergence allows to prove the equivalence between two minimum problems, one

characterised by a functional F an another characterised by a functional Fǫ. Some problems are in

fact originally formulated through a functional F and the Γ-convergence allows its approximation

by means of a sequence of more tractable functional Fǫ. The free discontinuity problems are

problems of this kind. On the other hand, there are other problems originally formulated by a

functional Fǫ that is difficult to handle numerically when the parameter ǫ is small. In this case the

Γ-convergence is used to prove that the original functional can be approximated by a parameter-

free functional. The homogenization problems are classical examples of this kind. The usefulness of

proving such an equivalence lies on the fact that, depending on the cases, one of the two functional

is more tractable than the other, both from an analytical and a numerical viewpoint.

It is worth to mention, even only as a simple list, some of the more relevant problems that can

be faced and solved by using the Γ-convergence technique. The examples include:

the gradient theory of phase transition;

homogenization problems;

Regarding the study of free discontinuity problems, the main advantage of the SBV space is that it does notinclude pathological functions, with Dcu 6= 0, that are functions with Dau = Dju = 0 but with Du 6= 0 sincethe presence of the Cantor part. The simplest example of function of bounded variation which does not belongto SBV is the Cantor-Vitali function: the absolutely continuous part of its derivative is 0, the set Su is empty,but Dcu is a non trivial measure whose support is the Cantor middle third set.

Another relevant advantage comes from a regularity theorem that guarantees that if u ∈ SBV (Ω) is a minimiserof the Mumford and Shah problem, then:

u ∈ C1(Ω \ Su).

Moreover, for u ∈ SBV (Ω):u ∈ W 1,1(Ω) ⇐⇒ H1(Su) = 0

and being H1(Su) = 0 it is possible to integrate Su on the whole of Ω.

14


dimension reduction;

continuous limits of difference schemes;

approximation of Free Discontinuity Problems.

We report in the following just the definition of Γ-convergence and a relevant result related to

the Γ-convergence, while for an introductory treatment on the theory of Γ-convergence and for a

parade of examples we refer to (Braides, 2002a).

Let (X, d) be a metric space and let f, fǫ : X → [0,+∞] be functions. The sequence of functions

(fǫ) Γ-converges to f if the following two conditions are satisfied:

(i) for any sequence (xǫ) ⊂ X converging to x the following holds:

limǫ→0

inf fǫ(xǫ) ≥ f(x);

(ii) for any x ∈ X there exists a sequance (xǫ) ⊂ X converging to x such that

limǫ→0

sup fǫ(xǫ) ≤ f(x).

The function f is uniquely determined by (i), (ii) and is denoted by Γ− limǫ→0 fǫ, moreover if (xǫ)

is a converging sequence to x so that

limǫ→0fǫ(xǫ) = limǫ→0fǫ(x),

then its limit is a minimum point for f .

1.3 Variational Segmentation: The Mumford and Shah

Model

One of the best known and studied free discontinuity problem is the model introduced by Mumford

and Shah (1989) for image segmentation. The model is presented here to review the main aspects

related to the formulation of such a problem in a variational framework. The Mumford and Shah

functional is defined by:

MS(u,K) =

∫

Ω\K

(u − g)2dx + λ

∫

Ω\K

|∇u|2dx + αH1(K ∩ Ω), (1.5)

where Ω ⊂ R2 is a bounded open set, g ∈ L∞(Ω) is the input data, λ and α are given strictly

positive parameters. H1 is the 1-dimensional Hausdorff measure. The problem is to minimise MS

in the set of admissible pairs (u,K):

A =

(u,K) : K ⊂ Ω closed, u ∈ W 1,2loc (Ω \ K)

. (1.6)

15


The solution of the problem results in a joint smoothing and edge detection effect, i.e. a piecewise

smoothed image u outside the set of contours K coming from the sharp discontinuities, the ”edges”,

of g. The pair (u,K) that minimise MS is called an optimal pair, the solution is therefore called an

optimal segmentation. The existence theorem of minimisers of the Mumford and Shah functional

over the class of admissible pairs (u,K) has been proved by De Giorgi, Carriero and Leaci in

(De Giorgi et al., 1989b).

Intuitively, K is expected to be a set of smooth curves. Mumford and Shah conjectured the

existence of a minimal segmentation made of a finite set of C1 curves, and each curve may end

either as a crack tip, a free extremity, or in triple junctions, that is three curves meeting at their

end points with a 2π/3 angle between each pair.

The study of MS remains particularly difficult since the intrinsic interaction between the two-

dimensional integrals in u, and the one-dimensional term H1(K ∩ Ω). This interaction actually

makes the Mumford and Shah functional a free discontinuity problem. In order to apply the

direct methods of the Calculus of Variations it is necessary to define a topology that ensures the

compactness of the minimising sequences and the lower semicontinuity of the functional. This can

not be easily nor directly achieved when dealing with the Mumford and Shah problem of the form

(1.5), in fact the map:

K 7→ H1(K)

is not lower semicontinuous with respect to the Hausdorff metric or any compact topology. We

refer to (Ambrosio et al., 2000), (Morel and Solimini, 1995) and (Aubert and Kornprobst, 2002)

for a more rigorous treatment of these topics. As it is usual in the Calculus of Variations, the

key idea is to enlarge the class of admissible pairs for MS functional, such that in this wider

class it is possible to define a suitable relaxed functional with properties of compactness and lower

semicontinuity as required by the Weierstrass theorem. Thank to the work of De Giorgi a weak

formulation can be defined dropping the requirement that K has to be a closed set and to allow it

to be the jump set Su of a function u in the SBV space. With this respect it is possible to define

the weak formulation of the Mumford and Shah functional as

MSw(u) =

∫

Ω

(u − g)2dx + λ

∫

Ω

|∇u|2dx + αH1(Su) u ∈ SBV (Ω). (1.7)

Here it is very important to remark several properties that make the weak formulation MSw

much more tractable with respect to the MS. They are:

working in the SBV space allows the definition of a proper topology, that guarantees lower

semicontinuity and compactness and hence the existence of a minimiser;

the domain of the volume energies can be extended entirely to Ω;

the set K is replaced by the set Su that does not contain only closed curves, i.e. not only

boundaries are allowed to exist in K;

16


the set Su is no longer related to the discontinuities of the given image g but to the discon-

tinuities of the solution u;

the functional now depends only on the unknown u.

Various results prove the existence of a minimiser for the weak formulation of the Mumford

and Shah functional, see (Ambrosio, 1989a), (Ambrosio, 1989b) and (De Giorgi et al., 1989a).

Despite the existence theory in the SBV space, it is not often feasible, to compute analytically

nor numerically the minimum of the Mumford and Shah problem. To satisfy such a need many

approximations techniques have been applied in order to define a suitable sequence of functionals

Γ-converging to the weak formulation MSw. We refer to (Braides, 1998) and to (Ambrosio et al.,

2000) for a detailed review of many of them. In the following, the approximation proposed by

Ambrosio and Tortorelli, which is the one used in the thesis, is briefly introduced.

The approximation introduced by Ambrosio and Tortorelli concerns a family of functionals de-

pending on two variables (u, υ), where the second variable υ is related to the set Su.This ap-

proximation takes advantage of a theorem proved by Modica and Mortola (1977), which proves a

variational approximation of the term H1(Su) by the quadratic, elliptic functionals defined by

MMǫ(υ) =

∫

Ω

(

ǫ |∇υ|2

+W (υ)

ǫ

)

dx υ ∈ W 1,2(Ω), (1.8)

where W (t) = (1 − t)2 is a ”single-well” potential. Ambrosio and Tortorelli proved in (Ambrosio

and Tortorelli, 1992) that the sequence of functionals

AT ǫ(u, υ) =

∫

Ω

(

(u − g)2 + λυ2 |∇u|2

+ α

(

ǫ |υ|2

+(1 − υ)2

4ǫ

))

dx. (1.9)

Γ-convergences to the functional MSw, being ǫ the convergence parameter. Here υ ∈ W 1,2(Ω),

uυ ∈ W 1,2(Ω) and 0 ≤ υ ≤ 1.

The Ambrosio and Tortorelli approximation, as for other approximations, is numerically tracta-

ble. The solution of the minimum problem related to AT ǫ can be explicitly computed by means

of the classical methods in the Calculus of Variations, i.e. deriving the associated Euler equation

that can be disctretised by a Finite Differences Method. An alternative numerical implementation

can be derived applying a Finite Element Method directly to the functional. This second approach

does not require to derive the associated Euler Equation and allows to impose weaker conditions

on the regularity of the solution.

17

2 The Mumford and Shah Model

This Chapter is fully dedicated to describe the image segmentation functional model proposed by

Mumford and Shah, (Mumford and Shah, 1989). The original formulation of the Mumford and

Shah functional is reported and some of its peculiarities are commented. The need to develop a

weak formulation of the original model is underlined and the weak formulation is reported and

commented in order to clarify the need to develop and work with an approximating model. In

particular, details on the Ambrosio and Tortorelli approximation by elliptic functionals are given.

Subsequently, and on the basis of this analysis, an extension of the Mumford and Shah functional

is detailed. The extension takes into account, aside the length of the discontinuity set K, a new

term that controls the curvature of the discontinuity set K.

For both the models, the Euler equations are derived and discretised using a Finite Differences

Method (FDM). Application of gradient descent is then reported to allow to solve numerically the

Euler equations.

Finally a drawback related to the order of the term controlling the ”smoothness” of the solution

u is reported. Minimising a first order term requires the solution to be as ”flat” as possible. For

uniformly sloped signals, or signals presenting some kind of trend without discontinuities, the

solution may follows the data if the term controlling the solution smoothness is non constrained

too much while, as opposite, the solution may present a sequence of ”steps”, i.e. jumps, when the

constrain on the smoothness is increased over a certain threshold. This behaviour is known as the

”gradient effect”. The Blake and Zisserman functional, presented in Chapter 3, has been therein

considered and studied since it is a second order functional, i.e. it is not affected by the ”Gradient

effect”. Despite this chapter is dedicated to image segmentation, i.e. to two dimensional domains,

for consistency with the next chapter the Gradient effect is therein presented in dimension one.

Section 2.1 treats the same matter as section 1.3 in a slightly wider manner; the following sections

treat some generalisations and alternative formulations.

2.1 The Mumford and Shah Model for Image Segmentation

2.1.1 The Mumford and Shah Functional

The Mumford and Shah functional is probably one of the best known models in image segmentation.

It has been proposed by David Mumford and Jayant Shah in (Mumford and Shah, 1989). The

functional is related to earlier models introduced in a discrete setting by D. Geman and S. Geman

and by Blake and Zisserman (1987), as reported by Ambrosio et al. (2000) and many others.

2. The Mumford and Shah Model

Given an open bounded set Ω ⊂ R2, a function g ∈ L∞(Ω) and g : Ω 7→ R and strictly positive

parameters λ, α, the Mumford and Shah functional for image segmentation is defined by

MS(u,K) =

∫

Ω\K

(u − g)2dx + λ

∫

Ω\K

|∇u|2dx + αH1(K ∩ Ω), (2.1)

where u is a piecewise smoothed image outside a set K of contours derived by the sharp disconti-

nuities of g, i.e the homogeneous regions boundaries. The space X of the function u is a Sobolev

space W 1,2(Ω \K), H1 denotes the one-dimensional Hausdorff measure of the set K which should

be made of smooth curves and the regions boundaries are expected to be of finite perimeter.

The minimisation of the Mumford and Shah functional requires through the first term the

solution u to be as close as possible to the given data g within each homogeneous region. The

second term controls the smoothness of the solution u by asking its gradient to be as small as

possible within each homogeneous region, i.e. outside the edge set K. The third term imposes

that the discontinuity set K has minimal length being the length measured by the Hausdorff one-

dimensional measure. Requiring the length of the set K to be as short as possible it is somehow

equivalent to request the K curves to be as smooth as possible.

Mumford and Shah conjectured the existence of the solution and some relevant properties of the

set K. The conjecture states:

Conjecture 2.1 (Mumford and Shah) There exists a minimiser (u,K) of MS such that the

edges (the discontinuity set K) are the union of a finite set of C1,1 embedded curves. Moreover,

each curve may end either as a crack tip (a free extremity, i.e. K looks like a half-line) or in triple

junction, that is, three curves meeting at their endpoints with 2π/3 angle between each pair.

This conjecture has not been yet fully proved and only partial, but meaningful, results are

available. The problem is of particular difficulty because of the intrinsic interaction between the

two-dimensional terms and the one-dimensional term: the length of the set K. We refer to (Am-

brosio et al., 2000), (Morel and Solimini, 1995) and to (Aubert and Kornprobst, 2002) for the

statements and the proofs of the available results and for a review of the open problems. In Chap-

ter 5 numerical test are reported to show practically some of the edges properties conjectured by

Mumford and Shah.

The direct methods of the Calculus of Variations can not be immidiatly applied to (2.1) since

H1(K ∩ Ω) is not lower semicontinuous. This motivates the definition of a relaxed problem, the

weak formulation of the Mumford and Shah problem, allowing to prove that:

a minimum exists, but it is not unique;

the set of minimal segmentation is compact;

the edge set K is countably (H1) rectifiable, i.e. it is made, up to a set of H1-measure zero,

of a countable family of curves with finite length.

20


2.1.2 Weak Formulation in SBV

In order to overcome the remaining difficulties, in particular the lack of lower semicontinuity, a

relaxed formulation of the Mumford and Shah functional has been defined. The new formulation,

known as ”weak formulation”, involved the space of Special Functions of Bounded Variation and is

based on the idea of removing the requirement that K is a set of closed curves and to replace K

by the discontinuity set Su of u ∈ SBV (Ω). In this way the weak formulation is:

MSw(u) =

∫

Ω

(u − g)2dx + λ

∫

Ω

|∇u|2dx + αH1(Su). (2.2)

Working in the SBV space allows, moreover, to extend the domain of the two-dimensional integrals

in u from Ω \ K to the entire image domain Ω.

The existence of a weak solution on SBV for the minumum problem

minu∈SBV (Ω)

MSw(u)

has been proved by Ambrosio, see (Ambrosio, 1989a), (Ambrosio, 1989b) and (De Giorgi et al.,

1989a), by using a compactness and lower semicontinuity result. Moreover in (De Giorgi et al.,

1989b) it is proved that

infu∈SBV (Ω)

MSw(u) ≤ inf(u,K)∈A

MS(u)

being the class A of the admissible pairs (u,K) defined by the conditions:

K ⊂ Ω, u ∈ W 1,2(Ω \ K).

2.1.3 Approximations of the Weak Formulation

Even if in the SBV space a general existence theory is available, exact computation of solutions

of MSw can not be performed. In fact the weak formulation is not yet differentiable, that is the

Euler equation can not be derived. This requires the definition of approximating functionals whose

computation can be performed. A commonly used method is to approximate MSw by a sequence

Fǫ of regular functionals defined on Sobolev spaces, the Γ-convergence of Fǫ to MSw as ǫ tends

to 0 guarantees the ”equivalence” of the two functionals.

Four main classes of approximating functionals have been proposed :

approximation by elliptic functionals;

approximation by finite-differences and by finite-elements schemes;

approximation by introducing second order singular perturbations;

approximation by introducing a non-decreasing continuous function (non-local terms).

21


The first two approaches are briefly introduced hereafter, while for more details on all of them

we refer to (Ambrosio and Tortorelli, 1990), (Chambolle, 1995; Chambolle and Dal Maso, 1999;

Gobbino, 1998a; Gobbino and Mora, 1999) , (Bellettini and Coscia, 1994a; Carriero et al., 1996),

(Braides and Dal Maso, 1997) and to (Braides, 1998; Aubert and Kornprobst, 2002; Ambrosio

et al., 2000).

(A) Approximation by elliptic functionals

Here the set Su is modeled by means of an auxiliary function υ(x). This function approxi-

mates (1 − χSu) where χSu

is the characteristic function of Su, i.e. υ(x) ≈ 0 if x ∈ Su and

υ(x) ≈ 1 otherwise. The Ambrosio and Tortorelli approximation is:

AT ǫ(u, υ) =

∫

Ω

(u − g)2dx + λ

∫

Ω

(υ2 + oǫ) |∇u|2dx + α

∫

Ω

(

ǫ |∇υ|2

+(1 − υ)2

4ǫ

)

dx. (2.3)

This is the approximation model used in this thesis and is discussed afterward.

(B) Approximation by finite-differences schemes

This kind of approximation is perhaps the most natural one from a numerical point of view.

The method considers, in fact, u(x) as a discrete function defined on a mesh of step-size h > 0

and defines FDh as a discrete version of the Mumford and Shah functional. Chambolle,

following earlier ideas of Blake and Zisserman, proposed the following discrete functional:

FDh(uh) = h2∑

k,l

(

uhk,l − gh

k,l

)2+λ

h2∑

k,l

Wh

(

uhk+1,l − uh

k,l

h

)

+ h2∑

k,l

Wh

(

uhk,l+1 − uh

k,l

h

)

,

(2.4)

where Wh(t) = min(t2, α/h).

