Phase unwrapping via Grid Cuts

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 14 (2015) pp 34120-34131

© Research India Publications. http://www.ripublication.com

34120

Very Fast Phase Unwrapping Via

Grid-Cuts: G-PUMA

Sharoze Ali Research Scholar, Department of ECE,

KL University, Vaddeswaram, Guntur, Andhra Pradesh, India

[email protected]

Habibulla Khan

Professor & Dean (SA), Department of ECE,

KL University, Vaddeswaram, Guntur, Andhra Pradesh, India.

[email protected]

Idris Shaik

Assistant Professor, Department of ECE,

Bapatla Engineering College, Bapatla, Andhra Pradesh, India

[email protected]

Firoz Ali

Head of Department, Department of EEE, Nimra College of Engineering & Technology,

Vijayawada, Andhra Pradesh, India.

[email protected]

Abstract

The problem of Phase unwrapping (PU) can be solved by

various approaches.Here, we present a very fast algorithm for

two-dimensional (2D) phase unwrapping. Even for larger

images, the new algorithm “G-PUMA” quickly finds solution

than other methods. Previously, we published a robust to

noise, low complexity and fast phase unwrapping algorithm

(I-PUMA) for both convex and non convex potentials. But the

response times were slow especially while unwrapping phase

of larger images. Here an improved and faster version of

phase unwrapping algorithm is presented. The objective is to

extend our previous phase unwrapping algorithm and achieve

faster runtimes while unwrapping phase of larger images.

Results show that the proposed algorithm G-PUMA works

40% to 80% faster than the José's proposed phase unwrapping

via graph cut (PUMA). In addition, G-PUMA consumes less

memory of about 10 to 40% than PUMA.

Keywords—Phase Unwrapping, PUMA, Interferometry SAR,

Magnetic Resonance Imaging, Graph cuts

I. INTRODUCTION

Phase unwrapping (PU) [1] is one of the well-known and

widely adopted techniques for phase estimation. Estimation of

anabsolute (true, ϕ) phase from the measured phase (wrapped,

principle, ψ) is a key problem for many imaging techniques.

For instance, inremote sensing applications [2] like Synthetic

aperture radar (SAR) or Sonar (SAS), phase difference

between the terrain and the radar is captured by two or more

antennas.The measured phase by SAR or SAS is in the

interval of [-π, π].By using the Phase unwrapping process, we

can obtain the absolute phases from the measured one (ψ).

Similarly for MRI (Magnetic Resonance imaging), PU

technique is used to determine magnetic field deviation maps,

chemical shift based thermometry, and to implement BOLD

contrast based venography. PU also acts as a necessary tool

for the three-point Dixon water and fat separation. In optical

interferometry, phase measurements are used to detect objects

shape, deformation, and vibration.

Phase unwrapping (PU) [1] is however, an ill-posed problem,

if no further information is added. There are different methods

mailto:[email protected]






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to unwrap the phase and can be broadly classified as Path-

following, Minimum-Lp-normand Bayesian or regularization

methods.

Bayesian methods [3], [4], [5] are also known as statistical

methods. Bayesian methods depend on a data observation

mechanism model and utilize prior knowledge of the phase.

Bayesian approaches can be optimal from the information-

theoretic point of view but they are unable to restore

uniqueness of the solution.

Path following algorithms apply line integration schemes over

the wrapped phase image, and basically rely on the

assumption that Itoh condition holds along the integration

path. Wherever this condition fails different integration paths

may lead to different unwrapped phase values. This approach

is not a global approach as it does not make use of all

observed phase to determine the phase. These methods are

less robust to noise as it does not follow the global approach.

Minimum norm [6], [7] is a global minimization approach

where all the observed phases are utilized to compute the

solution. It find’s the phase solution (ϕ) for which the LP norm

of the difference between absolute phase differences and

wrapped phase differences is minimized. If p=2, then we have

Least square (L2) method. The disadvantage of least square

method is they tend to smooth even the discontinuities. L1

deals well with dis-continuities when compared to L2

solution.The main advantage of minimum norm method when

compared with the Bayesian method is, it does not require any

prior knowledge of the phase.

During the last three decades, redundant of algorithms were

proposed based on the above methods for Phase unwrapping.

Among them, PUMA belongs to Minimum norm method is

one of the best algorithm for unwrapping. Most of the

Minimum norm algorithms faces difficulty while unwrapping

phase especially at discontinuities, but PUMA even though

belongs to Minimum Norm method, it unwraps the phase

properly even at discontinuities. PUMA [8] is the first

technique which uses graph cut as an optimization technique

for phase unwrapping.

