Superquadric Recovery in Range Images via Region Growing influenced by Boundary Information...

Superquadric Recovery in Range Images via Region Growing influenced by Boundary InformationMaster-Thesis

Christian Cea Bastidas

Christian Cea BastidasMaster Thesis

Contents Motivation and Objectives

Superquadric and Rim

Overview of the Proposed Solution

Superquadric Fitting and Rim Fitting

Proposed and Alternative Solution

Evaluation Methodology and Comparison

Summary


Motivation

1. may be modeled with a low fitting error, using a type of surface called Superquadric

2. are not covered in the image by another object

To solve the Segmentation and Recovery Problem which

consists in extracting from a 3D image the objects that:


Objectives

1. To develop an algorithm which solves the stated problem by completing the solution of the existing algorithm Seed Generation.

2. To compare the improved solution with that of the well known approach Recover-and-Select Segmentation ( Leonardis, 1990 ).








Summary


Superquadric : Modeling Element

Superquadrics, a generalization of the quadric, were chosen as Modeling Object because:

1. They possess a simple mathematical formulation

2. The presence of superquadric-like objects is recurrent in many applications.

3. Its representation capacity can be easily incremented by means of Deformations.


Superquadric : Definition Parametric representation

Observation : Superellipsoids, a special type of Superquadric has been considered, which are closed and connected.


Superquadric : Examples (I)

The number of edges increases as distance themselves from 1.


Superquadric : Transformations

• Euclidean Transform

• Global Deformations ( Bending and Tapering )

6 new parameters => Superquadric needs 11 parameters


Superquadric : Examples (II)

A cylinder along its circular and parabolic deformations


Rim : Definition

The 3D points in a range image are collected by a laser sensor located on a certain plane. The normal to this plane corresponds to the Viewing Point.

Assumption : The distance between the laser sensor and the objects is supposed to be large


Rim : Example (I)

The rims have been drawn for the objects in the image.Viewing point is (0,1,0) ( Axis Y )


Rim : Superquadric Rim

A parametric representation of the rim is derived from a more operative definition :

=> Rim equation

Important Property : It permits to sample the rim efficiently !


Rim : Example (II) Rim of a superquadric in general position

Viewing Point = (0,1,0)








Summary


Solution Part 1 : Seed Generation

Input : Range Image Output : - Seeds - Edge Map

Seed : Set of points which belong with high probability to a single object

Edge Map : Points on the rims and edges ( All sets are Undistinguishable ! )

Seed Generation+

Edge Detection


Solution Part 2 : Region Growing(1)

Output : Recovered ObjectsInput : - Seeds - Edge Map

Region Growing

Key Idea : Alternate fitting of the superquadrics to the range image with the fitting of the superquadric rims to the edge map.


Solution Part 2 : Region Growing(2)

SuperquadricFitting

Rim Fitting

Rim Filter

Superquadric is fitted to O

Rim is fitted to the Edge Map using the rim of as first estimate

O* : set of points in the range image whose projection on the plane XZ is inside the Rim Projection

O ( A Seed )

O* ( Recovered Object )

Stop?

O = O*

Yes

NoIf O* ~ O or Big Error Fitting the Stop!








Summary


SQ Fitting : The Problem (I)

Restriction : The points in S come from the visible part of the object ( Self-Occlusion )

The problem can be stated as follows:


SQ Fitting : The Problem (II)Minimization Problem : Preliminary Formulation

Self-Occlusion →


SQ Fitting : The Problem (II)Minimization Problem : Preliminary Formulation

Self-Occlusion →


SQ Fitting : Objective Functions

3 Alternatives for the function :

Standard Euclidean Distance (SED)- There does not exist closed mathematical formula- S does not contain necessarily the closest point to an

arbitrary point because of the Self-Occlusion. ← Unfeasible

Radial Euclidean Distance (RED)- It has a closed mathematical formula- A good approximation to SED.

Modified Algebraic Distance (MED) ← Selected- Closed mathematical formula and simple derivatives- Broadly used in the literature


SQ Fitting : SED and REDBut SED and RED are more intuitive error measuresSED

RED


SQ Fitting : The Problem (III)

Definitive Formulation (Solina and Bajcsy )

Using a modified algebraic distance for :

where

and reverse the effect of deformations and euclidean transformation respectively.