It is relevant to note here that the two previous approximations actually approach the approx-

imations problem from two distinct and different points. The first one is in fact a continuous

approach while the second is, indeed, a discrete approach. Both the methods do approximate the

same free discontinuity problem, that is the weak formulation of the Mumford and Shah model,

and the approximation, for both of them, is proved by means of the Γ-convergence. The first term

in (2.3) is related, in the discrete model, to the first term of (2.4) and the same holds for the last

two terms in (2.3) and the last in (2.4). Chambolle in (Chambolle, 1995), proved the convergence

of the discrete model to the continuous model. The convergence, once more, has been established

by means of the Γ-convergence theory, which is in fact used in different contexts to solve problems

related to the continuous limits of difference schemes.

The approximation of Ambrosio and Tortorelli is the first one appeared in literature and is

one of the more commonly used in image analysis. The approximation is twofold: on one hand

the auxiliary variable υ is introduced and, on the other hand, the measure of the length of the

22


discontinuity set Su is approximated by a sequence of elliptic functionals of the form:

MMǫ(υ) =

∫

Ω

(

ǫ |∇υ|2

+W (υ)

ǫ

)

dx υ ∈ W 1,2(Ω), (2.5)

where 0 ≤ υ ≤ 1 and W (t) can be chosen as W (t) = t2(1 − t)2, i.e. a ”double-well” potential or

as W (t) = (1 − t)2, i.e. a ”single-well” potential. The first choice forces the set Su to be made

of only closed curves, i.e. strictly only boundaries are allowed to exist in Su. Since in the weak

formulation of the Mumford and Shah functional curves of the set Su are not necessary boundaries,

it is necessary to choose W (t) = (1 − t)2 instead (Ambrosio et al., 2000).

The heuristic explanation of the Γ-convergence result is the following: on one hand υǫ is forced

to stay very close to 1 as ǫ tends to 0, because the potential W (t) is positive and vanishes only for

t = 1; on the other hand, the factor υ2ǫ in front of |∇uǫ|

2must go to zero near to discontinuities

of u to keep the Dirichlet integral bounded. Hence, υǫ is forced to make transitions between 0

and 1 near to discontinuities of u. The transitions are sharper and sharper as ǫ tents to 0. With

this respect the balance between the potential W (t) and the Dirichlet integral in the third term

of (2.3) shows that the energy of cheapest transitions is proportional to H1(Su). The proof of

the Γ-convergence implies that any limit point of minimisers (uǫ, υǫ) of AT ǫ is a pair (u, 1), with

u ∈ SBV (Ω) a minimiser of the week formulation of the Mumford and Shah functional.

The term oǫ in (2.3) is a non negative infinitesimal faster that ǫ ant it ensures C1 regularity of

the minimisers of AT ǫ, (Ambrosio and Tortorelli, 1992). If oǫ is equal to 0 the approximation still

holds.

It is important to underline that the Ambrosio and Tortorelli approximation presents some

relevant improvements with respect to the weak formulation of the Mumford and Shah functional.

In particular, in the AT ǫ functional all of the integrals are defined on the same bi-dimensional

domain. Moreover, the presence of elliptic functionals, as approximation of the Hausdorff one-

dimensional measure of the discontinuity set Su, permits to implement numerically and solve

automatically the functional.

An intuitive explanation of the convergence of AT ǫ to MSw can be found in (March and Dozio,

1992).

2.1.4 Drawbacks of the Mumford and Shah Model

It is worth to mention some drawbacks of the Mumford and Shah model that are directly related

to the structure of the functional and in particular to the geometric term controlling the length of

the discontinuities curves. The firsts limits have been known since the original work of Mumford

and Shah, and are explicitly declared in the Mumford and Shah conjecture 2.1. Discontinuities

curves can only end presenting a free extremity or a triple junctions, moreover the discontinuities

curves can meet the domain boundary only orthogonally.

Another important drawback is that the minimisation of the discontinuity term, i.e. of the

curves length, prevents the existence of corners in the solution. In practice this means that corners

23


are rounded out: the greater the constrain on the length of the discontinuities curves the greater

the rounding effect where the curves present high curvature values, e.g. corners. In Chapter 4

some numerical evidences of the effects of the nature of the geometric term are reported. In the

next section a formulation of the Mumford and Shah functional where a new term controls the

curvature of the discontinuity curves is reported. Even if neither this formulation explicitly permits

the recover of corners, the roundness effect is indeed reduced by the presence of the curvature

dependent term. A practical evidence on this is reported in Chapter 5.

Another relevant drawback is related to the order of the term controlling the smoothness of the

solution u, i.e. the Dirichlet integral. Controlling the smoothness with a first order term, e.g.

using the gradient, yields to the occurrence of the so call ”Gradient effect” that will be described

in the last section of this chapter.

2.2 Controlling the Curvature in the Mumford and Shah

Model

To better describe the nature of the discontinuities curves the extension of the Mumford and

Shah functional presented in this section introduces a new term controlling the curvature of the

discontinuities curves. This extension was originally proposed by Nitzberg et al. (1993). March

and Dozio (1997) presented an approximating functional including, along with the terms presented

so far, a curvature-dependent term for the segmentation with smooth boundaries. Here, it is

important to note that, due to the particular choice on the structure of the function W (t), the

discontinuities curves forming the set Su are only closed and smooth curves, i.e. boundaries. With

this respect it has to be noted, moreover, that this model can not be used for the recovery of corners

or junctions since the discontinuities are described by mean of level curves of a smooth function,

i.e. non-intersecting closed smooth curves. The boundaries are here modelled as deformable snakes

that minimise a functional depending on the lowest order intrinsic measures on curves: the length

and the curvature. Moreover, this model combines aspects of the snake model with a region based

segmentation method.

In the Mumford and Shah functional the smoothness of the solution u is measured by the integral

of its squared gradient. This naturally suggests to consider an equivalent integral, in dimension

one, to control the smoothness of the contours:

∫

K

[

α + β(dϕ/ds)2]

ds =

∫

K

(α + βk2)ds, (2.6)

where β is a positive weight, ϕ is the tangent orientation along the contour, and k = dϕ/ds is the

contour curvature. The weak formulation of the complete functional is now:

MSkw(u,K) =

∫

Ω

(u − g)2dx + λ

∫

Ω

|∇u|2dx +

∫

Su

(α + βk2)ds. (2.7)

24


The minimisation of the third term requires the boundaries to be as short and smooth as possible.

The parameter α and β control, respectively, the elasticity and the rigidity of the curves, as usual

in the snake model. An existence theory of minimisers of functionals of the type MSkw has been

studied by Bellettini, Dal Maso and Paolini and by Mantegazza, see (March and Dozio, 1997) for

more details and references.

Here the discontinuities are supposed to be smooth and the one-dimensional Hausdorff measure

is simply the total length,∫

ds of the set of curves.

As for the Mumford and Shah functional, it is necessary to define an approximating functional

of MSkw to permit its numerical treatment. The definition of the approximating functional is

based on some considerations about the relationship between the geometric term in the Mumford

and Shah functional and its associated term appearing in the Euler equation associated to the

functional. The Ambrosio and Tortorelli approximation of the Mumford and Shah functional with

the curvature term is reported in section 2.3.1 along with the expression of the associated Euler

equation.

A further extension of the Mumford and Shah functional, recently introduced by Braides and

March in (Braides and March, 2006a) considers, beyond the terms controlling the length and the

curvature of the discontinuity set, a new term controlling the so called ”end points” of the curves.

The aim of this model is to recover corners and junctions of the discontinuity curves and to avoid

some geometrical constrains that the original formulation of the Mumford and Shah functional

imposes on the discontinuity set.

2.3 Euler Equation and Gradient Flow

One of the classical methods in the Calculus of Variation, aimed to explicitly compute the values

of the minimisers, consists in deriving and solving the Euler equation associated to the minimum

problem. To compute the Euler equation it is necessary to derive the first variation of the functional.

Let consider the general minimum problem:

min F(u) : u ∈ X , (2.8)

where

F(u) =

∫

Ω

f (x, u(x),∇u(x)) dx, (2.9)

Ω ⊂ Rn is a bounded open set, and X is a suitable function space. Here F is called variational

integral. The integrand of F is commonly known as Lagrangian, or variational integrand, or

Lagrange function. Now, let u∗ be a minimiser of the functional F(u), and let us consider φ ∈ C10 (Ω)

so that for t ∈ R small u∗ + tφ is admissible for the functional F , in fact u∗ + tφ belongs to X. It

is therefore possible to compute the functional F in u∗ + tφ, obtaining:

F(u∗) ≤ F(u∗ + tφ) (2.10)

25


since u∗ minimises the functional F . Hence, for u∗ and φ fixed, F(u∗ + tφ) depends only on the

real variable t which obviously admits a minimum for t = 0. Therefore it has to be:

d

dt(F(u∗ + tφ))|t=0 = 0. (2.11)

This equation is called the ”null first variation” of the functional F . Using equation (2.9), and by

some straightforward manipulation one obtains:

d

dt(F(u∗ + tφ))|t=0 =

∫

Ω

fu (x, u∗(x),∇u∗) · φ(x)dx +

∫

Ω

fp (x, u∗(x),∇u∗) · φ′(x)dx = 0, (2.12)

where fu and fp are the first derivative vectors of f with respect to the variables u = (u1, . . . , un)

and p = (p1, . . . , pn) and the symbol · indicates the euclidean scalar product in Rn.

Now, if fu ∈ C0 and fp ∈ C1, by integrating by parts it is possible to obtain:

∫

Ω

(

fu (x, u∗(x),∇(u∗)) −d

dxfp (x, u∗(x),∇(u∗))

)

· φ(x)dx = 0. (2.13)

Since the above equality must be satisfied for any function φ and using φ ∈ C10 (Ω), by the funda-

mental lemma of the calculus of variations it must be:

fu (x, u∗(x),∇(u∗)) −d

dxfp (x, u∗(x),∇(u∗)) = 0. (2.14)

This equation is called ”Euler equation” and represents a necessary condition for u∗ to be a min-

imiser of the functional F(u).

A more rigorous and more general derivation of the Euler equation and the fundamental lemma

of the calculus of variations can be found in (Dacorogna, 2004; Giaquinta and Hildebrandt, 1996)

and in (Spitaleri et al., 1999).

In general, the solutions of the Euler equation are not necessarily minima of the variational

problem; they are solutions of the minimum problem when the integrand functions are convex.

2.3.1 The Euler Equation Associated to the Elliptic Approximation of

the Mumford and Shah Model without and with the Curvature

Term

In this section the functionals AT ǫ and AT kǫ are considered and the procedure for the derivation

of the associated Euler equations is detailed.

Let us consider the functional

AT kǫ (u, υ) =

∫

Ω

(

(u − g)2 + λυ2 |∇u|2

+ α

(

ǫ |∇υ|2

+W (υ)

4ǫ

))

dx, (2.15)

where, without loosing in generality, oǫ has been chosen to be 0 and the potential W (t) is not

26


expressed in its explicit form to keep the treatment more general, i.e. valid also for the AT kǫ

functional. Then∂

∂uAT k

ǫ (u, υ) = −2λ∇ · (υ2∇u) + 2(u − g) (2.16)

and∂

∂υAT k

ǫ (u, υ) = −αν + 2λυ |∇u|2, (2.17)

where the term ν

ν = 2ǫ∆υ −Wυ(υ)

ǫ(2.18)

is the term associated to the geometric term in the AT ǫ functional.

Mumford and Shah (1989), proved that the geometric term in MS yielded a term representing

the contour curvature k in the Euler equation associated to MS. As reported by March and Dozio

(1992), De Giorgi then proposed to approximate the term∫

Ωk2dx controlling the discontinuity

boundaries curvature in MSkw with the sequence of functionals:

DGkǫ (υ) =

1

ǫ

∫

Ω

[

2ǫ∆υ −Wυ(υ)

ǫ

]

dx. (2.19)

Let us consider now the functional

MSkw(u,K) =

∫

Ω

(u − g)2dx + λ

∫

Ω

|∇u|2dx +

∫

Su

(α + βk2)ds. (2.20)

Following the above considerations the Ambrosio and Tortorelli approximation becomes:

AT kǫ (u, υ) =

∫

Ω

(

(u − g)2 + λυ2 |∇u|2

+ α

(

ǫ |∇υ|2

+W (υ)

4ǫ

)

+ β1

ǫ

(

2ǫ∆υ −Wυ(υ)

ǫ

))

dx,

(2.21)

where W (t) differs from the one appearing in AT ǫ. In particular here the ”double-well” potential

W (t) is chosen as W (t) = (1 − t2)2 which is quartic in t while in AT ǫ, W (t) is quadratic in t.

The subscript t on Wt(t) denotes the derivative with respect to t. Deriving the Euler equation

associated to AT kǫ (u, υ) yields:

∂

∂uAT k

ǫ (u, υ) = −2λ∇ · (υ2∇u) + 2(u − g) (2.22)

and∂

∂υAT k

ǫ (u, υ) = −αν + 2λυ |∇u|2− 2

β

ǫ2νWυυ(υ) + 4β∆ν, (2.23)

where ∆ denotes the Laplacian operator.

An intuitive explanation of the convergence of the geometric term appearing in AT kǫ to the

geometric term appearing in MSkw can be found in (March and Dozio, 1997).

The last two terms in the Euler Equation (2.23), associated to the functional AT kǫ appear only

when the discontinuity boundaries curvature is taken into account in the model, while the first two

terms come from the AT ǫ functional. The equation (2.22) remains the same for both of the two

27


functionals, AT ǫ and AT kǫ .

The minimisation of both the functionals AT ǫ and AT kǫ implies to find the pairs (u, υ) solutions

of the systems of the associated Euler equation, that is:

0 = −2λ∇ · (υ2∇u) + 2(u − g)

0 = −αν + 2λυ |∇u|2

(2.24)

and, respectively,

0 = −2λ∇ · (υ2∇u) + 2(u − g)

0 = −αν + 2λυ |∇u|2− 2 β

ǫ2 νWυυ(υ) + 4β∆ν ,(2.25)

with Neumann boundary conditions.

Both the systems are made of two non-linear coupled equations in the unknown pair (u, υ).

in order to solve numerically the systems (2.24) and (2.25) an appropriate strategy has to be

chosen. One of the more common choices is to reinterpret the problem of finding the zeros of the

Euler equation as a new evolution problem, i.e. a time dependent problem, which steady state is

seek. This can be done introducing the parabolic equations associated to the first variation of the

functionals, so that the systems:

∂u∂t = µu

(

2λ∇ · (υ2∇u) − 2(u − g))

∂υ∂t = µυ

(

αν − 2λυ |∇u|2) (2.26)

and, respectively,

∂u∂t = µu

(

2λ∇ · (υ2∇u) − 2(u − g))

∂υ∂t = µυ

(

αν − 2λυ |∇u|2

+ 2 βǫ2 νWυυ(υ) − 4β∆ν

) (2.27)

are associated to the Euler equations (2.24) and (2.25). The two systems can now be considered

as systems of gradient descent equations, where the parameters µu and µυ control the rates of

descent. In general, the link between the Euler equation an its associated parabolic equation

can be formulated by means of the application of the so called ”Gradient Flow” theory. For an

introduction to the Gradient Flow theory and its application to the analysis of the Mumford and

Shah functional, we refer to (Hirsch and Smale, 1974), (Gobbino, 1998b), (Evans, 1998), (Gobbino

and Mora, 1999).

It is worth to notice that the first equation of both of the systems, (2.26) and (2.27), is an

anisotropic diffusion equation1, being the anisotropy given by υ2.

The choice of the parameters µu and µυ can be made in order to approximate a Newton-type

1

Another important image processing problem involving anisotropic diffusion equations is the generalised edge-detection theory proposed by Perona and Malik (1990). We refer to (Kawohl, 2004) for a treatment on the linkbetween the Perona and Malik problem and the Mumford and Shah problem.

28


descent method, such as the non-linear Jacobi method and the Gauss-Seidel method.

In particular this leads to chose the parameters as

µu =

(

∂Φu

∂u

)−1

, µυ =

(

∂Φυ

∂υ

)−1

, (2.28)

where

Φu = −2λ∇ · (υ2∇u) + 2(u − g) on both the systems (2.26) and (2.27),

Φυ = −αν + 2λυ |∇u|2

on the first system (2.26),

Φυ = −αν + 2λυ |∇u|2− 2

β

ǫ2νWυυ(υ) + 4β∆ν on the second system (2.27).

2.4 Finite Differences Discretisation

The two gradient descent systems (2.26) and (2.27) can now be implemented and solved numer-

ically. One possibility is to discretise the equations on a uniform grid using finite differences

approximations.

Let the domain Ω be a the rectangle (0, n)× (0,m) and let x = ih and y = jh be a discretisation

of Ω with step size h > 0. Let ui,j be the approximate value of u(i, j), υi,j an approximation of

υ(i, j) and gi,j an approximation of g(i, j), with i = 0, . . . , n and j = 0, . . . ,m. Given this notation

the systems of gradient descent equations in the discrete form become:

ut+1i,j = ut

i,j − µuΦu(uti,j , υ

ti,j)

υt+1i,j = υt

i,j − µυΦυ(uti,j , υ

ti,j)

(2.29)

and

ut+1i,j = ut

i,j − µuΦu(uti,j , υ

ti,j)

υt+1i,j = υt

i,j − µυΦυ(uti,j , υ

ti,j) ,

(2.30)

where Φu = −2λ∇· (υ2∇u)+2(u− g) on both the systems and Φυ = −αν +2λυ |∇u|2

on the first

system and Φυ = −αν + 2λυ |∇u|2− 2 β

ǫ2 νWυυ(υ) + 4β∆ν on the second system. Here, t is the

discrete ”time” or the iteration step and i, j are the position indexes on the mesh sets over Ω and

the parameters µu and µυ are given in (2.28). We assume that the discrete time step is equal to 1.