Later on, several algorithms belong to different approaches of

phase unwrapping uses graph cut as an optimization step. For

instance, algorithms like [3], [4], [5] use graph cuts as a

optimization step.But these algorithm requires prior step and

consume more memory, time than PUMA Method. For

instance, the algorithm [5] faces issue while unwrapping

especially at discontinuities. Among all available algorithms,

PUMA is the optimum solution in both the aspects of

discontinuity preserving and utilizing system resources.

However, computation speed and memory consumption often

times limit the effective use of PUMA especially for large

images. Recent research on optimization techniques [11], [12]

shows that there is still considerable need and room for

improvement of the PUMA algorithm.

Previously, we presented a fast two-dimensional (2D) phase

unwrapping algorithm (I-PUMA) [14].In our previous paper,

we have reduced the complexity of the PUMA algorithm and

the result is significant reduction in time and complexity of

the algorithm. However, the speed can be further increased

especially for larger graphs. Here, we continue our previous

work showing an improved version of our previous 2D

unwrapping algorithm.

In our new approach, we show that significant improvements

in speed can be achieved by adopting the memory

optimization techniques, instead of reducing the complexity of

the algorithm. G-PUMA is a descent algorithm of PUMA and

I-PUMA. The significant advantage of the proposed method is

increase in speed whether the profile is of any variety or too

noisy or too big. Our algorithm performs well either if clique

potential is convex or non-convex.

We evaluate the performance gain on a variety of typical

problems. Our new algorithm G-PUMA shows improvement

in speed of about 40% to 80% and 10% to 40% reduction in

memory consumption compared to existing approach PUMA.

From the experiments in Section V, our proposed method

achieves faster running times than PUMA unwrapping

method.

The remaining of the letter is organized as follows. Section II,

we present the PUMA phase unwrapping algorithm.In Section

III, we present the cache efficient Gridcut method; Section IV,

we briefly presents our new algorithm G-PUMA.In Section V,

a set of experiments and results to compare G-PUMA with

PUMA in all aspects and we conclude this letter in Section

VI.

II. PUMA : PHASE UNWRAPPING VIA MAX

FLOW

Phase unwrapping (PU) is the process of recovering the

absolute phase from the wrapped phase, formally as

ψ

(1)

Phase unwrapping via graph cuts [8] is a novel technique

which uses graph cut as an optimization step. PUMA

technique mainly consists of two algorithms and it is

classified according to its clique potential, as an energy

minimization framework for PU. The clique is a set of sites

that are mutually neighbors. If the clique potential is greater

than one, then such cliques are named as Convex potential

(p≥1) and PUMA has an exact energy minimization

algorithm.



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Fig. 1. Representation of the site (i, j) and its first-order neighbours

along with the variables h and v signalling horizontal and

vertical discontinuities, respectively.

For non-convex clique potentials (p<1), PUMA has an

approximation solution owing to its discontinuity preserving

ability. Both algorithms solve optimization problems by

computing sequence of binary optimization, each one solved

by graph cut techniques.

Let us define the energy for a site of (i, j) as shown in Fig.1 as

(2)

(

) ∑

( ) (

)

(2)

Where is an image of integers, denoting multiples, the

so-called wrap-count image of ψ, is the clique potential, a

real-valued function, 1 and (.) h and (.) denote pixel

horizontal and vertical differencesgiven by (3) to (6). Our

purpose is to find the integer image that minimizes energy

(2), being such that π ψ

[ ( )

]

(3)

[ ( )

]

(4)

(5)

(6)

A. ENERGY MINIMIZATION BY A SEQUENCE OF

BINARY OPTIMIZATIONS: CONVEX POTENTIALS

By using the proof of Equivalence between Local and Global

Minimization, Convergence Analysis and Mapping Binary

Optimizations onto Graph Max-Flows, the author’s [8]

rewrites the energy equation (2) as (7)

| ∑

[ ( ) ]

[ ( ) ]

(7)

Authors of PUMA, for the sake of simplicity, rename the

equation (7) to (8)

| ∑ ( )

(8)

The minimization of the equation (7) with respect to is now

mapped onto a max-flow algorithm. For graph construction,

Authors[8] exploits a one to one map existing between the

energy function (7) and the cuts on a directed graph with non-negative weight, the graph has two specials vertices,

namely the source ‘s’ and the sink ‘t’. The number of vertices

Ѵ is 2 + NxM (two terminals, the source and the sink, plus the

number of pixels or nodes).An s-t cut or min-cut is a partition

of vertices Ѵ into two disjoint sets S and T, such that s S

and t T with min cost.Cost of the cut is the sum of costs of

all edges between S and T.

As per [9], a function of class of functions is graph

representable, i.e., there exists a one-to-one relation between

configurations Є {0, 1} MN and s-t cuts on that

|ψ graph, if and only if it satisfy the regularity

condition (9).