SQ Fitting : Type of Problem

The formulated problem :

For this kind of problem, Levenberg-Marquardt Algorithm is specially suitable.

Corresponds to a Nonlinear Least Squares Minimization:


SQ Fitting : LM Algorithm

Iterative Procedure defined by :

holds for Nonlinear Least Squares Minimization

Requisites :

1. The initial estimate => An Ellipsoide ( Rosenfeld and Kak )

2. The derivatives in order to evaluate and


SQ Fitting : Examples

- The original points ( in pink ) lies on the visible part of the object

- Rounded objects are more easily fitted than objects with edges.


Rim Fitting : The Problem (I)There exist 2 mayor differences respect to the SQ Fitting :

1. The real rim of an object cannot be isolated from the Edge Map

=> Objective Function = Sum of the distances from each point in a

sampling of the SQ Rim to the Edge Map.

Important Assumptions :

- Edge Map contains enough points for each rim

- The points on the rim sampling must be uniformly distributed.


Rim Fitting : The Problem (I)There exist 2 mayor differences respect to the SQ Fitting :

1. The real rim of a object cannot be isolated from the Edge Map

=> Objective Function = Sum of the distances from each point in a

sampling of the SQ Rim to the Edge Map.

Important Assumptions :

- Edge Map contains enough points for each rim

- The points on the rim sampling must be uniformly distributed.

2. The fitting is done in 2D: Rim Sampling and Edge Map are projected

onto the plane XZ before computing the distances.

Reasons :

- Efficiency

- Rim Filter needs only the Rim Projection


Rim Fitting : The Problem (II)Mathematical Formulation


Rim Fitting : Examples

The estimate comes from aSuperquadric fitting a small region=> stays far from the real Rim

The estimate comes from aSuperquadric fitting a big region=> evolves nearly into the real Rim

Case 1 Case 2








Summary


Region Growing : Algorithm (1)Stop Criterion

Parameters


Region Growing : Algorithm (2)

Output


Recover-and-Select Segmentation:Part 1: Seed Generation and Expansion

1. Partition the image into nxn regions2. Fit a superquadric to each region3. Add new points to a region if they are well approximated by the associated superquadric4. Go to 2. until no more suitable points are available

Seed Generation+

Region Expansion


Recover-and-Select Segmentation:Part 2: Model Selection

Model Selection

A subset M’ of the generated models M is selected by solving a Binary Quadratic Programming Problem :

The idea is to minimize the information quantity I needed to represent the image :

Information I = Bits for SQ parameters + Bits for Error Fitting + Bits for Free Points

m : decision binary vector








Summary


Evaluation Methodology : Reference SolutionThe Ideal Solution has 2 parts, one related to the Segmentation and the second one to the Modelling :

1. The objects are segmented manually from the image and their points are stored as sets

2. For each object , the superquadric with the best fitting is found. Thus the set constitutes the second part of the solution.

As the best fitting cannot be guarranteed, then the Modelling part is replaced by the from the SQ Fitting.

The Segmentation part continues being the ideal one.


Evaluation Methodology : Indexes1. Each object O in the solution is matched manually with an object O* in the reference solution.

2. Then the Solution Quality is evaluated in three aspects :

Modelling

Segmentation

Time

belongs to the solution and is the superquadric fitted to O.

: Convex hull of the projection of the set onto the plane XZ

3. Finally each index is averaged over the objects exposed in the image weighting with the size of each set O ( |O| )


Algorithm Comparison : ImagesThe difficulty in solving the recovery problem for an image depends on :

1. Number of Objects (No)

2. Average size of the Objects (Size) [ small, medium, large]

3. Shape of the Objects (Shape) [rounded, box-like, mixed]

4. Percentage exposed objects or closely exposed (%E.O.)

The algorithms were tested using 8 images with the following characteristics :


Algorithm Comparison : Index 1

The superquadrics from Alg 1model better the objects than Alg 2.

The exception is the Image 4.