Considering the equations of system (2.26), the discretisation can be carried out considering

29


central finite differences, that is to set:

∂u

∂x

∣

∣

∣

∣

i,j

=ui+1,j − ui−1,j

2hi = 0, . . . , n and j = 0, . . . ,m , (2.31)

∂u

∂y

∣

∣

∣

∣

i,j

=ui,j+1 − ui,j−1

2hi = 0, . . . , n and j = 0, . . . ,m , (2.32)

(∆u)i,j =1

h2(ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j) i = 0, . . . , n and j = 0, . . . ,m . (2.33)

The same expressions can be written for the function υ. By setting

∇ · (υ2∇u) =∂

∂xi

(

υ2 ∂u

∂xi

)

+∂

∂yj

(

υ2 ∂u

∂yj

)

. (2.34)

and deriving the discrete approximation of the terms µu and µυ given in (2.28) the finite difference

discretisation of the system (2.26) becomes:

ut+1i,j =

1h2 ui,j + 1

λgi,j

1λ + 1

h2 υi,j

υt+1i,j =

2αǫh2 υi,j + α

2ǫ8αǫh2 + α

2ǫ + 2λ|∇u|2i,j,

(2.35)

where the discrete ”time” index t, the iteration index, has been dropped on the right hand side to

avoid cumbersome notation, and where

ui,j = υ2i+1,jui+1,j + υ2

i−1,jui−1,j + υ2i,j+1ui,j+1 + υ2

i,j−1ui,j−1,

υi,j = υ2i+1,j + υ2

i−1,j + υ2i,j+1 + υ2

i,j−1,

υi,j = υi+1,j + υi−1,j + υi,j+1 + υi,j−1,

|∇u|2i,j =1

4h2

[

(ui+1,j − ui−1,j)2 + (ui,j+1 − ui,j−1)

2]

.

The expression of the discrete approximation of the terms µu and µυ has been omitted, the dis-

cretisation can be however obtained in a straightforward way.

It is possible to derive a slightly different approximation of u starting from the expansion:

∇ · (υ2∇u) = υ2∆u + 2υ∇υ · ∇u.

A discrete approximation of the equation in u in the systems (2.26) and (2.27) can then be written

as

ut+1i,j = ui,j + µu

λυ2i,j(∆u)i,j +

λ

2h2zi,j [(υi+1,j − υi−1,j)(ui+1,j − ui−1,j)+

+(υi,j+1 − υi,j−1)(ui,j+1 − ui,j−1)] − ui,j + gi,j , (2.36)

30


where the discrete ”time” index t has been dropped on the right hand side to avoid cumbersome

notation, and (∆u)i,j is the discrete approximation of the Laplacian of u as in (2.33). Substituting

the discrete approximation of the parameter µu the approximation becomes:

ut+1i,j =

2υi,j(∇υ)i,j · (∇u)i,j + 1h2 υ2

i,j ui,j + 1λgi,j

1λ + 4

h2 υ2i,j

, (2.37)

where

ui,j = ui+1,j + ui−1,j + ui,j+1 + ui,j−1. (2.38)

The discrete approximation of the equation in υ in the system (2.27) is:

υt+1i,j = υi,j + µυ

−4β(∆υ)i,j + ανi,j + 8βǫ−2(3υ2i,j − 1)νi,j +

−λ

2h2υi,j

[

(ui+1,j − ui−1,j)2 + (ui,j+1 − ui,j−1)

2]

, (2.39)

where the discrete approximation for νi,j is:

νi,j = 2ǫ(∆υ)i,j −4

ǫ(z3

i,j − zi,j). (2.40)

The Laplacian (∆υ)i,j is defined as in (2.33), and from the expansion:

∆ν = 2ǫ∆2υ −1

ǫ∆Wυ = 2ǫ∆2υ −

4

ǫ

[

(3υ2 − 1)∆υ + 6υ|∇υ|2]

, (2.41)

it is possible to derive the discrete approximation of (∆υ)i,j

(∆υ)i,j = 2ǫ(∆2)i,jυ −4

ǫ

(3υ2i,j − 1)(∆υ)i,j +

3

2h2υi,j

[

(υi+1,j − υi−1,j)2+

+(υi,j+1 − υi,j−1)2]

, (2.42)

where (∆2υ)i,j denotes the biharmonic operator:

(∆2υ)i,j =1

h4[−8(υi+1,j + υi−1,j + υi,j+1 + υi,j−1) + 2(υi+1,j+1 + υi+1,j−1+

+υi−1,j+1 + υi−1,j−1) + υi+2,j + υi−2,j + υi,j+2 + υi,j−2 − 20υi,j ] . (2.43)

31


Substituting the discrete approximation of the parameter µυ the approximation becomes:

zt+1i,j =

[

2αε

h2υi,j +

8α

ευ3

i,j +8βε

h4(8υi,j − 2υi,j − υi,j) −

32β

ε(3υ2

i,j + 1)(∆υ)i,j+

+128β

εh2υi,j(3υ

2i,j − 1) +

128β

ε3υ3

i,j(3υ2i,j − 2)

] [

8αε

h2+

4α

ε(3υ2

i,j − 1) + 2λ |∇u|2i,j +

+32β

εh2

(

h2

ε2+ 5

ε2

h2− 4

)

−96β

ε|∇υ|

2i,j −

192β

ευi,j(∆υ)i,j +

384β

εh2υ2

i,j

(

1 −h2

ε2

)

+480β

ε3υ4

i,j

]−1

,

(2.44)

where υi,j is as in (2.38).

It is relevant to note here that in both of the systems if the the array υi,j is kept fixed in the

first equation, the discrete equation for u correspond to a linear system for the n × m unknowns

ui,j . In the same manner, in both of the systems if the the array ui,j is kept fixed in the second

equation, the discrete equation for υ correspond to a linear system for the n × m unknowns υi,j .

Moreover, it can be intuitively understood that the grid must resolve the width of the transition

region of the function υ. As the transition region shrinks as ǫ → 0, this means that the grid step h

must goes to zero faster than ǫ. From a more analytical point of view, Bellettini and Coscia (1994b)

proved that the Γ-convergence of the discrete approximation AT hǫ holds when the discretisation

step h is an infinitesimal faster than ǫ, i.e. when h = o(ǫ).

When numerical implementations are actually performed, i.e. when the discretisation step h is

given and fixed, the above condition imposes that h/ǫ < 1, (March and Dozio, 1997). In all the

experiments reported in Chapter 5 the discretisation step h is set equal to one.

We briefly mention here a relatively recent approach to the minimisation of the Mumford and

Shah functional. Chan and Vese (2001) presented a method to practically minimise the functional

of Mumford and Shah without neither the need of using a weak formulation nor an approximating

functional. The solution of the Mumford and Shah functional is computed numerically by means

of a level set algorithm. The key idea of the authors is to consider the set K as the zero level set

of an ad doc defined function ϕ(x), called a level set function.

2.5 The Gradient Effect in Dimension One

The Mumford and Shah functional, due to the presence of a first order term controlling the ”smooth-

ness” of the solution u, presents two main restrictions:

points where the function u is continuous with discontinuous first derivative, i.e. crease

points, can not be handled;

the functional is affected by the so call ”gradient” or ”ramp” effect.

32


For the seek of simplicity and for consistency with the treatment given in the next Chapter and

without loosing in generality, it is more convenient here to introduce the problem in dimension

one, where the formulation of the Mumford and Shah functional is:

MS(u,D) =

∫

(a,b)\D

(u − g)2dx + λ

∫

(a,b)\D

|u′(x)|2dx + α♯D, (2.45)

where D is the set of the discontinuities and ♯D is the measure of the counting of the discontinuity

points of u in D and u′(x) is the first derivative of u.

Following a work by Blake and Zisserman, (Blake and Zisserman, 1987), let consider the function

gm,h, defined on the interval (a, b) and depending on the positive parameters m and h:

gm,h(x) =

c if x ∈ [a, x)

c + mh x − m

h x if x ∈ [x, x + h]

c + m if x ∈ (x + h, b] ,

(2.46)

The function gm,h is continuous and is called a ”ramp” function, and it gives a quite clear example

of a function that generates a gradient effect.

The result of solving (2.5) exhibits a different behaviour depending on the value of g′ compared

with g′lim =√

βγ2

2 . The cases are:

if g′ < g′lim, the solution u presents the same features of the function g, i,e, u is a ”ramp”;

if g′ > g′lim, the solution u presents one discontinuity because a discontinuity is prefered to

a high value of the derivative;

if g′ ≫ g′lim, several discontinuities appear.

The function gm,h(x) and different solutions for u are plotted in figure 2.1.

Figure 2.1: Plot of the function gm,h(x) with a schematic representation of the Gradient effect.

33


To overcome such problem Blake and Zisserman introduced the second order functional:

BZ(u,D1,D2) =

∫

(a,b)\D1

(u − g)2dx + γ

∫

(a,b)\(D1∪D2)

|u′′|2dx + α♯D1 + β♯D2, (2.47)

where D1 and D2 are the set of discontinuity points of u and of u′ respectively, and ♯D1, ♯D2

are their respectively counting measures. The fact that the Blake and Zisserman functional is

not affected by the gradient effect can be intuitively understood noticing that the first derivative

of u does not appear any more in the functional, where, anyway, the counting measure of the

discontinuity points of u′ is controlled.

It is worth here to mention that the introduction of the second derivative of u in the functional

leads to new and relevant mathematical difficulties, among others, the difficulty in defining an

efficient weak formulation of the minimum problem in dimension n when n > 1.

The Blake and Zisserman functional is detailed in the next Chapter.

34

3 The Blake and Zisserman Model

In this Chapter the segmentation functional model of Blake and Zisserman is presented and de-

scribed in analogy with the previous Chapter on the Mumford and Shah model for image seg-

mentation, i.e. for a two dimensional signal. There are two main differences between the two

functionals. On one hand, the order of the term controlling the ”smoothness” of the signal: a

first order term, the gradient of the solution u, in the Mumford and Shah functional and a second

order term, the second derivative of the solution u, in the Blake and Zisserman functional. On the

other hand, in the Blake and Zisserman model ”crease” points of the discontinuity curves of the

set K are explicitly taken into account by means of a specific term in the functional. Following

the treatment given for the Mumford and Shah model the weak formulation of the Blake and Zis-

serman functional is reported along with its approximating functional. Again, the Ambrosio and

Tortorelli approximation by elliptic functional has been considered in order to allow the numerical

implementation of the Blake and Zisserman functional.

The Blake and Zisserman functional is discretised by finite elements, and the discrete minimum

problem is solved numerically . The solution of the Blake and Zisserman problem by means of the

Euler equation would also be possible, (Carriero et al., 2006), but it is not treated here.

In this work the Blake and Zisserman model has been applied to the segmentation of one di-

mensional signals. After a treatment in dimension two, given for the seek of generality and for

consistence with the Mumford and Shah functional treatment, the formulation of the Blake and

Zisserman functional in dimension one is reported and approached numerically . This choice is

strongly related to the capability of the Blake and Zisserman functional to segment one dimensional

signals overcoming the gradient effect and performing a ”less restrictive” signal smoothing with re-

spect to the smoothing performed by the Mumford and Shah functional. In fact, one dimensional

signals present very often trends which can not be correctly segmented by the Mumford and Shah

functional. Of course, the opportunity of applying such a functional to perform image segmentation

remains of great relevance. Anyway, while the segmentation of environmental photos and remotely

sensed images seems not to require specifically the features of the Blake and Zisserman functional,

being the results of the Mumford and Shah segmentation already of great quality and interest, the

Blake and Zisserman functional peculiarities could be relevant in segmenting environmental two

dimensional signals of different nature, such as lidar digital surface models. Aside in image seg-

mentation, the study of the minimisation of functionals depending on a second order bulk energy

and on a surface discontinuity energy, such as the functional proposed by Blake and Zisserman, is

of great interest in fracture theory, optimal partitions and elastic-plastic plates fields.

3. The Blake and Zisserman Model

3.1 The Blake and Zisserman Model for One Dimensional

Signal Segmentation

Blake and Zisserman (1987) introduced a variational principle for image segmentation in the context

of visual reconstruction, the proposed functional depends on second derivatives, free discontinuities

and free gradient discontinuities of the intensity levels of an image.

The formulation of the Blake and Zisserman functional in dimension two is:

BZ(u,K0,K1) =

∫

Ω\(K0∪K1)

(u−g)2dx+γ

∫

Ω\(K0∪K1)

|∆u|2dx+αH1(K0∩Ω)+βH1((K1\K0)∩Ω),

(3.1)

where Ω ⊂ R2 is a bounded open set, |∆u| denotes the euclidean norm of the Hessian matrix of u.

The positive real parameters γ, α, β and the bounded function g(x) are given, while K0,K1 ⊂ R2

are closed sets and u is a function such that u ∈ C1(Ω \ (K0 ∪ K1)) ∩ C0(Ω \ K0).

The existence of minimisers of the Blake and Zisserman functional has been proved by Carriero

et al. (1997), by regularisation of solutions of a weak formulation in dimension two under the

additional condition that g is bounded and measurable in Ω and:

0 < β ≤ α ≤ 2β. (3.2)

The condition β > 0 is necessary to assure compactness, while the condition β ≤ α ≤ 2β is

necessary to assure the lower semicontinuity of the sum of the two terms αH1(K0∩Ω)+βH1((K1 \

K0) ∩ Ω).

The weak formulation of the Blake and Zisserman functional is:

BZw(u) =

∫

Ω

(u − g)2dx + γ

∫

Ω

|∆u|2dx + αH1(Su) + βH1(S∇u \ Su), (3.3)

being BZw : GSBV 2(Ω) → [0,+∞], where the space of Generalised Special Functions of Bounded

Variation is a generalisation of the SBV space. We refer to (Ambrosio et al., 2000) for definitions

and treatment of these spaces.

3.2 The Ambrosio and Tortorelli Approximation

The effective numerical minimisation of BZw can not be directly performed due to the difficulties

of finding the free discontinuity set Su and the free gradient discontinuity set S∇u. In a framework

similar to the one developed for the elliptic approximation of the MSw functional, Ambrosio et al.

36


(2001) proposed to approximate BZw by the following sequence of functionals:

AT ǫ(u, s, σ) =

∫

Ω

(u − g)2dx + γ

∫

Ω

(s2 + δǫ) |∆u|2dx+

+ µǫ

∫

Ω

σ2 |∇u|2dx + βMMǫ(s) + (α − β)MMǫ(σ), (3.4)

where:

MMǫ(t) =

∫

Ω

[

ǫ |∇t|2

+(1 − t)2

4ǫ

]

dx. (3.5)

The two functions s and σ behave as the function υ in the Ambrosio and Tortorelli approximation

of the MSw. In particular, σ is related to the set Su and s is related to the set S∇u. The two

functions, σ and s, act respectively on the gradient of u and on the Hessian of u, so that where

the contribute of the derivatives is too large, σ and s can tent to zero to reduce the penalties. The

approximation takes place in the sense of the Γ-convergence as ǫ → 0, having:

limǫ→0

δǫ

ǫ3= 0 , lim

ǫ→0

ǫ

µǫ= 0. (3.6)

The expression of the Blake and Zisserman functional, of its weak formulation and of the elliptical

approximation in dimension one are now presented.

The formulation in dimension one is:

BZ(u,K0,K1) =

∫

(a,b)\(K0∪K1)

(u−g)2dx+γ

∫

(a,b)\(K0∪K1)

|u′′|2dx+α♯(K0∩Ω)+β♯((K1\K0)∩Ω),

(3.7)

where the symbol ♯ denotes the counting measure of the sets K0 and K1.

The weak formulation in GSBV 2 is:

BZw(u) =

∫

(a,b)

(u − g)2dx + γ

∫

(a,b)

|u′′|2dx + α♯(Su) + β♯(Su′ \ Su), (3.8)

where as in dimension two, Su indicates the set of jumps of u and Su′ the set of jumps of u′, i.e.

the crease points. Also in dimension one, the condition:

0 < β ≤ α ≤ 2β (3.9)

has to be respected to guarantee the existence of the solution.

The approximation of BZw by means of a sequence of elliptic functional becomes:

AT ǫ(u, s, σ) =

∫

(a,b)

(u − g)2dx + γ

∫

(a,b)

(s2 + δǫ) |u′′|

2dx+

+ µǫ

∫

(a,b)

σ2 |u′|2dx + βMMǫ(s) + (α − β)MMǫ(σ), (3.10)

37


where

MMǫ(t) =

∫

(a,b)

[

ǫ |t′|2

+(1 − t)2

4ǫ

]

dx (3.11)

and the conditions (3.6) on the parameters δǫ and µǫ have to be respected as in dimension two.

The functional AT ǫ(u, s, σ) is defined for g ∈ L2(a, b) and s, σ ∈ W 1,2(a, b) and u ∈ W 2,2(a, b).