(9)

(10)

(11)

(12)

(13)

So, for each energy term and

, authors construct an

“elementary" graph as shown in figure 2 (a) with four vertices



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{ }, where represents source and the sink,

common to all terms, and { } represents the two pixels

involved (v being the left (up) pixel and the right (down)

pixel).Finally, the directed edge { } is defined with the

weight of . Moreover, if

edge { } is defined with the weight

of or, otherwise, edge { } is defined with

the weight .Once energy is mapped on to the

graph, energy is easily minimized by using the max flow

/min-cut on the constructed graph. Among the available max

flow algorithms, authors of PUMA uses BK [10] Algorithm

(alpha expansion graph cut) for finding the min cut/maxflow.

PUMA runs for iterations to unwrap the phase of a profile,

where is the number of π multiples.

(a) (b)

Fig. 2. (a) Elementary graph for a single energy term, where and

represent source and sink, respectively, and represent the

two pixels involved inthe energy term. In this case and .(b) The graph obtained

at the end results from adding elementary graphs

B. ENERGY MINIMIZATION BY A SEQUENCE OF

BINARYOPTIMIZATIONS: NON-CONVEX

POTENTIALS

The PUMA Algorithm of Convex potential doesnot fits to

non-convex potential as it faces with the below two issues.

1) If the clique potential is nonconvex , it is not

possible, in general, to reach the minimum through

1-jump moves only.

2) With a general nonconvex , the condition of

regularity (9) does nothold for every horizontal and

vertical pair-wiseclique interaction. This means, it’s

not possible to mapenergy terms onto a graph and

energycannot be minimized via graph cuts.

Energy minimization of a non-convex potential is a NP Hard

problem. Even though, authors proved that the second issue

can be resolved by applying majorize minimize (MM)

concepts to energy function. That is where ever the pixel pairs

do not satisfy the regularity condition (9) then the edge weight

between pixels pairs are set to zero to satisfy the regularity

condition. With respect to the first referred issue, authors

extend the range of allowed moves. Instead of only 1-jumps

they now use sequences of s–jumps. In simpler terms if the

pixel belongs to Source (s) then it is increased by s instead of

1 as in convex algorithm. Finally authors [8] showed that the

non convex potentials can also be solved via graph cuts and

for uniformity sake they name the algorithm as PUMA.

(a) (b)

Fig. 3: (a) Graphs which have a grid-like structure. The double headed

arrows represent two oppositely oriented edges between the same

two nodes.(b) All 1-offset neighbours of (1, 1) pixel.

III. OPTIMIZATION TECHNIQUE : GRID-CUT

A. Previous Works

Globally optimal solutions can be quickly achieved by using

Graph cuts. For instance, many applications in vision and

graphics use min-cut algorithms either as a tool for computing

optimal hyper-surfaces or as an optimization technique in

energy minimization algorithms. Also, it can be useful for

inferring the maximum a posterior solution of a discrete MRF.

That is why these methods are at the core of many state-of-the

art algorithms and provide high-quality solutions in practice.

Graph cuts can be broadly divided into two main groups:

“push-relabel” and “augmenting path” methods.

Different researcher’s proposed different graph cut algorithms

on both of the above mentioned methods. Among them, the

one proposed by Boykov and Kolmogorov (BK) [10] belongs

to augmenting path method is the most popular algorithm, due

to its computational efficiency and out performs push –relabel

methods by considerable factors.The main advantage of this

method is, it is faster in reaching the Global Optimum

Solution than the other available methods like simulated

annealing, etc.



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Although it has been extensively used by the vision

community, it has no known polynomial time bound and the

algorithm performs poorly in practice on some non-vision

problems. Computation speed and memory consumption

oftentimes limit the effective use in applications requiring

high resolution grids or interactive response. To eradicate the

issues of the BK Algorithm and to process faster, authors of

[11] introduce the incremental breadth-first search (IBFS)

method. IBFS [11] is an extension of the BK algorithm. As

BK set out to always maintain the search trees (however

arbitrarily), IBFS also maintains the trees and make sure that

they first search breadth-wise.

IBFS graph cut provides a polynomial run time guarantee,

whereas BK Graph cut does not provides this. Empirical

testing of IBFS versus BK has shown that in the vast majority

of cases, IBFS outperforms BK with a speed increase of about

20 to 50% on a variety of vision instances. The IBFS

algorithm does not perform well than BK on some non-vision

instances. Also the pre-processing phase used in IBFS, is

handled inefficiently.

Voronoi based pre-flow push (VPP) [15] is also an attempt to

provide algorithm faster than BK for high resolution grids.

VPP graph cut not only exploits the structural properties

inherent in image based grid graphs but also combines the

basic paradigms of max-flow theory in a novel way. As per

[15] BK is slower by at least a factor of 2 on 70% of the

samples. But, in reality the VPP performed only comparable

or even worse than BK.