For images 5, 6, 7 and 8 themodels of Alg 1 are nearlyas good as those of the reference solution.


Algorithm Comparison : Image 4

The seed is completely contained in on one face of the box=> A seed should always contain points on “key sectors” of an object


Algorithm Comparison: Index 2

In general, Alg 1 excels insegmenting, except forimage 8.

Even for the image 8, thisindex is still good for Alg 1.


Algorithm Comparison : Image 8

The rim did not reach the bottomedges of the object.



Average Recovery Time (ART)Algorithm 1 : 35 sec.Algorithm 2 : 300 sec.

Only in one case Alg 2 wasfaster than Alg 1.



Image 6 is a important case because :

- Complexity of the Image - The seeds are not so big

Average Recovery Time = 52 sec.


Summary

1. The proposed solution showed a best performance both in efficacy and in efficiency in comparison with the classical approach.


Summary


2. The objective function used for the SQ fitting could be improved considering algebraic approximations to the Standard Euclidean Distance.


Summary


2. The objective function used for the SQ fitting could be improved considering algebraic approximations to the Standard Euclidean Distance.

3. The parameterization and sampling of the rim played a key role in the solution.


Summary

4. In the Rim Fitting the model error is measured only in a two-dimensional subspace. But it is feasible to compute this error in the original space.


Summary

4. In the Rim Fitting the model error is measured only in a two-dimensional subspace. But it is feasible to compute this error in the original space.

5. The performance of the proposed algorithm depends strongly on the edge map quality and to what extent the seeds contain points on key zones of the objects.


References


Superquadric and Rim : Sampling

Goal : To generate uniformly distributed points on the surface or curve.

Applications :

- Plotting and Visualization- Computation of the Closest Point from a given point to the SQ or Rim

( Used in Optimization Problems )

Mechanism :

- A SQ can be obtained by multiplying 2 superellipses - The rim can be obtained by multiplying 1 superellipse and 1 point. A Superellipse is a 2D curve which is easier to sample


Superellipse : Definition


Superellipse : Examples (I)

Superellipses with a=3 and b=9 for different values of ε.


Superellipse : Sampling (I)Problem : If ө is uniformly sampled then the resulting points on the superellipse are not uniformly distributed.• Two sampling mechanisms were tested:

Equal-Distance Sampling (Pilu and Fisher)- The superellipse is approximated using 2 models

which are easily parameterizable.

- Better distribution, but it returns fewer points than the

required number.

Angle-Center Parameterization (Löffelmann and Gröller)- The superellipse is represented in polar

coordinates (r,ω) and ω is uniformly sampled- It returns exactly the required number of points


Superellipse : Sampling (II)

The first mechanism showed the smaller interdistance deviation.


Superellipse : Sampling (III)

The ratio ρ between the required number of points and the obtained was fitted with a 2° polynom in є,b/a.

Then the required number of points is adjusted with :


SE : Sampling Comparison (I)If ө is uniformly sampled then the resulting points are not uniformly distributed.


SE : Sampling Comparison (II)Equal-Distance Sampling produces uniformly distributed points


SE : Sampling Comparison (III)Angle-Center Parameterization produces points not satisfactorily distributed


SE : Superquadric Sampling (I)

A uniformly distributed sampling for the superquadric is obtained by making the spherical product between the samplings of the superellipses and

Observation:


SE : Superquadric Sampling (II)

Superquadric Sampling based on Equal-Distance Sampling


is derived from the Rim Equation, = 0.001, R is the rotation matrix and is a constant

1. If

The point and the superellipse

2. If

The superellipse and the point

SE : Rim Sampling (I)A nearly uniformly distributed sampling for the rimcan be obtained as the spherical product between:


SE : Rim Sampling (II)

The sampling quality is acceptable The sampling quality is low


SE : Rim Sampling (III)

The problems appear when

and they are solved usingspecial parameterizationsfor the concerned rims

The figures show the attainedresults after applying the newparameterizations for thespecial rims.


Rim Anomalies

In the figure the discarded pointsappear in red

Date post:	18-Dec-2015
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Superquadric Recovery in Range Images via Region Growing influenced by Boundary Information...

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