3.3 Finite Elements Discretisation

The finite elements approximation is particularly well suited for the numerical discretisation of

variation problems. A fundamental feature of the FEM is that the integral of a measurable function

on an arbitrary domain Ω can be seen as a summation of integrals on an arbitrary set of non

overlapping sub-domains, Ωi, whose union is the original domain Ω. This allows to approach the

problem locally on a single sub-domain Ωh, the finite element, that is sufficiently small to allow

to represent the local behaviour of the solution by means of polynomial functions of some degree.

This allows, moreover, to focus the analysis on a prototype of the finite element so that the analysis

does not depend on the position of the fine element on the mesh discretising the domain Ω.

Let u be a function in an Hilbert space X, let a(·, ·) be a continuous bi-linear form and l(·) a

continuous linear form, the abstract variational problem of finding one element u so that:

u ∈ X,

a(u, v) = l(v) ∀v ∈ X,(3.12)

has one and only one solution. When the form a is symmetric, i.e. a(u, v) = a(v, u) ∀u, v inX the

solution of (3.12) is the solution of the minimum problem:

minv∈X

1

2a(v, v) − l(v)

, (3.13)

being equation (3.12) the Euler equation associated to the functional 12a(v, v) − l(v).

The finite elements method allows to approximate the solution of (3.12) by defining an analogous

problem in finite dimension sub-spaces of X. For each finite dimension sub-space Xn ⊂ X the

discrete problem becomes:

un ∈ Xn,

a(un, vn) = l(vn) ∀vn ∈ Xn,(3.14)

which admits one and only one solution, called discrete solution and the spaces Xn are called finite

elements spaces.

To each sub-space Xn it is associated a discrete solution un satisfying (3.14). The discrete

problem then converges to the continuous problem (3.12) if:

limn→+∞

‖ u − un ‖= 0, (3.15)

38


The solution of the discrete problem un, where (wk)k=1,...,d is a base of the space Xn and d is the

dimension of the space Xn can be written as

un =

d∑

k=1

xkwk,

and the vector of the coefficients (x1, . . . , xd) is the solution of the linear system:

d∑

k=1

a(wk, wk)xk = l(wm), 1 ≤ m ≤ d. (3.16)

From the numerical viewpoint the choice of the base (wk)k=1,...,d is especially important to build

a system matrix with the greatest number of null elements. This need can be satisfied imposing

the support of the functions wk to be as small as possible.

In the following the Ambrosio and Tortorelli approximation of the Blake and Zisserman functional

in the weak form in dimension one is discretised by means of finite elements.

Without loosing in generality it is possible to normalise the domain [a, b] to [0, 1]. It is then

necessary to define a mesh on the domain so that:

Mn+1 = 0 = t0 < t1 < . . . < tn−1 < tn = 1,

where n + 1 is the number of nodes of the mesh and the distance between two adjacent nodes is1n . Each element of the mesh is therefore an interval:

Ti =

[

i

n,i + 1

n

]

i = 0, . . . , n − 1,

and

Tn = Ti : i = 0, . . . , n − 1

is the domain discretisation. The finite elements space Xn is a finite dimension sub-space of

W 2,2(0, 1) × W 1,2(0, 1) × W 1,2(0, 1) and is hereafter denoted as Vn × Yn × Yn.

The functions s and σ belong to W 1,2(0, 1) and it is therefore sufficient to require the base of

the space Yn, where the functions s and σ are approximated, to be made of C0 functions. The

following functions ψi:

ψi(t) =

0 if t < ti−1

−n|t − ti| + 1 if ti−1 ≤ t ≤ ti+1

0 if t > ti+1 ,

(3.17)

with i = 0, . . . , n − 1 ,are piecewise linear and continuous on [0, 1], and the support of ψi is made

of no more than two adjacent elements of Tn, and each function ψi is uniquely determinate by the

39


values on the nodes i − 1, i, i + 1. For i = 0 and i = n, ψi are defined as

ψ0(t) =

−n(t − t0) + 1 if 0 ≤ t ≤ t1

0 if t > t1 .

(3.18)

ψn(t) =

0 if t < tn−1

−n(tn − t) + 1 if tn−1 ≤ t ≤ 1 .

As it is common in finite elements methods the functions ψi are called: shape functions or base

functions.

After the definition of the base functions ψi, the FEM requires to find in Yn two linear combi-

nations of the base functions that approximate the function s and σ:

sn(t) =n

∑

i=0

siψi(t), σn(t) =n

∑

i=0

σiψi(t).

The coefficients si and σi are unknown and uniquely determinate sn and σn

The given data g can be also approximated, using the base function ψ, by:

gn(t) =n

∑

i=0

uiψi(t).

Regarding the function u, it belongs to W 2,2(0, 1) and it is therefore sufficient to require the

functions in the finite elements space Vn to be C1. In this case, the base functions have to employ,

along with the values on the nodes, also the values of the first derivative on the nodes.

The base function φ1i has null first derivative in all the nodes and null value in all the notes

but the ith node, where the value is 1. The base function φ2i has null value in all the nodes and

null value of the first derivative in all nodes but the ith node, where the value is 1. The explicit

formulas of φ1i and φ2

i are:

φ1i (t) =

0 if t < ti−1

2n3|t − ti|(t − ti)2 − 3n2(t − ti)

2 + 1 if ti−1 ≤ t ≤ ti+1

0 if t > ti+1 .

(3.19)

φ2i (t) =

0 if t < ti−1

n2(t − ti)3 − 2n|t − ti|(t − ti) + t − ti if ti−1 ≤ t ≤ ti+1

0 if t > ti+1 .

Figures (3.1) shows the plot, in the interval [t − 1, t + 1], of the three base functions.

40


t

ψi(t)

ti ti+1ti−1

1

t

φ1i (t)

ti ti+1ti−1

1

t

φ2i (t)

ti ti+1ti−1

0.15

−0.15

Figure 3.1: Finite elements base functions.

41


The finite elements approximation un ∈ Vn of the function u then becomes:

un(t) =

n∑

i=0

(

u1i φ

1i (t) + u2

i φ2i (t)

)

,

where, again, the coefficients u1i and u2

i are unknown and uniquely determinate un.

Now the discretisation of the Ambrosio and Tortorelli approximation of the weak form of the

Blake and Zisserman functional in dimension one can be obtained computing:

AT ǫ(u, s, σ).

The discretisation is therefore:

AT ǫ(u, s, σ) = γ

∫ 1

0

n∑

i,j=0

sisjψiψj + δǫ

n∑

i,j=0

u1i u

1j (φ

1i )

′′(φ1j )

′′ + 2

n∑

i,j=0

u1i u

2j (φ

1i )

′′(φ2j )

′′+

+

n∑

i,j=0

u2i u

2j (φ

2i )

′′(φ2j )

′′

dt+

+ β

∫ 1

0

ǫ

n∑

i,j=0

sisjψ′iψ

′j +

1

4ǫ

n∑

i,j=0

(si − 1)(sj − 1)ψiψj

dt+

+ (α − β)

∫ 1

0

ǫ

n∑

i,j=0

σiσjψ′iψ

′j +

1

4ǫ

n∑

i,j=0

(σi − 1)(σj − 1)ψiψj

dt+

+ µǫ

∫ 1

0

n∑

i,j=0

σiσjψiψj

n∑

i,j=0

u1i u

1j (φ

1i )

′(φ1j )

′ + 2

n∑

i,j=0

u1i u

2j (φ

1i )

′(φ2j )

′ +

n∑

i,j=0

u2i u

2j (φ

2i )

′(φ2j )

′

dt+

+

∫ 1

0

n∑

i,j=0

u1i u

1jφ

1i φ

1j + 2

n∑

i,j=0

u1i u

2jφ

1i φ

2j +

n∑

i,j

u2i u

2jφ

2i φ

2j − 2

n∑

i,j=0

u1i gjφ

1i ψj − 2

n∑

i,j=0

u2i gjφ

2i ψj+

+n

∑

j,j=0

gigjψiψj

dt,

(3.20)

42


that with some manipulations becomes:

AT ǫ(u, s, σ) = γ

n∑

i,j,h,k

sisju1i u

1j

∫ 1

0

ψiψj(φ1h)′′(φ1

k)′′dt + 2n

∑

i,j,h,k

sisju1i u

2j

∫ 1

0


k)′′dt+

+

n∑

i,j,h,k

sisju2i u

2j

∫ 1

0


k)′′dt

+

+ δǫ

n∑

i,j

u1i u

1j

∫ 1

0

(φ1i )

′′(φ1j )

′′dt + 2

n∑

i,j

u1i u

2j

∫ 1

0

(φ1i )

′′(φ2j )

′′dt +

n∑

i,j

u2i u

2j

∫ 1

0

(φ2i )

′′(φ2j )

′′dt

+

+ βǫ

n∑

i,j

sisj

∫ 1

0

ψ′iψ

′jdt +

β

4ǫ

n∑

i,j

(si − 1)(sj − 1)

∫ 1

0

ψiψjdt+

+ (α − β)ǫ

n∑

i,j

σiσj

∫ 1

0

ψ′iψ

′jdt +

α − β

4ǫ

n∑

i,j

(σi − 1)(σj − 1)

∫ 1

0

ψiψjdt+

+ µǫ

n∑

i,j,h,k

σiσju1hu1

k

∫ 1

0

ψiψj(φ1h)′(φ1

k)′dt + 2n

∑

i,j,h,k

σiσju1hu2

k

∫ 1

0

ψiψj(φ1h)′(φ2

k)′dt+

+

n∑

i,j,h,k

σiσju2hu2

k

∫ 1

0

ψiψj(φ2h)′(φ2

k)′dt

+

+

n∑

i,j

u1i u

1j

∫ 1

0

φ1i φ

1jdt + 2

n∑

i,j

u1i u

2j

∫ 1

0

φ1i φ

2jdt +

n∑

i,j

u2i u

2j

∫ 1

0

φ2i φ

2jdt − 2

n∑

i,j

u1i gj

∫ 1

0

φ1i ψjdt+

− 2

n∑

i,j

u2i gj

∫ 1

0

φ2i ψjdt +

n∑

i,j

gigj

∫ 2

0

ψiψjdt.

(3.21)

43


The quantities:

∫ 1

0


k)′′dt,

∫ 1

0


k)′′dt,

∫ 1

0


k)′′dt,

∫ 1

0

(φ1i )

′′(φ1j )

′′dt,

∫ 1

0

(φ1i )

′′(φ2j )

′′dt,

∫ 1

0

(φ2i )

′′(φ2j )

′′dt,

∫ 1

0

ψ′iψ

′jdt,

∫ 1

0

ψiψjdt,

∫ 1

0

ψiψj(φ1h)′(φ1

k)′dt,

∫ 1

0

ψiψj(φ1h)′(φ2

k)′dt,

∫ 1

0

ψiψj(φ2h)′(φ2

k)′dt,

∫ 1

0

φ1i φ

1jdt,

∫ 1

0

φ1i φ

2jdt,

∫ 1

0

φ2i φ

2jdt,

∫ 1

0

φ1i ψjdt,

∫ 1

0

φ2i ψjdt, (3.22)

are the analogous of the coefficients a(wk, wm) of the abstract problem (3.16) and are characterised

by the presence of many zeroes. In particular all of the quantities in (3.22) are null if one index

differs from any of the others of more than one unit.

Once the finite elements discretisation (3.21) is computed it is necessary to chose a minimisation

algorithm to implement numerically the functional model of Blake and Zisserman.

One possibility is to search the zeroes of the equation:

∇AT ǫ(u, s, σ) = 0, (3.23)

being

∇AT ǫ(u, s, σ) =

(

∂AT ǫ

∂u1l

,∂AT ǫ

∂u2l

,∂AT ǫ

∂sl,∂AT ǫ

∂σl

)

l=0,...,n

, (3.24)

where the subscript l indicates the element with respect to which the derivatives are computed.

Now, to compute numerically the zeroes of the equation (3.23), one possibility is to define a

new time dependent problem, i.e. a dynamic problem, which steady states are searched. The new

44


problem is defined by applying gradient descent to the parabolic equations associated to (3.23):

∂u1l

∂t = µu1∂AT ǫ

∂u1l

∂u2l

∂t = µu2∂AT ǫ

∂u2l

∂sl

∂t = µs∂AT ǫ

∂sl

∂σl

∂t = µσ∂AT ǫ

∂σl

,

(3.25)

where the parameters µu1 , µu2 , µs and µσ can be set to define a Newton-type descent method, i.e.

by setting:

µu1 =

(

∂

∂u1l

∂AT ǫ

∂u1l

)−1

, µu2 =

(

∂

∂u2l

∂AT ǫ

∂u2l

)−1

,

µs =

(

∂

∂sl

∂AT ǫ

∂sl

)−1

, µσ =

(

∂

∂σ1l

∂AT ǫ

∂σl

)−1

.

In the following the expressions of the terms appearing in (3.25) are given.

•∂AT ǫ

∂u1l

= γ

2

n∑

i,j,h

sisju1h

∫ 1

0


l )′′dt + 2

n∑

i,j,k

sisju2k

∫ 1

0

ψiψj(φ1l )

′′(φ2k)′′dt+

+2δǫ

n∑

i

u1i

∫ 1

0

(φ1i )

′′(φ1l )

′′dt + 2δǫ

n∑

j

u2j

∫ 1

0

(φ1l )

′′(φ2j )

′′dt

+

+ 2µǫ

n∑

i,j,h

σiσju1h

∫ 1

0

ψiψj(φ1h)′(φ1

l )′dt + 2µǫ

n∑

i,j,k

σiσju2k

∫ 1

0

ψiψj(φ1l )

′(φ2k)′dt+

+ 2

n∑

j

u1j

∫ 1

0

φ1l φ

1jdt + 2

n∑

j

u2j

∫ 1

0

φ1l φ

2jdt − 2

n∑

j

gj

∫ 1

0

φ1l ψjdt. (3.26)

•∂AT ǫ

∂u2l

= γ

2n

∑

i,j,h

sisju2h

∫ 1

0


l )′′dt + 2

n∑

i,j,k

sisju2k

∫ 1

0

ψiψj(φ2l )

′′(φ2k)′′dt+

+2δǫ

n∑

i

u1i

∫ 1

0

(φ1i )

′′(φ2l )

′′dt + 2δǫ

n∑

j

u2j

∫ 1

0

(φ2l )

′′(φ2j )

′′dt

+

+ 2µǫ

n∑

i,j,h

σiσju1h

∫ 1

0

ψiψj(φ1h)′(φ2

l )′dt + 2µǫ

n∑

i,j,k

σiσju2k

∫ 1

0

ψiψj(φ2l )

′(φ2k)′dt+

+ 2

n∑

i

u2l

∫ 1

0

φ1i φ

2l dt + 2

n∑

j

u2j

∫ 1

0

φ2l φ

2jdt − 2

n∑

j

gj

∫ 1

0

φ2l ψjdt. (3.27)

45


•∂AT ǫ

∂sl= γ

2

n∑

i,h,k

siu1hu1

k

∫ 1

0

ψiψl(φ1h)′′(φ1

k)′′dt + 2

n∑

i,h,k

siu1hu2

k

∫ 1

0


k)′′dt+

2

n∑

i,h,k

siu2hu2

k

∫ 1

0


k)′′dt

+ 2βǫ

n∑

i

si

∫ 1

0

ψ′iψ

′ldt +

β

2ǫ

n∑

i

(si − 1)

∫ 1

0

ψiψldt.

(3.28)

•∂AT ǫ

∂σl= γ

2n

∑

i,h,k

σiu1hu1

k

∫ 1

0


k)′′dt + 2n

∑

i,h,k

σiu1hu2

k

∫ 1

0


k)′′dt+

2

n∑

i,h,k

σiu2hu2

k

∫ 1

0


k)′′dt

+ 2βǫ

n∑

i

σi

∫ 1

0

ψ′iψ

′ldt +

β

2ǫ

n∑

i

(σi − 1)

∫ 1

0

ψiψldt.

(3.29)

•∂

∂u1l

∂AT ǫ

∂u1l

= γ

2

n∑

i,j

sisj

∫ 1

0

ψiψj(φ1l )

′′(φ1l )

′′dt + 2δǫ

∫ 1

0

(φ1l )

′′(φ1l )

′′dt

+

+ 2µǫ

n∑

i,j

σiσj

∫ 1

0

ψiψj(φ1l )

′(φ1l )

′dt + 2

∫ 1

0

φ1l φ

1l dt. (3.30)

•∂

∂u2l

∂AT ǫ

∂u2l

= γ

2

n∑

i,j

sisj

∫ 1

0

ψiψj(φ2l )

′′(φ2l )

′′dt + 2δǫ

∫ 1

0

(φ2l )

′′(φ2l )

′′dt

+

+ 2µǫ

n∑

i,j

σiσj

∫ 1

0

ψiψj(φ2l )

′(φ2l )

′dt + 2

∫ 1

0

φ2l φ

2l dt. (3.31)

•∂

∂sl

∂AT ǫ

∂sl= γ

2

n∑

h,k

u1hu1

k

∫ 1

0

ψlψl(φ1h)′′(φ1

k)′′dt + 4

n∑

h,k

u1hu2

k

∫ 1

0


k)′′dt+

+2

n∑

h,k

u2hu2

k

∫ 1

0


k)′′dt

+ 2βǫ

∫ 1

0

ψ′lψ

′ldt +

β

2ǫ

∫ 1

0

ψlψldt. (3.32)

46


•∂

∂σl

∂AT ǫ

∂σl= µǫ

2

n∑

h,k

u1hu1

k

∫ 1

0


k)′′dt + 4

n∑

h,k

u1hu2

k

∫ 1

0


k)′′dt+

2

n∑

h,k

u2hu2

k

∫ 1

0


k)′′dt

+ 2(α − β)ǫ

∫ 1

0

ψ′lψ

′ldt +

α − β

2ǫ

∫ 1

0

ψlψldt.