In order to increase the computational efficiency and to

decrease the memory consumption especially for high

resolution grids and large size images, authors of [12]

proposed the Grid cut algorithm. The idea of Grid cut

algorithm is a cache efficient implementation of BK algorithm

presented by Boykov and Kolmogorov [10]. GridCut

implementation uses the full combination of the compact

residual graph representation (CR), the structure of arrays

(SoA), and the blocked array layout (BLK) techniques which

will be explained in detail in next section.

B. Grid Structure

Generally graphs for energy minimization looks like as in

figure 3(a). A graph G is a pair G = (V, E) where V is called

the set of nodes and ‘E’ is called the edge set consisting of

pairs (u, v), u, v ε V as shown in Figure 3(a). A graph ‘G’ is a

collection of Nodes ‘V’ and they are connected with edges

‘E’, if they are neighbours. For instance for an image of size

‘M’ X ‘N’ where ‘M’ represents rows and ‘N’ represent

columns, we can arrange the nodes into an ‘M’ X ‘N’ grid

with double edges between the every two neighbour’s in the

grid. As shown in Figure 3(b), if one of the nodes is located at

the position (1, 1) then the neighbours of such node are

accessed by adding or subtracting one. In other words, all of

the neighbours of (1, 1) are at one- step away from (1, 1).

C. Compact residual graph Representation (CR)

In graph cut theory, residual graph is constructed by replacing

each edge with a pair of oppositely oriented edges that

represent either the possibility of pushing additional flow

along the edge in the flow network, or diminishing the flow

against the direction of the edge. As shown in Figure 4(a) for

each pair of neighbouring nodes, there exists its own capacity

‘c1’edge between the two nodes and during the max flow

computation an additional flow “f1” is added to the edge.

There is a possibility that the additional flow can be pushed

along one of the additional edge or pushing back against the

other edge as shown in the figure 4(b). This process is call the

augmenting the flow between the neighbouring nodes. Finally,

the two residual capacity can be represented by r1=c1-f1+f2

and r2 =c2-f2+f1 which means when we augment along this

edge, the amount of flow we are augmenting is subtracted

from this residual capacity and added to the oppositely

oriented residual edge. This compact representation of the

residual graph is shown in Figure 4(c).This causes to lose the

exact distribution of the flow on each edge, but it will provide

information about the minimum cut and max flow.

(a) (b)

(c)

Fig. 4. (a) The flow network has a pair of sibling edges, each with is

own capacity and current flow. (b) The resulting residual graph will

in this case contain two pairs of parallel edges.(c) We can merge

these parallel edges back into a pair of residual sibling edges.

We can combine this representation with the simplifications

that the grid structure allows us. In every node, the residual

capacity of every outgoing edge is stored. When we augment

along an edge, some of its residual capacity is transferred to

its sibling edge. Since all nodes have the same local

neighbourhood, this enumeration can be applied to every node

consistently as in figure 5(a). We can quickly access this edge

by using a lookup table that maps each edge index to the

index of its corresponding reverse direction as in figure5(b).



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D. Cache friendly indexing of the nodes

The pattern of data access in the max flow algorithm is when

some node is visited during the computation, it is likely that

the nearby nodes are accessed in the next step. As transfers

between cache and memory are executed in blocks (cache

lines) with fixed size (typically 64 bytes), we can rearrange

such arrays into a blocked layout with fixed size (typically 64

bytes) as shown in Figure 5(c). In order to improve the cache

efficiency, the idea proposed by the authors [12] is to arrange

the fields in such a way that spatially close nodes on the grid

become spatially close in memory. When a field from the

block is accessed for the first time, a cache miss occurs and

the field is loaded into the cache along with fields of other

nodes lying in the same block. With this behaviour, fields of

nearby nodes can be quickly accessed in further steps of the

algorithm.

E. Structure splitting

BK maintains search trees with marking heuristic requiring

for each node to store the tree membership tag, a reference to

a parent node, an estimation of the distance to its terminal, and

a time-stamp indicating when the distance was computed.

From the optimization point of view, the key observation is

that these fields are typically not accessed at the same time

during the max-flow computation. Similarly IBFS, also

maintain trees like parent map, residual capacity between each

node, distance labelling and the {S, T, FREE} –labelling. If

we were to store this data as an array of node-structs, all these

different data fields are stored in an interlaced fashion in the

memory which often decreases the cache performance as the

unnecessary data will be transferred to the cache.

Generally frequently accessed fields are typically referred as

HOT and unused fields are referred as COLD. In order to

make graph cut a cache efficient, space is preserved for HOT

fields and avoid transfer of COLD fields. Instead of storing all

nodes in a single array of structures with all fields packed

together, it’s better to split the individual fields into separate

arrays using the structure-of-arrays layout (SOA). With this

layout the data can be naturally split into hot and cold portion.

Finally, the aim of authors [12] is to ease the memory

bandwidth by employing a compact graph representation with

cache-friendly memory layout that exploits the regular

structure of grid-like graphs.

Gridcut is tested on all instances and it out performs than BK,

IBFS and VPP. For the common class of graphs, Gridcut data

structure allows for 3 to 12 times higher grid resolutions and

3to 9 fold speedup compared to the existing approaches.

When compare it with BK, the reduction in memory

consumption ranges from 2X to 6X for 32-bit and 3X to 12X

for 64-bit modes. The average speed up over all instances and

CPUs is 2.7X for 32-bit and 4.4X for 64-bit modes.

(a)

edge REV [edge]

1 3

2 4

3 1

4 2 (b)

(c)

Fig. 5. (a) Enumeration of outgoing edges and reverse edge. (b) Lookup using the REV table. (c) Blocked array layout.

IV. PHASE UNWRAPPING VIA GRID CUT : G-

PUMA

PUMA is a two-step phase unwrapping process. Firstly, the

elementary graphs are constructed for a site by using the

energy equation

ψ and secondly, it is minimized by the

graph cut optimization techniques. These two steps together

are run for ‘k’ iterations to unwrap the phase. PUMA [8] is

one of the novel technique which unwraps phase even at dis-

continuities for both convex and non-convex potentials. The

only disadvantage of PUMA algorithm is it consumes lot of

time and memory to unwrap phase of larger images. This

often time limits the utilization of PUMA algorithm. If we

take a closer look at the algorithm, most of the time spent by

the algorithm during the calculation of maxflow algorithm. In-

order to overcome the difficulty and to increase the speed of



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PUMA Algorithm, we have to adopt either of the below two

approaches.

1) To reduce the complexity of the PUMA Algorithm

2) To use the cache efficient techniques while

calculating the max flow.

In-order to overcome the difficulty and to increase the speed

of PUMA Algorithm, we have to adopt either of the below

two approaches

1) To reduce the complexity of the PUMA Algorithm

2) To use the cache efficient techniques while

calculating the max flow.

In our previous approach, I-PUMA [14] we have followed the

first approach of reducing the complexity of the PUMA

Algorithm. We have reduced the complexity of the algorithm

to . By doing so, we were successfully being able to

reduce the complexity of PUMA algorithm and the runtimes

were 20 to 40 % faster than PUMA. But the runtimes can be

further improved especially while un-wrapping phase of larger

images. When we deeply analyse the PUMA algorithm for

further improvement, especially the optimization part, the

below issues were faced by both of the algorithms PUMA and

I-PUMA.

1) During the computation of the max flow algorithm,

there will be frequent transfer of data between

memory and CPU. Also constant updates are

required, which causes heavy stress on the memory

bandwidth.

2) More time consumed while finding the connectivity

information using pointers on every time. The

Pointers are generally used to represent the general

graphs and it provides the connectivity information.

These pointers often comprise the majority of the

graph’s memory footprint, in particular on 64-bit

CPUs where single pointer occupies eight bytes.

3) Cache is poorly handled.

The run times will be very faster if we are able to solve the

above three issues. As per our second approach, we are able to

solve the above three stated issues where we have employed

the cache efficient techniques during the optimization step of

PUMA. In this section, we introduce our new, fast and robust

unwrapping algorithm “G-PUMA”.

All the above three mentioned issues will play a crucial role in

finding min cut of large graphs. The first issue (stress on

memory bandwidth) is resolved by employing a compact

graph representation with cache-friendly memory layout that

exploits the regular structure of grid-like graphs which

reduces the stress on the memory bandwidth. Secondly

(calculation of connectivity information), by exploiting prior

knowledge of the graph structure we can eliminate the need

of pointers altogether by determining connectivity information

on the fly. Thirdly, Cache can be efficiently utilized by

segregating the memory fields as Hot and Cold depending

upon their accessibility (Structure splitting) and by dividing

the given graph into blocks (Blocked array layout).

Among the available, the best cache efficient graph cut

technique is Grid Cut. But grid cut can be applied only to

graphs like grid structured. The graph constructed by PUMA

algorithm is also a 2-D grid. So, it satisfies the basic

requirement of Gridcut and with slight modification in PUMA

we can easily utilize Gridcut as an optimization technique for

PUMA. We name our new approach as G –PUMA where ‘G’

represents the Gridcut technique.

Like PUMA algorithm, G-PUMA algorithm also comprises of

twosteps. In G-PUMA the sequences of steps to unwrap the

phase for both convex and non-convex will be remain same.

As a First step, we have to construct the elementary graphs by

using the energy equations. The power of PUMA lies in its

energy minimization framework and shows greater

attenuation to noise, so we have not modify or changed the

energy equation

ψ of PUMA.Once the elementary graph is

constructed, the minimization of energy equation with respect

to is now mapped onto a max-flow problem.