(3.33)

47

Part II

Numerical Applications

4 Software Implementations

In this Chapter the software implementing the numerical algorithms described in section (2.4) and

(3.3) are described. All of the software has been written in the C language.

Given the difference between the discretisation methods applied to solve numerically the Mum-

ford and Shah problem in dimension two, a finite difference method, and the Blake and Zisserman

problem in dimension one, a finite element method, two distinct projects have been developed. In

both the cases a C library has been developed implementing the functions that actually perform

the segmentation. Two different C programs, which basically implement the data I/O and use the

functions defined within the two C libraries have been then implemented. The program for images

segmentation has been written as a GRASS1 (Geographical Resources Analysis Support System)

GIS module, while the program for signals segmentation is a standalone program.

The libraries code has been documented directly within the source code following the Doxygen2

specifications. This permits both to easily and properly comment the source code and to produce

automatically the software documentation in many file formats from the same source code, avoid-

ing the need to separately and independently comment the source code and write the software

documentation. The two C programs have been instead normally commented within the source

code, the programs usage documentation is provided in HTML format and the essential usage

documentation is available in-line.

All of the source code developed is released under the GNU3 General Public License, version 2.

4.1 The seglib Library for Images Segmentation

The library seglib implements the Mumford and Shah model and the Mumford and Shah model with

the curvature term. The library functions solve the discrete gradient descent problem associated to

the finite differences discretisation of the Euler equation associated to the Ambrosio and Tortorelli

approximation of the two functional models. The library implements, for both of the models, the

nonlinear Jacobi and the nonlinear Gauss-Seidel iterative methods.

The general structure of the seglib library is:

headers inclusion

preprocessor macros definition

1http://grass.itc.it2http://www.doxygen.org3http://www.gnu.org

4. Software Implementations

segmentation functions implementation

Six different segmentation functions have been implemented. Definitions and tasks of these

functions are detailed. The notation is kept as simple and essential as possible to facilitate the

descriptions of the main aspects of the implementation. The names of the variables is kept coherent,

whenever possible, to the name of the corresponding quantities appearing in the mathematical

formulas. The data type of variables is specified only when non standard data types have been

used.

The segmentation functions are:

ms n(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

This function implements the equations of system (2.35)

and the Gauss-Seidel iterative method.

ms o(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

The function implements the equations of system (2.35)

and the Jacobi iterative method.

ms t(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

The function implements the equations of system (2.35), with the first equation replaced by

(2.36)


msk n(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

The function implements the equations (2.36), (2.39)


msk o(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

The function implements the equations (2.36), (2.39)

and the Jacobi iterative method.

msk t(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc)

The function implements the equations of system (2.35), with the second equation replaced

by (2.39)


Where, the variable z stores the solution υ and the variable kepsilon the inverse of the parameter

ǫ appeared in the Ambrosio and Tortorelli approximation. The variable mxdf stores the maximum

difference, over the entire domain, between the values of the solution u at two consecutive iteration

steps. Provided a value for the convergence threshold of the iterative method, tol, the variable

mxdf can be used to verify a convergence criterion, ze.g. mxdf < tol. The variables nr and nc

store the number of rows and of columns of the input image.

An example of the code implementing a call to tsegmentation functionshe function ms n(),

and a convergence check, could be:

52


iter = 0; mxdf = 0; ms_n(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc);

while((mxdf > tol) && (iter <= max_iter))

mxdf = 0;

ms_n(g, u, z, lambda, kepsilon, alpha, mxdf, nr, nc);

iter = iter + 1;

where the variable iter is the iteration counter and max_iter stores the maximum number of

iterations permitted. The fourth line of the example performs a new call to the function ms n()

until the convergence criterion is satisfied, i.e. when the value of mxdf becomes smaller of the

convergence threshold tol or when the maximum number of iterations is exceeded.

The general structure of one of the segmentation functions is:

variables definition

set of boundary conditions

solution computation

for j = 1:nr-1

for i = 1:nc-1

uti,j = [...]

zti,j = [...]

diffi,j = uti,j - ut-1i,j

mxdf = max(mxdf, diffi,j)

end

end

Where, i and j are the row and column indexes and the superscript t indicates the iteration

step. The variable diffi,j stores, for the running element i,j, the difference between the value of

the solution at the running iteration t and at the previous iteration t-1.

The Neumann boundary conditions are imposed by set the values of the boundary nodes to the

values of the closest inter-nodes.

The first two lines within the for loops are the equations of the gradient descent system as-

sociated to the approximation of the functional implemented in the considered segmentation

function. We recall here that the equations for ui,j, in (2.35) and (2.36), are linear if the variable

zi,j is kept fixed during the running iteration. The same is true for the equations for zi,j, in (2.35)

and (2.39). This permits, for each point i,j on the image, to directly compute the value of ui,j

and zi,j at each iteration step.

4.1.1 The GRASS GIS module r.seg

The GRASS module r.seg, by accessing the seglib library functions, provides an end-user program

to perform image segmentation within the GRASS GIS framework. The choice of developing a

GRASS module rather that and stand alone program depends mainly on the advantage of inte-

grating an image analysis code within a GIS. A GIS is, by design, the natural software framework

53


where ”environmental” images can be efficiently and properly handled. This does not however pre-

vent the possibility of using such a framework to manage and analyse non-environmental images,

e.g. all of the segmentation tests reported in Chapter (5), both environmental and not, have been

performed within the GRASS GIS. Moreover, developing a program as a GRASS module permits

to exploit the GRASS GUI (Graphical User Interface), i.e. to avoid the development of an ad hoc

program GUI. As for an image analysis program this is particularly convenient since the necessary

tools for data I/O and for image visualization are already available within GRASS. The GRASS

GIS provides two different display drivers, i.e. two different way of handling visualization, and

permits the I/O of many different image formats. The r.seg module code, and the seglib code

as well, fulfills the GRASS ”SUBMITTING” requirements, delivered with the GRASS source code

and available on the GRASS website4.

The r.seg program structure is as follow:

headers inclusion

main()


GRASS environment initialisation [G_gisinit() ]

GRASS module initialisation [G_define_module()]

GRASS module parameters, options and flags definition and description

GRASS module parameters, options and flags parser [G_parser() ]

input requirements checks

memory allocation, variables initialisation and fill up

call to the seglig functions and convergence loop

output writing

freeing allocated memory

where G_gisinit(), G_define_module() and G_parser() are functions of the GRASS library

set.

An example of a GRASS module parameter, named in_g, definition and description could be:

parm.in_g = G_define_option() ;

parm.in_g->key = "in_g";

parm.in_g->type = TYPE_STRING;

parm.in_g->required = YES;

parm.in_g->description = "a description of the parameter meaning";

Where parm.in_g->key stores the name of the parameter for the command line use of the GRASS

module, parm.in_g->type define the parameter data type, parm.in_g->required set the param-

eter to be required or optional for the program execution.

At the time of writing, aside of the powerful command line user interface, GRASS provides

a native default tcl/tk GUI. In figure (4.1) the tcl/tk GUI of the r.seg module is shown.

The GRASS development team is hardly working on a new GUI based on wxpython 5. Figure

4http://grass.itc.it5http://www.wxpython.org

54


(4.2) shows the wxpython GUI of the r.seg module. It is worth to note that the GUI of all of the

GRASS module are generated automatically from few lines included in the module source code and

that these lines are merely the definitions and the textual descriptions of the module parameters

and options, i.e. any graphical oriented code need to be written.

Figure 4.1: r.seg native tcl/tk GRASS GUI.

55


Figure 4.2: r.seg new wx-python GRASS GUI.

4.2 The seglib1d Library for One Dimensional Signals

Segmentation

The library seglib1d implements the Blake and Zisserman model in dimension one. The library

functions solve the gradient descent equations associated to the finite elements discretisation of

the Ambrosio and Tortorelli approximation of the Blake and Zisserman functional. The library

implements the non-linear Gauss-Seidel iterative method.

The general structure of the seglib1d library is:

headers inclusion


setup function implementation

56


system minimisation function implementation

single segmentation functions implementation

All the data structures used in the library and the library functions are declared and defined

in the file seglib1d.h. In the following definitions and tasks performed by each of the library

functions are detailed. As for the previous section, the used notation is simple and essential. The

names of the variables is very close to the name of the corresponding quantities appearing in the

mathematical formulas. The data type of variables is specified only when non standard data types

have been used.

The setup function is:

setup(sfi, n, pos)

The function initialises the variables representing the integrals of the finite elements base

functions products integrals reported in (3.22) and initialises a variable which serves to man-

age the relative position of the indexes of the base functions products integrals.

The function has to be explicitly called in the main program before the call to the system

minimisation function. The memory for the variables sfi and pos has to be allocated in

the main program.

The system minimisation function is:

minimize(g, np, prm, apx, sfi, pos, mxdf)

The function allocates the memory for and initialises the variable TC and then calls the

single segmentation functions, which are detailed later on.

The function has to be explicitly called in the main program after the call to the setup

function.

The single segmentation functions are:

minimize u1(g, np, prm, apx, sfi, pos, TC, mxdf)

The function implements the first equation of the system (3.25), and the Gauss-Seidel method.

minimize u2(g, np, prm, apx, sfi, pos, TC, mxdf)

The function implements the second equation of the system (3.25), and the Gauss-Seidel

method.

minimize sgm(g, np, prm, apx, sfi, pos, TC, mxdf)

The function implements the third equation of the system (3.25), and the Gauss-Seidel

method.

minimize s(g, np, prm, apx, sfi, pos, TC, mxdf)

The function implements the fourth equation of the system (3.25), and the Gauss-Seidel

method.

The variable sfi is a pointer to a shp_fun_int structure. This structure is designed to store the

values of the base functions products integrals. The variable pos is a pointer to a pos_idx_jhk

57


structure. This structure basically stores position indexes, its use is described later on. The

variable n is the number of finite elements the domain is discretised by and np is the number of

points of the input signal. The variable prm is a func_params structure. The structure stores the

Blake and Zisserman parameters, i.e. α, β, ǫ, λ, δǫ and µǫ. The variable apx is a pointer to a

appx_u_s_sigma structure. The structure stores the solutions, i.e. u1, u2, σ and s. The variable

TC is a combs structure. The structure stores indexes which use is detailed later on. The variable

mxdf stores the maximum difference, over the entire domain, between the values of the solution u1

computed at two successive iteration steps.

Each single segmentation function solves the correspondent equation of the gradient descent

system (3.25) associated to the minimisation of the finite elements discretisation of the Ambrosio

and Tortorelli approximation of the Blake and Zisserman functional.

The general structure of each of the single segmentation functions is:


solution computation loop for the first domain point

solution computation loop for the inner domain points

solution computation loop for the last domain point

Before describing the structure of the solution computation loops some considerations on

the structure of the base functions products integrals and of the summations terms appearing in

system (3.25) have to be made to give reason of some implementation choices made to optimise

the memory requirements and the numerical iterations.

Observing the structure of the base functions products integrals it is possible to note that the

number of possible combinations of the indexes is of the order of n4 for the four indexes integrals

and of n2 for the two indexes integrals. This would require the definition, in the code, of vector

variables of n4 and n2 entries. Actually, the base functions have, by design, a very small support.

This means that the base functions products integrals admit not null values only on very few

elements of the domain. For example, let a(i, j) be one of the two indexes integrals in (3.22), then:

a(i, j) = 0 if |i − j| ≥ 2

and therefore a(i, j) is potentially not null only when:

a(i, i), a(i, i + 1) i = 0;

a(i, i − 1), a(i, i), a(i, i + 1) i = 1, . . . , n − 1;

a(i, i − 1), a(i, i) i = n;

It is now possible to note that the combinations of the indexes i, j that do not produce null integrals

are actually only 3, that is when j = i, j = i − 1 and j = i + 1. In figure (4.3) the geometry of

the not null configurations is sketched. Since in (3.26) to (3.33) there are summations, involving

two indexes base functions products integrals, running either on the index i and on the index j,

58


in figure (4.3) the not null configurations are reported for both the indexes. It has to be note

however how the geometry of the configuration are actually the same; the colors should help to

see the equivalences. In figure (4.3) the configurations have been identified by the c index. In

the following we will refer to this index as the configuration index. When the values of the base

functions products integrals are computed, equivalent configurations yield equal values. This means

that, since only 3 not null configurations exist, the base functions products integrals can assume

only 3 different values. Therefore, to store the values of the integrals it is sufficient to define vector

variables of only 3 entries.

Figure 4.3: Sketch of the indexes combinations that give no null two indexes base functions productsintegrals.

It is important to note here that the points i = 0 and i = n have to be treated in a slightly

different manner. In fact when the base functions products integrals are computed for these points,

and in particular when i = j = 0 and when i = j = n, the products exist only on a single finite

element of the domain instead of two, as it is when i = j and i = 1, . . . , n − 1. In these two cases,

even if from the geometrical viewpoint the configuration is i = j, i.e. c = 2, the values of the

integrals differ from the values of the integrals on the same configuration computed for the inner

points i = 1, . . . , n−1. This impose to store two more entries in the vector variables that store the

values of the two indexes base functions products integrals on the external nodes, i = 0 and i = n.

Similar considerations can be done for the four indexes base functions products integrals, leading

to a set of only 15+2 = 17 configurations, i.e. the variables representing the four indexes base func-

tions products integrals are arrays of 17 entries. Figure (4.4) sketched the 15 main configurations.

The missing two configurations are indeed equivalent to the configuration 8 in figure (4.4). The

need of using two more configurations permits to correctly handle the two external nodes where

the values of the shape functions products integrals differ from the values of the same integrals for

any of the inner nodes.

In figure (4.4) the configurations are defined on the first of the four sets reported. The first set

of configurations correspond to the case of a fixed i index, while the others correspond respectively

to the case of a fixed j, h and k index. For these last 3 indexes, the name of the configurations, i.e.

the value of the configuration index c, is derived from the definition given on the first set. We recall

that the need of keeping one index fixed arises from the structure of the summations appearing in

(3.26) to (3.33).

59


Figure 4.4: Sketch of the indexes combinations that give no null four indexes base functions prod-ucts integrals.

60


It is now possible to give the general structure of the solution computation loop that, for the

inner points is:

for p = 1:np-2

S_2 = Sd_2 = S_4 = Sd_4 = 0

for c = 1:3

i = p

j = p+(c-2)

S_2 = S_2 + xi * sfi_2(c) + xj * sfi_2(c)

end

c=2

Sd_2 = Sd_2 + sfi_2(c)

for c = 1:15

i = pos.i[c]

j = pos.j[c]

h = pos.h[c]

k = pos.k[c]

S_4 = S_4 + xi * xj * xh * sfi_4(c) + xi * xj * xk * sfi_4(c)

end

for q = 5:11

i = pos.i[c]

j = pos.j[c]

c = TC.h[q]

Sd_4 = Sd_4 + xi * xj *sfi_4(c)

end

Xt = Xt-1 - ( S_2 + S_4)/(Sd_2 + Sd_4)

diffp = Xt - Xt-1

end

mxdf = max(diffp)

Where, np is the number of points of the given signal and p the position index. The quantity x

represents one of the functions u1, u2, σ and s in the finite elements discretisation. The quantities

S_2 and Sd_2 represent the terms involving two indexes base functions products integrals in the

equations (3.26) to (3.33). The variable sfi_2() generically represent the two indexes shape

functions products integrals. The quantities S_4 and Sd_4 represent the terms involving four

indexes base functions products integrals in the equations (3.26) to (3.33). The variable sfi_4()

generically represent the two indexes shape functions products integrals.

The reported structure represents the iterative procedure that allows to numerically solve the

system (3.25). We recall here that each equation in the system (3.25) can be considered as a linear

equation if only one unknown is solved for, on the basis of the same considerations given for the

discrete gradient decent system associated to the Mumford and Shah involving the equations (2.35)

and (2.39).

61


Each iteration step cycles over all the points of the given signal and, for each point, the compu-

tation of the summations is made by cycling over the configurations c that can be defined for the

running point.

The expression for S_2, within the first for loop in c, represent the cycle that allow to compute

the summations involving two indexes base functions products integrals. There is no need of a for

loop to compute the value of the quantity Sd_2 since the associated terms in equations (3.30) to

(3.33) depend only on base functions products integrals with equal indexes. In the same manner,

the expression for S_4, within the second for loop in c, and Sd_4, within the for loop in q,

represent the cycles that allow to compute the summations involving four indexes base functions

products integrals. Here it is important to note that the last for loop is no longer controlled by

the configuration index c, but by the counter q. This change is necessary since the computation

of S_4d involves summations with the index h and the index k fixed and equals, see (3.30) to

(3.33). In this case it is no longer possible to use c as a for loop counter because it is no longer

sequential. See the value of c in the configuration set for h and k in figure (4.4), and in particular

the combinations underlined by a green line. The for loop variable is therefore controlled by the

counter q and the variable TC.h gives the correct configuration index c being the relation between

q and c sketched in figure (4.4).