For construction of graph we have followed the approach of

PUMA, where the vertices and edges corresponding to each

pair of neighbouring pixels are build first, and then join these

elementary graphs together based on the additive theorem.

(a) (b)

Fig. 6. (a) 2-D /4C neighboring connectivity. (b)2-D /8C neighboring

connectivity

Once the energy is represented onto the graph, we have to use

the max flow algorithm for minimizing the equation. As we

have followed the same energy minimization approach as in

PUMA for both Convex and non-convex algorithms, our

algorithm G-PUMA also stops for ‘K’ iterations.

The main limitation of Gridcut optimization technique is

hidden in the fact that it supports only graphs which has grid-

like topology. Grid like topology is a graph topology as in

Figure 3(a), where nodes can be stored in memory in such a

way that neighbours can be addressed by a set of fixed offsets.

The graphs constructed by the PUMA will not have all pixel



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interactions required to represent a grid-like topology. So we

cannot directly use the Grid cut on the graph constructed by

the PUMA algorithm. For instance, the required pixel

interactions to represent 2-D graphs are left, right, up and

down (4-neighbours) as shown in figure 6 (a). Whereas the

graphs constructed by the PUMA will only provide the

information of right and down pixel interactions.

In order to use the same graphs of PUMA on Gridcut, either

we have to construct the full grid or we have to set zero

capacity for edges which are not provided by PUMA and treat

them as dummy nodes. If we follow the second approach of

padding, then the performance of algorithm is degraded [12].

So we have adopted the first option of constructing the full

grid.

The grid can be fully constructed from either of the below two

options.

1) To calculate the other pixel interactions which are

not provided by PUMA. (or)

2) To reuse the same graph for calculating the pixel

interactions which are not provided by the PUMA.

Purge the missing Pixel interactions from its 1- offset

neighbour and pad the edges with Zeros.

We have adopted the reuse approach to construct the full grid

as the first approach is more time consuming. For instance, if

a node in 2-D Grid is located at A (1, 1) location, then its right

and down pixel interaction are provided by the graph

constructed by PUMA. The LEFT pixel interaction of the A(1,

1) node is nothing but the right interaction of its left neighbour

B(1, 0). Similarly, the UP interaction of the A(1, 1) node is

equal to the down interaction of its up neighbour C(0, 1). By

doing this for every pixel and adding them, we can construct

the full grid. But for an edge which does not have pixel

interaction, we have padded them with zeros.

Once we constructed the elements of the graph, we have

added them based on the additive theorem and provided as

input to the min cut solver. But, BK Algorithm [10] works on

node number whereas Grid Cut works on Co-ordinate’s. So,

we have represented these elementary graphs in terms of co-

ordinates instead of node numbers and provided as input to

the Gridcut solver. We have followed the similar method of

graph construction for the other Grid like structures as in

figure 6(b).

We have analyse the coding parts of PUMA and we have

identified that there are some unnecessary calls between

various programming parts of PUMA and by properly tunning

it we can make the algorithm bit faster. For Instance, in

PUMA along with the edge weights, the authors of PUMA [8]

also sending the other supplementary information related to

the pixel interactions like node number of the starting pixel

and ending pixel, etc to maxflow mex-part of coding. In our

algorithm, we have avoided passing such supplementary

information, as the same can be extrapolated during the max

flow computation.

As per [12] the run times will be faster if we use the

optimization techniques while compiling the matlab mex’s.

So, we have also make use of full compiler optimizations

technique (/Ox in Visual Studio) while compiling the Mex.

We have in-cooperated all these above mentioned changes in

PUMA and named it as G-PUMA. From the experiments in

Section V, We have attained 40-75% faster in run times for

both 32-bit and 64-bit machines. Also our new algorithm

consumes less memory than PUMA algorithm of about 20-

40% for 32-bit and 40-60% for 64-bit machines. Like PUMA,

G-PUMA also deal well with discontinuity and unwraps the

phase faster than PUMA.

(a) (b) (c)

(d) (e) (f)

(f) (g) (h)

(i) (j) (k)

Fig. 7(a), (d), (f), (i)Interferogram’s of shear ramp ratio of 1, 3/4, 1/2, 2*1/5

respectively.7(b), (e), (g), (j) unwrapped by PUMA algorithm for

interferogram’s of (a), (d), (f), (i) respectively.7(c), (f), (h), (k) unwrapped by

G-PUMA algorithm for interferogram’s of (a), (d), (f), (i) respectively.