For each running configuration the variable pos returns the values to add to the index p to obtain

the values of the indexes i, j, h and k. For example, the configuration c=8, see figure (4.4), has

i=j=h=k and therefor the variable p returns 0 for all of the indexes so that, for the current point

p, we have: i=j=h=k=p. The values of the indexes i, j, h and k are then used to access correct

position on the vector variable x.

Once the expression for S_2, Sd_2, S_4, Sd_4 have been computed it is therefor possible to

compute the new value of the function x. The difference between the new value and the previous

one is eventually computed and the maximum value on the entire domain is then stored in the

variable mxdf, that can be used in the main program to implement a convergence criterion.

The structure of the solution computation for the first and for the last domain points are very

similar to the one presented for the inner points. The main difference regard the values of the for

loops indexes, since the first and the last point do not present symmetric configurations.

In the numerical implementation, the base functions products integrals have been named, within

62


the library source code, according to the following notation:

∫ 1

0ψiψj(φ

1h)′′(φ1

k)′′dt = sfi->ppf11dd[c]∫ 1

0ψiψj(φ

1h)′′(φ2


0ψiψj(φ

2h)′′(φ2


0(φ1

i )′′(φ1

j )′′dt = sfi->f11dd[c]

∫ 1

0(φ1

i )′′(φ2


∫ 1

0(φ2

i )′′(φ2


∫ 1

0ψ′

iψ′jdt = sfi->ppd[c]

∫ 1

0ψiψjdt = sfi->pp[c]

∫ 1

0ψiψj(φ

1h)′(φ1

k)′dt = sfi->ppf11d[c]∫ 1

0ψiψj(φ

1h)′(φ2


0ψiψj(φ

2h)′(φ2


0φ1

i φ1jdt = sfi->f11[c]

∫ 1

0φ1


∫ 1

0φ2


∫ 1

0φ1

i ψjdt = sfi->pf1[c]∫ 1

0φ2

i ψjdt = sfi->pf2[c]

The values of the base functions products integrals have been computed by hand, the values are

the same reported in (Piazza, aa 1998–1999), where the integrals had been computed using the

Mathematica software.

4.2.1 The sigseg Program

The sigseg program accesses the siglib1d functions and provides a command line program to

perform segmentation of one dimensional signals. Running the program is however quite simple,

in fact the program requires only the name of a parameters file, the name of the file containing

the input signal and the name of the output file to be input on the command line. The input and

the output files are in text format. The input parameters file contains the Blake and Zisserman

parameters, α, β, ǫ, λ, δǫ, µǫ and two parameters used to perform a solution convergence check.

The two parameters are tol and mxit, the first parameter set the convergence threshold and the

second is the maximum number of iterations. At least one of the two parameters has to be set

greater than zero. If both of them are set, the program will terminate as soon as one of the

two convergence conditions is satisfied. If one parameter is leaved to zero it is not considered in

the convergence check and the program terminates as soon as the convergence condition on the

other parameter is satisfied. The program, at the execution end, gives the number of iterations

performed and the maximum difference, on the entire domain, between the values of the solution

at two successive iteration steps.

The input file has to contain a single column, each row of the column contains a single signal

value. The output file contains four columns. Each column contains one value, and in particular:

the first column contains the solution u1, the second column the solution u2, the third column the

solution σ and the last column the solution s. The content of the output file can be easily plotted

using gnuplot, MatLab or Ocatave programs, the use of spreadsheets is of course possible. All

the tests reported in Chapter (6) have been carried out using the sigseg program and all the plots

have been made with gnuplot.

The general structure of the program sigseg is:

63


headers inclusion

preprocessor constants definition


main()


input syntax check

read input parameters file

count the number of lines of the input signal file

read input signal file

memory allocation

initialisation of variables u1, u2, sgm, s

setup() function call

minimize() function call and convergence loop

write output

To properly use the seglib1d library functions a general program, such as sigseg, must:

include the seglib1d.h file;

define the variable sfi to store the shape functions products integrals,

the name of the variable is free, the data type is shp_fun_int;

define the variable apx to store the solutions for u1, u2, σ and s,

the name of the variable is free, the data type is appx_u_s_sigma;

define the variable pos to store the relation between the position indexes and the combination

index,

the name of the variable is free, the data type is pos_idx_jhk;

define the variable prm to store the Blake and Zisserman parameters,

the name of the variable is free, the data type is func_params;

define the variable tol to store the convergence threshold,

the name of the variable is free, the data type is double;

define the variable mxit to store the maximum number of iterations,

the name of the variable is free, the data type is double;

allocate the memory for the variables g, u1, u2, sgm and s;

call the seglib1d setup() function;

call the seglib1d minimize() function within a convergence loop;

64


An example of a convergence loop could be: iter = 0; mxdf = 0;

minimize(g, np, prm, apx, sfi, pos, &mxdf);

while((mxdf > tol) && (iter <= max_iter))

mxdf = 0;

minimize(g, np, prm, apx, sfi, pos, mxdf);

iter = iter + 1;

where iter is the iteration counter and max_iter stores the maximum number of iterations. At

the fourth line, a new call to the function minimize() is performed until the value of mxdf becomes

smaller of the convergence threshold tol or the maximum number of iterations is exceeded.

An example of the input parameters file is: #alpha

<value>

#

beta

#<value>

#

#epsilon

<value>

#

#lambda

<value>

#

#lambda_epsilon

<value>

#

#mu_epsilon <value>

#

#tolerance

<value>

#

#max_nof_iter <value>

Lines beginning with # are mandatory and considered as comments, i.e. ignored. The order of the

parameter is mandatory too, while the content of the comment lines is optional.

65

5 Segmentation of Images

This Chapter contains a series of images segmentation results obtained using the developed code,

all of the images have been analysed within the GRASS GIS using the r.seg module.

The series of images includes a set of synthetic images designed specifically to give practical

evidence of the features of the Mumford and Shah functional, and of its extension, that can

be theoretically foreseen and illustrated in the Chapter (2). The effects on the solution of the

weights value variability and of the weights relative ratios along with the practical evidence of the

rounding corners effect have been studied. The capability of the Mumford and Shah model with

the curvature term to reduce the rounding corners effect has also been investigated and proved

practically. The practical occurrence of the constrains on the nature of the end points of the

discontinuity boundaries previews in the Mumford and Shah conjecture has also been inspected.

Some of these effects have also been stressed on real pictures. Moreover, on real images, the

effect of respecting the condition that the mesh size must resolve the transition interval ǫ of the

function υ has also been treated along with the effect of not respecting such a condition.

Finally, the Chapter presents a set of segmentations of real ”environmental” images. The segmen-

tation benefits have been emphasised comparing the results of an unsupervised image classification

technique applied both to original images ant to the segmented version of the same images.

The r.seg module requires as input data, aside the input image g and the values of the param-

eters, an initial data of the solution u and an initial data for the function υ. In all of the tests the

input image g has been corrupted by adding a synthetic noise and this noisy image has been used

as initial data of the solution, whereas an image with a constant value of 1 has been used as initial

data for the function υ. This choice is particularly convenient from the numerical point of view, in

fact the solution, even if piecewise smoothed, has to remain as close as possible to the input data,

and the function z has to present transitions to 0 only in correspondence of the discontinuities of

the solution u, i.e. υ has to remain close to 1 almost everywhere on the image domain.

Before presenting the results of the segmentation tests we note that the Mumford and Shah

model can be seen as a multi-scale model. In fact, if the weight α is small the segmentation will be

”fine grained”, allowing the existence of long boundaries many small homogeneous regions will be

detected. When the weight α is large the segmentation will be a ”coarse segmentation”, opposite

considerations now hold. Moreover, the ramp effect threshold is (α/4λ3)1/4 and the resistance to

noise α/λ2, (Blake and Zisserman, 1987).

Finally, we recall here the theoretical condition, (Bellettini and Coscia, 1994b; March and Dozio,

1997), that holds between the discretisation step h and ǫ, that is: h/ǫ < 1.

From a direct experience of the author on numerical tests, confirmed by a direct communication

5. Segmentation of Images

of prof. Rccardo March1, in practice the condition on the discretisation step can be violated,

indeed numerical tests shown that it is even convenient to use a smaller value of ǫ. Violating the

condition h/ǫ < 1 leads to better results, the function υ presents in fact sharper transitions, i.e.

a finer detection of the discontinuities, that induce a better detection of homogeneous regions.

Some numerical results are reported in the following to give evidence of such a fact. A further,

less heuristic, step in the direction of finding solutions with sharp edges, as it can be obtained

by violating the condition h/ǫ < 1, has been formulated by Aubert et al. (2006). The authors

introduced a function ϕ that preserves the edge sharpness by controlling the gradient of υ.

5.1 Segmentation of Synthetic Images

5.1.1 Test Image A

Figure 5.1: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ see table (5.1).

1Istituto per le Applicazioni del Calcolo ”Mauro Picone” - CNR

68


Figure (5.1) presents on top left the original image, on top right the noisy image, the one actually

segmented, on bottom left the solution u and on the bottom left the function υ; the segmentation

has been performed with the parameters reported in table (5.1). In the following the same noisy

image have been segmented with different parameters values to observe the effect of their variability

on the solution and to confirm therefore the correct behaviour of the developed segmentation code.

Table 5.1: Parameters of the segmentation of figure (5.1).

Test aim To evaluate the effect of the variability of the weights

Image size 600 x 600 pixel

Image range [1, 4]

Added noise range [-0.2, 0.2] a

Image colours yellow: 1; green: 2; red: 4

Discontinuity colours white: 1: black: 0

α 0.5

λ 20

kǫ 5 b

n. of iterations 15

a Using the rand() function of the module r.mapcalc.b To have sharper transition zones.

In figure (5.2) the rounding corner effect can be easily observed both in the solution u and in the

function υ, in particular on the squares corners between the green and the yellow regions which

present a smaller difference in the values of the input data with respect to the difference between

the red and the yellow regions whose boundaries are geometrically consistent.

In figure (5.3) the effect of imposing a large weight on the term controlling the length of the

discontinuities is reported. Observing (5.3) it can be understand how the adopted weights limit

the existence of the smallest discontinuity of u, the difference between the value of u on the green

areas and the value of u on the yellow area is in fact smaller than the differences between the values

on the yellow and red areas, see the values reported in table (5.3). It is more convenient, from the

minimisation point of view, to consider the yellow and the green areas as one respect to introduce

a discontinuity between the two regions.

69


Figure 5.2: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Forcing short discontinuity see table (5.2).

70



Test aim To evaluate of the effect of the variability of the

weights

Image size 600 x 600 pixels

Image range [1, 4]

Added noise range [-0.2, +0.2] a



α 50

λ 600

kǫ 2 b

n. of iterations 150c

a Using the rand() function of the module r.mapcalc.b To have sharper transition zones.c Stopped before convergence, to have an intermediate solution.


Test aim To evaluate of the effect of the variability of the

weights


Image range [1, 4]

Added noise range [-0.2, +0.2] a



α 100

λ 600

kǫ 4 b

n. of iterations 29


71


Figure 5.3: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Forcing very short discontinuity see table (5.3).

72


5.1.2 Test Image B

Figure 5.4: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Using the Mumford and Shah with the curvature term weight set to zero see table(5.4).

Figure (5.4) presents on top left the original image, on top right the noisy image, the one actually

segmented, on bottom left the solution u and on the bottom left the function υ; the segmentation

has been performed with the parameters detailed in table (5.4). In the following figures the weights

are sequentially increased to produce the rounding corner effect. The parameter β, weighting the

curvature term, is kept fixed and equal to 0 except in the last figure (5.7).

It is worth to recall here that the elliptic approximation of Mumford and Shah model with the

curvature term involve a function W (t) which differs from the one used in the elliptic approximation

of the original model. The main difference is that choosing W (t) = (1− t2)2 yield the existence of

only closed edges. Moreover, this choice permits the function υ to range between [−1, 1] instead

of [0, 1]. The capability of υ to avoid the existence of large values of ∇u is preserved, being still

possible that υ = 0. The effect of choosing W (t) = (1 − t2)2, i.e. to have υ ∈ [−1, 1] is observable

73



Test aim To prove the capability of the Mumford and Shah

model with the curvature term to reduce the round-

ing corners effect.


Image range [0, 10]

Added noise range [-2, +2] a

Image colours yellow: 0; red: 10

Discontinuity colours white: 1: black: -1

α 3

β 0

λ 0.3

kǫ 0.99

n. of iterations 10

a Using the rand() function of the module r.mapcalc.

clearly on the lower right part of figures (5.4), (5.7), (5.6) and (5.7), where the black areas have

υ = −1 and the white ones υ = 1, being the discontinuity of u concentrate along the zero level

curve of υ.

74


Figure 5.5: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Using the Mumford and Shah with the curvature term weight set to zero and forcingshort boundaries see table (5.5).

75





ing corners effect.


Image range [0, 10]

Added noise range [-2, 2] a



α 40

β 0

λ 0.3

kǫ 0.99

n. of iterations 200b

a Using the rand() function of the module r.mapcalc.b Stopped before convergence, to have an intermediate solution.




ing corners effect.


Image range [0, 10]




α 40

β 0

λ 0.3

kǫ 0.99


a Using the rand() function of the module r.mapcalc.b Stopped before convergence, to have an intermediate solution.

76


Figure 5.6: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Using the Mumford and Shah with the curvature term weight set to zero and forcingvery short boundaries see table (5.6).

77


Figure 5.7: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Using the Mumford and Shah with the curvature term weight greater than zero andforcing very short boundaries see table (5.7).

78





ing corners effect.


Image range [0, 10]




α 40

β 4

λ 0.3

kǫ 0.99

n. of iterations 29


79


5.1.3 Test Image C

Figure 5.8: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Forcing very short boundaries see table (5.8).

80



Test aim To give evidence of the Mumford and Shah conjec-

ture about triple junctions.


Image range [0, 10]


Image colours yellow: 0,light blue: 5 , red: 10


α 400

λ 200

kǫ 1.75 b

n. of iterations 200c

a Using the rand() function of the module r.mapcalc.b To have sharper transition zones.c Stopped before convergence. Convergence is not easily reachable due to the weights

values.

Figure 5.9: 3D visualization of the discontinuity function υ colored using the values of the solutionu of segmentation of figure (5.8) see (5.8).

81


5.1.4 Test Image D

Figure 5.10: Segmentation of a synthetic image. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Forcing very short boundaries see table (5.9).

Figure (5.10) presents on top left the original image, on top right the noisy image, the one

actually segmented, on bottom left the solution u and on the bottom left the function υ; the

segmentation has been performed with the parameters detailed in table (5.9).

It is important to note that the Mumford and Shah conjecture can not be easily reproduced

in practice. In fact it is necessary to impose particularly high values to the weights to observe

the theoretically foreseen behaviours of the discontinuity lines. Moreover it is not possible to

distribute on four contiguous regions, such as the four central squares in figure (5.10), equivalent

differences in the value of the input signal so that to obtain, along the region boundaries, equivalent

jumps. These two conditions imply that the discontinuity between the two contiguous regions

presenting the smallest difference of the input data can not be preserved by the segmentation

process. Practically, the same conditions yield the impossibility to observe practically the appearing

of the third discontinuity line connecting the two discontinuity lines visible at the top right of figure

82



Test aim To give evidence of the Mumford and Shah conjec-

ture about triple junctions.


Image range [1, 3]

Added noise range [-0.2, 0.2] a

Image colours yellow: 1,light blue: 2 , red: 3


α 100

λ 300

kǫ 2 b

n. of iterations 100 c

a Using the rand() function of the module r.mapcalc.b To have sharper transition zones.c Stopped before convergence, to have an intermediate solution.

(5.11) and dividing the two contiguous regions presenting the smallest difference of the input data.

At the bottom of the same figure, this line has been manually added in red.

In figure (5.12) a 3 dimensional view of the solution u of the segmentation processes, obtained

with the parameters detailed in table (5.9), is reported. It is easily to note how the large values

imposed to the weights have permitted only the biggest discontinuities, between yellow and red

regions, to be preserved, while the smaller discontinuities, between yellow and light blue and

between red and light blue, have been smoothed out.

83


Figure 5.11: Zoom of the segmentation of the synthetic image of figure (5.10). Upper left: theoriginal image; upper right: the noisy image; lower left: the segmented image u; lowerleft: the discontinuity function υ. Forcing very short boundaries see table (5.9).

Figure 5.12: 3D visualization of the discontinuity function υ colored using the values of the solutionu of segmentation of figure (5.10) see (5.9).

84


5.2 Segmentation of Real Images

5.2.1 Lenna

Figure 5.13: Segmentation of a real image of Lenna. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Respecting the condition h/ǫ ≪ 1 see table (5.10).




85



Test aim To prove the effect of varying the value of the pa-

rameter ǫ.


Image range [0, 255]


Image colours grey levels


α 5

λ 1

kǫ 0.8 b

n. of iterations 30

a Using the rand() function of the module r.mapcalc.b To have smooth transition zones.



rameter ǫ.






α 200

λ 1

kǫ 0.99

n. of iterations 62


86


Figure 5.14: Segmentation of a real image of Lenna. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Respecting the condition h/ǫ < 1 see table (5.11).

87


Figure 5.15: Segmentation of a real image of Lenna. Upper left: the original image; upper right: thenoisy image; lower left: the segmented image u; lower left: the discontinuity functionυ. Violating the condition h/ǫ < 1 see table (5.12).

88




rameter ǫ.






α 200

λ 1

kǫ 4 b

n. of iterations 22


89


5.2.2 Aeroplane

Figure 5.16: Segmentation of a real image. Upper left: the original image; upper right: the noisyimage; lower left: the segmented image u; lower left: the discontinuity function υ.Respecting the condition h/ǫ < 1 see table (5.13).




90




rameter ǫ.






α 200

λ 1

kǫ 0.99

n. of iterations 51




rameter ǫ.






α 200

λ 1

kǫ 10 b

n. of iterations 45


91


Figure 5.17: Segmentation of a real image. Upper left: the original image; upper right: the noisyimage; lower left: the segmented image u; lower left: the discontinuity function υ.Violating the condition h/ǫ < 1 see table (5.14).

92


5.3 Segmentation of Real ”Environmental” Images

5.3.1 Pebbles

Figure 5.18: Segmentation of a real image. Left: the original image; right: the segmented image usee table (5.15).

Figure (5.18) presents on the left the noisy image, the one actually segmented, and on the right

the solution u; the segmentation has been performed with the parameters detailed in table (5.15).


Test aim To approximate a piecewise constant solution u.






α 300

λ 2

kǫ 10

n. of iterations 307


Figure (5.19) presents on the left the noisy image, the one actually segmented, and on the

right the solution u; the segmentation has been performed with the parameters detailed in table

(5.16). Here the weights have been set up to approximate a piecewise constant solution. Figure

93


Figure 5.19: Segmentation of a real image. Left: the original image; right: the segmented imageu. Forcing a piecewise constant segmentation see table (5.16).

(5.20) presents on the left the noisy image while on the right the solution shown in figure (5.19) is

reported once its background has been removed and a thin version of the discontinuity curves has

been over-layered. The thinned version has been obtained by means of the GRASS GIS module

r.thin.

Figure 5.20: Segmentation of a real image. Left: the original image; right: the almost piecewiseconstant segmented image u with edges over-layed . Forcing a piecewise constantsegmentation see figure (5.19) and table (5.16).

94



Test aim To approximate a piecewise constant solution u.






α 1750

λ 30

kǫ 10


a Using the rand() function of the module r.mapcalc.b Stopped before convergence. Convergence is not easily reachable due to the

weights values.

95


5.3.2 Braided River-bed

Figure 5.21: Segmentation and classification of a real image. Upper left: the original image; upperright: the segmented image u; lower left: classification in two classes of the originalimage; lower left: classification in two classes of the segmented image u see table(5.17).

Figure (5.21) presents on top left original image, the one segmented, on top right the solution u of

the segmentation performed with the parameters detailed in table (5.17). On bottom left the result

of the application on the original image of an unsupervised image classification algorithm where

only two classes have been asked to be identified and on bottom right the results of the application

on the solution of the segmentation of the same unsupervised image classification algorithm. The

two classes requested have to roughly distinguish between wet and dry zones of a braided bed-river.

The unsupervised classification has been performed using the GRASS GIS module r.maxlik which

implements a maximum likelihood discriminant analysis classifier.

In figure (5.22) the thinned version of the discontinuities, obtained by means of the GRASS GIS

r.thin module, has been over-layered on the solution u of the segmentation process.

96



Test aim To prove the advantages of image segmentation

when image classification has to be performed.






α 150

λ 10

kǫ 0.99


a Using the rand() function of the module r.mapcalc.b Stopped before convergence. Convergence is not easily reachable due to the very low

discontinuities on the input image.

Figure 5.22: Segmentation of a real image.The segmented image u with edges over-layed see table(5.17).

97


5.3.3 High Resolution Ortho-Photo

Figure 5.23: Segmentation of a real image. Upper: the noisy image; lower: the segmented image usee table (5.18).

Figure (5.23) presents on the top the noisy image, the one actually segmented and on the bottom

the solution u of the segmentation performed with the parameters detailed in table (5.18). Figure

(5.24) shows on the top the selection of two portions, of different extension, of the noisy image and

on the bottom the same two portions of the segmented image.

Figure (5.25) reports on the top the solution u of the segmentation process and on the bottom

the discontinuity function υ. Figure (5.26) reports, for the two portions displayed in figure (5.24),

the solution u of the segmentation process on the top and on the bottom the discontinuity function

υ.

98



Test aim To prove the advantages of image segmentation

when image classification has to be performed.






α 500

λ 1

kǫ 10

n. of iterations 75


On figure (5.27) the results of the application of an unsupervised classification algorithm on the

original image and on the solution u of the segmentation procedure. The classification has been

performed with the same setup and requiring the classification of the same number of classes on

both the images. The classification has been carried out by means of the use of the GRASS GIS

i.maxlik module and six classes have been asked to be distinguished. Figure (5.28) reports the

classification results on the same portions displayed in figure (5.24) and (5.26).

99


Figure 5.24: Zoom of the segmentation of the real image of figure (5.23). Upper: two regions ofthe noisy image; lower: two regions of the segmented image u see table (5.18).

100


Figure 5.25: Segmentation of a real image. Upper: the segmented image u; lower: the discontinuityfunction υ see table (5.18).

101


Figure 5.26: Zoom of the segmentation of the real image of figure (5.25). Upper: two regions of thesolution image u; lower: two regions of the discontinuity function υ see table (5.18).

102


Figure 5.27: Segmentation and classification of a real image. Upper: classification of the originalnoise-free image; lower: classification of the segmented image u see table (5.18).

103


Figure 5.28: Zoom of the segmentation and classification of a real image see figure (5.27). Upper:two regions of the classification of the original noise-free image; lower: two regions ofthe classification of the segmented image u see table (5.18).

104

6 Segmentation of One Dimensional

Signals

This Chapter contains a series of one dimensional signals segmentation results obtained using the

developed code, all of the signals have been analysed within the sigseg program.

The Chapter includes a set of synthetic signals designed specifically to permit the practical

confirmation of the features of the Blake and Zisserman functional introduced in the Chapter (3).

The occurrence of the Gradient effect and, Blake and Zisserman functional’s capability to overcome

such a drawback and to segment noisy signals have been studied.

The Chapter presents also a set of segmentations of real signals arising in Geodesy. In particular

coordinates time series of GNSS (Global Navigation Satellite System) permanent stations have

been treated. The effects on the solution of the weights value variability and of the weights relative

ratios have been studied on such signals. The capability of the variational segmentation model to

detect jumps within a signal and to simultaneously smooth the signal has been investigated too.

This capability is of particular interest in the analysis of GNSS stations’ coordinates time series.

In fact, GNSS stations’ coordinates time series are used to derive sites reference coordinates. Sites

reference coordinates are then used both to properly materialise ”dynamic”, i.e. modern, reference

systems and to estimate the parameters of the transformations, commonly a seven-parameters

transformation, necessary to convert coordinates between different reference frames. In both the

cases, being the reference coordinates usually obtained by means of a linear fitting, the presence

of signal discontinuities is particularly unwanted and harmful.

We recall here the expression of the approximation AT ǫ(u, s, σ) of the Balke and Zisserman as

in equation (3.10).

AT ǫ(u, s, σ) =

∫

(a,b)

(u − g)2dx + λ

∫

(a,b)

(s2 + δǫ) |u”|2dx+

+ µǫ

∫

(a,b)

σ2 |u′|2dx + βMǫ(s) + (α − β)Mǫ(σ),

where

Mǫ(t) =

∫

(a,b)

[

ǫ |∇t|2

+(1 − t)2

4ǫ

]

dx.

We recall also the relation between the parameters α and β that has to be respected to guarantee

solution existence:

0 < β ≤ α ≤ 2β.

6. Segmentation of One Dimensional Signals

It is relevant to note as setting λ = 0, δǫ, β = 0 and considering, with a little abuse of notation, µǫ

as a normal weight, i.e. not respecting the condition (3.6), the functional AT ǫ(u, s, σ) reduces to

the functional AT ǫ(u, υ) as in equation (2.3), i.e. the Ambrosio and Tortorelli approximation of

the Mumford and Shah functional. This permits to analyse one dimensional signals by means of

the functional of Blake and Zisserman or of Mumford and Shah depending on the analysis needs.

6.1 Segmentation of Synthetic One Dimensional Signals

6.1.1 Occurrence and overcoming of the Gradient Effect

The input signal g considered here, presents one single jump and one single ”crease” point.

-4

-3

-2

-1

0

1

2

0 20 40 60 80 100

"g.dat""u.dat"

"sgm.dat"

Figure 6.1: Occurrence of the Gradient effect with one single step see table (6.1).

Given the choice made on the values of the parameters reported in table (6.1), the segmentation

model adcaptioopted is actually the Mumford and Shah one. The Gradient effect is clearly visible

in figure (6.1), the solution u does present an evident step which is not actually present on the

input signal g. In figure (6.1) it is also plotted the function σ which is almost everywhere close to

1 and present two transitions to 0 where the solution u presents the two jumps.

106


Table 6.1: Parameters of the segmentationof figure (6.1).

Test Aim occurrence of the Gradien effect

N. of points 100

α 0.02

β 0

λ 0

δǫ 0

µǫ 0.02

ǫ 0.01

By reducing the value of the weight α, i.e. allowing more discontinuities of u to exist, and by

increasing the value of the weight µǫ, i.e. enforcing the condition on the first derivative of u, it is

possible to observe the appearance of more than one step, as shown in figure (6.2). In table (6.2)

are reported the values of the weights used to obtain the segmentation of figure (6.2).

In figure (6.2) it is also plotted the first derivative u′ of the solution u. The values of u′ have

been normalised to permit the overlaying and to observe the relation between the its graph and

the function σ. As expected where the first derivative of u increases the function σ goes to 0 to

annul the impact of the first derivative integral on the overall value of the functional. Moreover,

having increased the value of the weight µǫ the slope of the steps in figure (6.2) is smaller with

respect to the slope of the step in figure (6.1).

The segmentation of the same signal g obtained by applying the Blake and Zisserman model is

plotted in figure (6.3). The values of the weights are reported in table (6.3).

As expected, in figure (6.3) the Gradient effect does not appear anymore. The function σ is

almost everywhere close to 1 and presents just one transition to 0 where the input data g, and the

solution u present the jump. The functions s present instead two transitions to 0 in correspondence

of the discontinuities of the first derivative of the solution, one at the signal jump position and one

at the beginning of the signal ramp.

107


-4

-3

-2

-1

0

1

2

0 20 40 60 80 100

"g.dat""u.dat"

"sgm.dat""u’.dat"

Figure 6.2: Occurrence of the Gradient effect with several steps see table (6.2).

Table 6.2: Parameters of segmentation offigure (6.2).

Test Aim occurrence of the Gradien effect

N. of points 100

α 0.001

β 0

λ 0

δǫ 0

µǫ 0.01

ǫ 0.01

108


-4

-3

-2

-1

0

1

2

3

0 20 40 60 80 100

"u.dat""g.dat""s.dat"

"sgm.dat"

Figure 6.3: Overcome of the Gradient effect occurred in figure (6.1) and (6.2) by a Blake andZisserman segmentation see table (6.3).

Table 6.3: Parameters of segmentation offigure (6.3).

Test Aim overcoming of the Gradien effect

N. of points 100

α 0.00004

β 0.00002

λ 0.000001

δǫ 0.00001

µǫ 0.000001

ǫ 0.01

109


6.1.2 A Complex Signal and a Complex Noisy Signal

In this section the input signal g presents two jumps and three ”crease” points.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200


"sgm.dat"

Figure 6.4: Blake and Zisserman segmentation of a complex synthetic signal see table (6.4).

The segmentation obtained by applying the Blake and Zisserman model is plotted in figure (6.4);

in table (6.4) the adopted parameters are reported. The solution u follows the signal g correctly

depicting all of its features. In particular in the interval [40, 80] the input signal g has been obtained

discretising the parabola of equation y = 200x2 −160x+28 that would not be correctly segmented

by the Mumford and Shah model since the relevant weight induced by the values of its derivatives.

In figure (6.4) it is easily observable the correct behaviour of the two functions σ and s. The former

is, in fact, almost everywhere close to 1 and presents transitions to 0 only in correspondence of

the input signal jumps. The value of the function σ is related to the value of the first derivative

of the solution u. In fact σ is equal to 1 wherever the solution u is constant and presents a value

lesser that 1 in correspondence of the input signal ramp and parabola. The function s presents

presents five transitions to 0, two of them are in common with the transitions of σ, i.e. on the

jumps correspondence, whereas the other three appear in correspondence of the crease points of

the solution u, i.e. the points where the solution first derivative, u′, presents discontinuities.

Figure (6.5) plots the segmentation of the a noisy input signal g which structure is the input

110


Table 6.4: Parameters of segmentation of figure (6.4).

Test Aim segmentation of a signal with jumps and crease points

N. of points 200

α 0.0002

β 0.0001

λ 0.00000001

δǫ 0.0001

µǫ 0.000001

ǫ 0.005

signal g plotted in figure (6.4). In table (6.5) the weighs values used to perform the Blake and

Zisserman segmentation are given. It is relevant to note how the solution u is actually much

smoother than the noisy data g whereas all of the original features of the noise-free signal are still

correctly depicted. This second fact confirmed observing how also the behaviours and the main

features of the functions σ and s are in this case very close to the ones obtained by the segmentation

of the noise-free signal, reported in figure (6.4).

Table 6.5: Parameters of segmentation of figure (6.5).

Test Aim segmentation of a noisy signal with jumps and crease points

N. of points 200

Added noise range [-0.375, +0.375]a

α 0.005

β 0.0025

λ 0.000000035

δǫ 0

µǫ 0.000015

ǫ 0.005

a Using the rand() function of OpenOffice Calc

.

111


-5

-4

-3

-2

-1

0

1

2

3

4

5

0 50 100 150 200


"sgm.dat"

Figure 6.5: Blake and Zisserman segmentation of a complex and noisy synthetic signal see table(6.5).

112


6.2 Segmentation of GNSS Coordinates Time Series

The data analysed in this section are the coordinates time series of two IGS (International GNSS

Service) sites: the SYOG site and the SANT site. We refer to the IGS website1 for further details.

In all of the figures of this section the x axis represents a time coordinate, and each position on the

axis corresponds to a GPS week. The y axis represents a spatial coordinate, and in particular the

decimal part, in millimeters, of the GNSS site East coordinate. The UTM coordinates are obtained

converting the geocentric coordinates, (X,Y,Z), to the ellipsoidal coordinates, (lat, long, hell) on

the GRS80 ellipsoid and then the ellipsoidal lat, long coordinates to the plane coordinates N,E.

All of the above operations have been performed using the cs2cs program provided by the Proj4

project2. The geocentric X,Y,Z coordinates have been extracted from the sinex files downloaded

from the products directory of the IGS website.

The signals analyses reported in the following have been exclusively performed to test the de-

veloped software and are by no means aimed to the actual study of the coordinates time series.

In particular in the section 6.2.2 the effects on the segmentation solution of the variability of the

weights has been investigated on such signals.

6.2.1 Detection of a Singular Jump

In this section the results of different segmentations performed on the same coordinate time series

are presented. The differences arising from the different values of the weights adopted for the

segmentations. The values of the parameters are omitted for all of the tests presented.

All the tests of this section are performed on the East coordinate time series of the SANT site.

In figure (6.6) the result of a Mumford and Shah segmentation is reported along with the plot of

the function σ. The solution is clearly close to a piecewise constant segmentation and the function

σ presents as many transitions to 0 as many steps are present on the solution u. The appearing of

such a many steps is a clear occurrence of the Gradient effect.

In figure (6.7) the solution u has been obtained applying the Blake and Zisserman model. Com-

pared to the solution in figure figure (6.6) here the segmentation is no more piecewise constant.

Looking at the values of the functions σ and s it is possible to observe that the solution here

presents only one jumps, in correspondence of the GPS week 1400. Moreover, since the function

s is almost everywhere constant and close to 1 the solution is ”almost” linear before and after the

jump. This gives reason of the commonly adopted procedure to derive reference coordinates from

GNSS coordinates time series by means of a linear fitting model.

1http://www.igs.org2http://www.remotesensing.org/proj

113


396

398

400

402

404

406

408

410

1200 1250 1300 1350 1400

"u.dat""g.dat"

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1200 1250 1300 1350 1400

"sgm.dat"

Figure 6.6: Occurrence of the Gradient effect on a real signal: a GNSS site coordinate time series.

114


396

398

400

402

404

406

408

410

1200 1250 1300 1350 1400

"u.dat""g.dat"

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1250 1300 1350 1400

"sgm.dat"

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1200 1250 1300 1350 1400

"s.dat"

Figure 6.7: Blake and Zisserman segmentation of a real signal: a GNSS site coordinate time series.