34128

(a) (b) (c)

Fig. 8(a) Interferogram’s of High phase rate Gaussian Hill with a

quarter set to zero.8(b) Unwrapped by PUMA algorithm for

interferogram of (a) respectively.8(.c) unwrapped by G- PUMA

algorithm for interferogram’s of (a) respectively

V. EXPERIMENTAL RESULTS

To check the uniqueness and power of G-PUMA verses

PUMA Algorithm, we have tested both the algorithms on

some simulation profiles and real data. Survey do to know

effect of five factors are discontinuity (mask), noise, PSNR,

elapsed time and memory. As PUMA is a cache efficient

technique and depends on the CPU, number of bits and cache

memory, so we have tested the both of algorithm’s on

different machines (32-bit and 64 -bit).

A. Discontinuity

Phase unwrapping at discontinuities is a critical one with the

available information as in wrapped image. PUMA algorithm

though belongs to Minimum norm method, it unwraps the

phase correctly even at discontinuities. Because of the

discontinuity preserving ability, PUMA is one of popular

algorithm. Our algorithm, the newly developed G–PUMA

algorithm also owns to discontinuity preserving ability and it

is tested on interferogram’s of shear and ramp, Gaussian and

mask, as in figure 7 and 8 respectively.

Both the algorithms are executed on the interferogram’s of

shear and ramp, Gaussian and mask. We can see that both the

algorithms are equally powerful while unwrapping at

discontinuity, for shear ramp ratio of 1, 1/2, 3/4, 2*1/5

interferogram’s as in Figure 7.Even for High phase rate

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 9(a), (d), (g) Interferogram’s of High phase rate Gaussian Hill a

quarter set to zero treated with noise of coefficients 0.6, 0.8 & 1.0

respectively. 9(b), (e), (h) unwrapped by PUMA algorithm for

interferogram of 9(a), (d), (g) respectively.9(c), (f), (i) unwrapped by

G- PUMA algorithm for interferogram’s of 9(a), (d), (g) respectively.

Gaussian Hill with a quarter set to zero as in figure 8(a), the

unwrapping results ofboth algorithms are equal as in figures

8(b) and 8(c) respectively. As per [13], PUMA out performs

CUNWRAP algorithm especially while unwrapping at

discontinuities. Therefore, G-PUMA also out performs

CUNWRAP while unwrapping at discontinuities.

B. NOISE

Random noise assigned with three different coefficients of

0.6, 0.8, 1.0 are sequentially added to the High phase rate

Gaussian Hill with a quarter of it set to zero interferogram.

These noisy profiles are provided as input to both the

algorithms. Noisy Interferogram’s of 0.6, 0.8, 1.0 coefficients

are shown in figure 9(a), 9(d) and 9(g) respectively. Each of

them did iterative running with both the algorithms. As shown

in figure 9, both the algorithms are equally unwrapped for all

interferogram’s of noise variance 0.6, 0.8, 1.0. I-PUMA [14]

has more incorrect shape at the peak of the Gaussian hill for

1.0 noise variance than PUMA, G-PUMA. Finally, G-PUMA

and PUMA are more noise resistant algorithms and unwraps

phase correctly than CUNWRAP and I-PUMA.

C. PSNR, MSE

MSE (mean square error) and PSNR are two parameters to

determine the robustness of the phase unwrapping algorithm.

Mean Square Error (MSE) is taken between the True Image

and unwrapped images. In order to check the robustness of

algorithms, we have treated both the algorithms under

different simulation profiles and noted down the MSR, PSNR

Values in the table I. G-PUMA and PUMA have provided

similar values of MSE and PSNR values, because G-PUMA is

just an cache efficient implementation of PUMA algorithm.

For noise variance of 1.0, both G-PUMA and PUMA records

low MSE and high PSNR values shows greater attenuation to

noise where as I-PUMA [14] and CUNWRAP [13] records

high MSE and low PSNR values. This shows that the G-

PUMA and PUMA algorithms perform well in unwrapping

phase of noisy profiles than I-PUMA and CUNWRAP

algorithms.

020

4060

80100

0

50

1000

10

20

30

40

50

60

Puma 3-D solution

020

4060

80100

0

50

1000

10

20

30

40

50

60

G-Puma 3-D solution

020

4060

80100

0

50

1000

10

20

30

40

50

Puma 3-D solution

020

4060

80100

0

50

1000

10

20

30

40

50

G-Puma 3-D solution

020

4060

80100

0

50

1000

10

20

30

40

50

Puma 3-D solution

020

4060

80100

0

50

1000

10

20

30

40

50

G-Puma 3-D solution



34129

(a) (b)

(c) (d)

(e) (f)

Fig.10(a). Wrapped phase image obtained from a simulated InSAR

acquisition from Long’s Peak, CO (data distributed with [1]).

(b)Wrapped Noise Phase (c) Quality Map of (a) distributed with [1].