115


6.2.2 Different Segmentations of the Same Time Series

In this section the results of different segmentations performed on the same coordinate time se-

ries are presented. The differences arise from the different values of the weights adopted for the

segmentations. The values of the parameters are omitted for all of the tests presented when not

explicitly necessaries to the comments or the understanding of the results.

All the tests of this section are performed on the East coordinate time series of the SYOG site.

In figure (6.8) are reported the results obtained using the Mumford and Shah model and the

Blake and Zisserman Model. The first instance of the application of the Blake and Zisserman model

has been performed imposing a weight λ much bigger than in the second instance, i.e. imposing

the first solution to be quasi piecewise linear and the second to be piecewise smooth, being the

Mumford and Shah solution piecewise constant.

535

540

545

550

555

560

1200 1250 1300 1350 1400

"MS_flt.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"BZ_lin.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"BZ_smt.dat""G.dat"

Figure 6.8: Comparison of a Mumford and Shah segmentation and of two different Blake andZisserman segmentation of a real signal: a GNSS site coordinate time series.

Figure (6.9) shows the effect on the segmentation of varying the value of the parameter δǫ; all

of the other coefficients are the same for all of the three segmentations. We recall here that this

parameter is an infinitesimal faster that the parameter ǫ and it has to respect the condition (3.6).

116


Intuitively this parameter permits the second derivative of the solution u to influence the overall

functional value even when the function s goes to 0, i.e. when the first derivative u′ present a

discontinuity. Being the value of δǫ smaller and smaller as long as the parameter ǫ goes to 0, the

influence of the integral of the second derivative is never to much relevant. Anyway, avoiding the

function s to completely annul the contribution of the second derivative, the parameter δǫ forces the

solution to be less ”linear”, and the bigger the value of δǫ the smoother the solution. In figure (6.9)

the first segmentation has been produced with a small value of δǫ, 10−6, the second segmentation

with the value 10−5 and the third segmentation with the bigger value, 10−4. The first solution

clearly presents a segmentation made of almost piecewise linear, the third result clearly smoother

and the second remains in the middle.

535

540

545

550

555

560

1200 1250 1300 1350 1400

"small l_e.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"med l_e.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"big l_e.dat""G.dat"

Figure 6.9: Different Blake and Zisserman segmentation of the same real signal: a GNSS sitecoordinate time series. Effect of varying the value of the parameter δǫ.

Figure (6.10) reports an analysis on the variation of the relative ratio of the weights β and λ.

These two weights control the number of the crease points and the value of the solution second

derivative u”. The three solutions plotted in figure (6.10) can be compared pair by pair in the

following way. The first solution has been obtained with β = 5 and λ = 10−5 and the second

117


535

540

545

550

555

560

1200 1250 1300 1350 1400

"many disc.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"mid disc.dat""G.dat"

535

540

545

550

555

560

1200 1250 1300 1350 1400

"few disc.dat""G.dat"

Figure 6.10: Different Blake and Zisserman segmentation of the same real signal: a GNSS sitecoordinate time series. Effect of varying the value of ratio of the parameters β andλ.

solution has been obtained with the same value of the weight β, i.e. 5, and with an smaller value

of λ, 10−7. The third solution has been obtained with the same value of λ of the second solution,

i.e. 10−7 and with β = 15, i.e. with a value greater than both the previous. Hence, the comparison

of the first and of the second solution allows to evaluate the effect of varying the weight controlling

the smoothness of the solution. It is clear as in the first solution the weight λ is too big with

respect to β and the segmentation approach a trivial solution, i.e. the input data g is almost fully

reproduced. Reducing the value of λ leads to the second solution. Here the ratio between β and

λ is more ”significant”, and the solution presents pieces where the control on the second derivative

is appreciable.

The comparison of the second and of the third solution allows to evaluate the effect of varying

the weight controlling the number of crease points. If the number of discontinuities of the first

derivative is forced to be as small as possible by increasing the value of β, the solution results

smoother since the control on the second derivative can act on longer signal intervals.

∗ ∗ ∗

118

References

During all of the time spent on this thesis, the author consulted and read several works which

contents allowed the understanding and the study of the more relevant aspects of the treated sub-

ject. Only a minor part of this material is cited in the text, whereas it appears of some relevance

and usefulness to provide a wider list of references, which, is in any cases, far from being exaustive.

G. Alberti, G. Bouchitte, and G. Dal Maso. The calibration method for the Mumford-Shah func-

tional. Comptes Rendus de l’Academie des Sciences Serie I-Mathematique, 329:249–254, 1999.

L. Ambrosio. A compactness theorem for a new class of functions of bounded variation. Bollettino

dell’Unione Matematica Italiana, 3-B:857–881, 1989a.

L. Ambrosio. Variational problems in SBV and Image Segmentation. Acta Applicandae Mathe-

maticae, 17:1–40, 1989b.

L. Ambrosio and N. Dancer. Calculus of Variations and Partial Differential Equations. Springer-

Verlag, New York, 1999.

L. Ambrosio and V. M. Tortorelli. Approximation of functionals depending on jumps by elliptic

functionals via gamma-convergence. Communications on Pure and Applied Mathematics, 43:

999–1036, 1990.

L. Ambrosio and V. M. Tortorelli. On the approximation of free discontinuity problems. Bollettino

dell’Unione Matatematica Italiana, 6-B:105–123, 1992.

L. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Discontinuity

Problems. Oxford mathematical monographs. Oxford University Press, Oxford, 2000.

L. Ambrosio, L. Faina, and R. March. Variational approximation of a second order free discontinuity

problem in computer vision. SIAM Journal on Mathematical Analysis, 32:1171–1197, 2001.

G. Aubert and P. Kornprobst. Mathematical Problems in Image Processing: Partial Differen-

tial Equations and the Calculus of Variations, volume 147 of Applied Mathematical Sciences.

Springer-Verlag, New York, 2002.

G. Aubert and P. Kornprobst. Mathematical Problems in Image Processing: Partial Differen-

tial Equations and the Calculus of Variations, volume 147 of Applied Mathematical Sciences.

Springer-Verlag, 2nd edition, 2006.

References

G. Aubert, L. Blanc-Feraud, and R. March. An approximation of the Mumford-Shah energy

by a family of discrete edge-preserving functionals. Nonlinear Analysis: Theory, Methods and

Applications, 64:1908–1930, 2006.

G. Bellettini and A. Coscia. Approximation of Functional Depending on Jumps and Corners.

Bollettino dell’Unione Matatematica Italiana, 8-B:151–181, 1994a.

G. Bellettini and A. Coscia. Discrete approximation of a free discontinuity problem. Numerical

Functional Analysis Optimization, 15(3-4):201–224, 1994b.

A. Blake and A. Zisserman. Visual Reconstruction. The MIT Press, Cambridge, 1987.

B. Bourdin. Image segmentation with a finite element method. Mathematical Modelling and

Numerical Analysis, 33(2):229–244, 1999.

B. Bourdin and A. Chambolle. Implementation of an adaptive finite-element approximation of the

Mumford-Shah functional. Numerische Mathematik, 85(4):609–646, 2000.

A. Braides. Gamma-convergence for beginners, volume 22 of Oxford lecture series in mathematics

and its applications. Oxford University Press, London, 2002a.

A. Braides. Approximation of free-discontinuity problems, volume 1694 of Lecture notes in mathe-

matics. Springer, Berlin, 1998.

A. Braides. Discrete approximation of functionals with jumps and creases. In Conference ”Homog-

enization and Multiple Scales”, Naples, 2001., 2002b.

A. Braides and V. Chiado Piat, editors. Topics on Concentration Phenomena and Problems with

Multiple Scales, volume 2 of Lecture Notes of the Unione Matematica Italiana. Springer, 2006.

A. Braides and G. Dal Maso. Non-local approximation of the Mumford-Shah functional. Calcolus

of Variations, 5:293–322, 1997.

A. Braides and A. Defranceschi. Homogenization of multiple integrals, volume 12 of Oxford lecture

series in mathematics and its applications. Oxford University Press, London, 1998.

A. Braides and R. March. Approximation by γ-convergence of a curvature-depending functional

in visual reconstruction. Communications on Pure and Applied Mathematics, 59:71–121, 2006a.

A. Braides and R. March. Approximation by Gamma-convergence of a curvature depending func-

tional in visual reconstruction. Communications on Pure and Applied Mathematics, 59:71–121,

2006b.

M. Carrierio, A. Farina, and I. Sgura. Image Segmentation in the Framework of Free Discontinuity

Problems. Quaderni di Matematica - Dip. di Matematica, Seconda UniversitA di Napoli Caserta,

16, 2004.

120

References

M. Carriero, A. Leaci, and F. Tomarelli. A second order model in image segmentation: Blake

& Zisserman functional, in Variational Methods for Discontinuous Structures, volume 25 of

Progress In Nonlinear Differential Equations And Their Applications. Birkhauser, Basel, 1996.

M. Carriero, A. Leaci, and F. Tomarelli. Strong minimizers of Blake & Zisserman functional.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (4), 25(1-2):257–285, 1997.

M. Carriero, A. Leaci, and F. Tomarelli. Calculus of Variations and image segmentation. Journal

of Physiology - Paris, 97(2):343–353, 2003.

M. Carriero, A. Leaci, and F. Tomarelli. Euler Equations for Blake & Zisserman Functional.

Unpublished, 2006.

A. Chambolle. Finite-differences discretizations of the Mumford-Shah functional. Mathematical

Modelling and Numerical Analysis, 33(2):261–288, 1999.

A. Chambolle. Image segmentation by variational methods: Mumford and Shah functional and

the discrete approximations. SIAM Journal on Applied Mathematics, 55(3):827–863, 1995.

A. Chambolle. Partial differential equations and image processing. In Image Processing, 1994.

Proceedings. ICIP-94., IEEE International Conference, 1994.

A. Chambolle and G. Dal Maso. Discrete approximation of the Mumford-Shah functional in

dimension two. Mathematical Modelling and Numerical Analysis, 33(4):651–672, 1999.

T. F. Chan and L. A. Vese. A level set algorithm for minimizing the Mumford-Shah functional in

image processing. In VLSM ’01: Proceedings of the IEEE Workshop on Variational and Level

Set Methods (VLSM’01), pages 161–168. IEEE Computer Society, 2001.

B. Dacorogna. Direct methods in the calculus of variations, volume 78 of Applied Mathematical

Sciences. Springer-Verlag, New York, 1989.

B. Dacorogna. Introduction to Calculus of Variations. Imperial College Press, 2004.

G. Dal Maso. An ntroduction to Gamma-convergence. Birkhauser, Boston, 1993.

E. De Giorgi. Gamma-convergenza e G-convergenza. Bollettino dell’Unione Matatematica Italiana,

A-14:213–220, 1977.

E. De Giorgi. Free discontinuity problems in calculus of variations. Frontiers in Pure and Applied

Mathemathics, a collection of papers dedicated to J.L. Lions on the occasion of his 60th birthday,

pages 55–62, 1991a.

E. De Giorgi. Some remarks on Gamma-convergence and least squares method. Composite Media

and Homogeneization Theory (ICTP, Trieste, 1990). Birkhauser, Boston, 1991b.

E. De Giorgi and L. Ambrosio. Un nuovo tipo di funzionale del calcolo delle variazioni. In Atti

Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), volume 82, pages 199–210, 1998.

121

References

E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. Rend. Sem. Mat. Brescia,

3:63–101, 1979.

E. De Giorgi and T. Franzoni. Su un tipo di convergenza variazionale. In Atti Accad. Naz. Lincei

Rend. Cl. Sci. Fis. Mat. Natur. (8), volume 58, pages 842–850, 1975.

E. De Giorgi, M. Carriero, and A. Leaci. Existence Theorem for a Minimum Problem with Free

Discontinuity Set. Archive for Rational Mechanics and Analysis, 108-4:195–218, 1989a.

E. De Giorgi, M. Carriero, and A. Leaci. Existence theorem for a minimum problem with free

discontinuity set. Arch. Rational Mech. Anal., 108:195–218, 1989b.

G. Del Piero and R. Lancioni, G.and March. A variational model for fracture mechanics: numerical

experiments. Unpublished, 2006.

D. Der Fakultt. On Variational Problems and Gradient Flows in Image Processing, 2005. URL

citeseer.ist.psu.edu/764153.html.

S. Esedoglu and R. March. Segmentation with depth but without detecting junctions. Journal of

Mathematical Imaging and Vision, 18:7–15, 2003.

L. Evans. Partial Differential Equations, volume 19 of Graduated Studies in Mathematics. American

Mathematical Society, 1998.

K. Falconer. Fractal Geometry: Mathematical Foundations and Applications. Jhon Wiley & Sons,

New York, 2nd edition, 2003.

M. Fornasier and R. March. Restoration of color images by vector valued BV functions and

variational calculus. Unpublished, 2006.

N. Fusco. An Overview of the Mumford-Shah Problem. Milan Journal of Mathematics, 71:95–119,

2003.

M. Giaquinta and S. Hildebrandt. Calculus of Variations I. Springer-Verlag, New York, 1996.

M. Gobbino. Finite difference approximation of the Mumford-Shah functional. Communications

on Pure and Applied Mathematics, 51(2):197–228, 1998a.

M. Gobbino. Gradient flow for the one-dimensional Mumford-Shah functional. Annali Scuola

Normale Superiore di Pisa, Classe di Scieze (4), 27(1):145–193, 1998b.

M. Gobbino and M. G. Mora. Finite Difference Approximation of Free Discontinuity Problems.

Preprint SISSA, Trieste, 1999.

GRASS Development Team. Geographic Resources Analysis Support System (GRASS GIS) Soft-

ware. ITC-irst, Trento, Italy, 2007. URL http://grass.itc.it.

122

References

M. W. Hirsch and S. Smale. Differential equations, dynamical systems and linear algebra. Academic

Press, 1974.

B. Kawohl. From Mumford-Shah to Perona-Malik in image processing. Mathematical Methods in

The Applied Sciences, 27:1803–1814, 2004.

S. Mahmood and B. S. Sharif. A nonlinear variational method for signal segmentation and recon-

struction using level set algorithm. Signal Processing, 86:11, 2006.

S. Mahmood and B. S. Sharif. Signal segmentation and densing algorithm based on energy opti-

misation. Signal Processing, 85(9):1845–1851, 2005.

C. Malpaga. Problemi unidimensionali di Calcolo delle Variazioni con discontinuita libere. Master’s

thesis, Faculty of Mathematic - University of Trento, aa 1997–1998.

R. March. Image Segmentation by Variational Methods. Scuola Estiva: ”Metodi di analisi ed

elaborazione delle immagini digitali” - S.Marco D’Alunzio, 4-9 Giugno 2001, 2001.

R. March and M. Dozio. A variational method for the recovery of smooth boundaries. Image and

Vision Computing, 15:705–712, 1997.

R. March and M. Dozio. Visual reconstruction with discontinuities using variational methods.

Image and Vision Computing, 10:30–38, 1992.

L. Modica and S. Mortola. Un esempio di Gamma-convergenza. Bollettino dell’Unione Matatem-

atica Italiana, B-14:285–299, 1977.

M. G. Mora and M. Morini. Local calibrations for minimizers of the Mumford-Shah functional

with a regular discontinuity set. Annales de l’institut Henri Poincare (C) Analyse non lineaire,

18(4):403–436, 1999.

J. M. Morel and S. Solimini. Variational methods in image segmentation, volume 14 of Progress

In Nonlinear Differential Equations And Their Applications. Birkhauser, Boston, 1995.

D. Mumford and J. Shah. Optimal Approximation by Piecewise Smooth Functions and Associated

Variational Problems. Communications on Pure Applied Mathematics, 42:577–685, 1989.

M. Negri and M. Paolini. Numerical minimization of the Mumford-Shah functional. Calcolo, 38:

67–84, 2001.

D. Nitzberg, D. Mumford, and T. Shiota. Filtering, Segmentation and Depth. Springer-Verlag,

Berlin, 1993.

P. Perona and J. Malik. Scale-space and edge detection using anisotropic diffusion. Transaction

on Pattern Analysis and Machine Intelligence, 12:629–639, 1990.

J. Petitot. An introduction to the Mumford-Shah segmentation model. Journal of Physiology-Paris,

97(2):335–342, 2003.

123

References

D. Piazza. Implementazione numerica di un modello del secondo ordine per l’analisi di segnali

monodimensionali. Master’s thesis, Faculty of Mathematic - University of Trento, aa 1998–1999.

T. J. Richardson. Scale independent piecewise smooth segmentation of images via variational

methods. PhD thesis, Massachusetts Institute of Technology. Dept. of Electrical Engineering and

Computer Science, 1990.

T. J. Richardson and S. K. Mitter. Scaling Results for the Variational Approach to Edge Detection.

MIT - LIDS Technical Reports, 1991.

F. Sanso and R. Rummel, editors. Geodetic Boundary Value Problems in View of the One Cen-

timeter Geoid, volume 65 of Lecture Notes in Earth Sciences. Springer Verlag, Berlin, 1997.

R. Spitaleri, R. March, and D. Arena. A multigrid finite difference method for the solution of Euler

equations of the variational image segmentation. Applied Numerical Mathematics, 39:181–189,

2001.

R. M. Spitaleri, R. March, and D. Arena. Finite difference solution of Euler equations arising in

variational image segmentation. Numerical Algorithms, 21:353–365, 1999.

124

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