(d) Image in (b) unwrapped by G-PUMA. (e) Image in (b)

unwrapped by PUMA. (f) True Phase of (a)

D. Elapsed Time (Speed)

As stated earlier that the speed of the PUMA algorithm suffers

while unwrapping phase of larger images. So there is a room

for improvement of speed the PUMA algorithm. In this paper,

we have resolved the main issue of PUMA. By using G-

PUMA, we are able to show that the runtimes are very faster

than PUMA. In order to check the effectiveness of G-PUMA,

we have considered different profiles and noted down values

in table II. The speed of the G-PUMA algorithm depends on

the processor, number of bits and cache memory, so we have

tested algorithms in both 32 and 64 bit machines.

Specification of the two computers in experimental are

1) Intel Core i3Processor, 64-bit, 4GB RAM, 400GB

HDD, 3MB Cache memory and LCD 17”.

2) Intel Core Duo2 Processor, 32-bit, 4 GB RAM, 500

GB HDD, 2 MB Cache memory and LCD 15”.

TABLE I. PSNR AND MSE RESULTS

Interferogram G-PUMA (in

decibels)

PUMA (in decibels)

MSE PSNR MSE PSNR

Guassian Hill

with zero noise

variance

91.69 +

0.14i

-19.62 -

0.01i

91.69 +

0.14i

-19.62 -

0.01i

With Noise of

Variance 0.6

1.7628-

2.02i

-2.46+0.04i 1.7628-

2.02i

-2.46+0.04i

With Noise of

Variance 0.8

1.544-

2.29i

-

21.88+0.065i

1.544-

2.29i

-

21.88+0.065i

With Noise of

Variance 1.0

1.5203-

2.532i

-21.81+0.07i 1.5203-

2.532i

-21.81+0.07i

Shear Ramp 1.3128e-

29

288.8182 1.3128e-

29

288.8182

TABLE II. ELAPSED TIME

Interferogram 32-BIT 64-BIT

G-

PUMA

(in

sec)

PUMA

(in

sec)

G-

PUMA

(in

Sec)

PUMA

(in

Sec)

High phase rate

Gaussian Hill

0.0636 0.1246 0.1053 0.2376

High phase rate

Gaussian Hill With

noise variance 0.6

0.1235 0.2662 0.1022 0.2381

High phase rate

Gaussian Hill With

noise variance 0.8

0.1141 0.2349 0.1043 0.2519

High phase rate

Gaussian Hill With

noise variance 1.0

0.1247 0.2290 0.1049 0.2206

High phase rate

Gaussian Hill with a

quarter set to zero.

0.1205 0.2210 0.2172 0.2497

High phase rate

Gaussian Hill with a

non-vertical and non-

horizontal aligned

sector set to zero.

0.7360 0.2559 0.3496 0.0858

Mean from 10 Records

Image Simulation’s are

1) High phase rate Gaussian Hill with out and with

noise variances (0.6, 0.8, 1.0)

2) Shear Ramp.

3) High phase rate Gaussian Hill with a quarter set to

zero.

4) High phase rate Gaussian Hill with a non-vertical

and non-horizontal aligned sector set to zero.

Interferogram

50 100 150 200 250 300 350 400 450

20

40

60

80

100

120

140

Wrapped noisy phase

Quality Map

0100

200300

400500

0

50

100

150

2000

50

100

150

G-Puma solution

0100

200300

400500

0

50

100

150

2000

50

100

150

Puma solution True phase



34130

TABLE III. ELAPSED TIME FOR DIFFERENT SIZES OF

IMAGE

Interferogram N=128 N=256 N=512

G-

PUMA

PUMA G-

PUMA

PUMA G-

PUMA

PUMA

Gaussian Hill

With noise

variance 0.6

0.2107 0.4924 2.3508 6.9326 8.1922 19.5057

Gaussian Hill

With noise

variance 0.8

0.2250 0.5384 2.3123 6.5099 8.2599 18.3168

Gaussian Hill

With noise

variance 1.0

0.2447 0.6406 1.8568 4.7621 6.5111 13.7413

Mean from 10 Records

Recapitulations of ET (elapsed Time, with tic-toc

function in Mat Lab) for simulation profiles are listed in table

II.As listed in the table II, our algorithm G-PUMA is able to

unwrap phase very faster than PUMA.As shown in the Figure

11, G-PUMA records very low elapsed times. In order to

check the effectiveness of the G-PUMA algorithm, we have

tested algorithm for an interferogram’s added with different

noise variance (0.6, 0.8 and 1.0). G–PUMA is faster even if

the interferogram’s is noisy. As noise variance increases the

difference between the elapsed times of PUMA and G-PUMA

increases. We have observed that G-PUMA is consistently

faster even if noise variance increases as shown in Figure 11.

The other way to check the effectiveness of G-PUMA is by

applying our algorithm on larger images. For instance, if size

of the image is increased by twice, then the time taken by the

algorithm to unwrap the phase will be more than twice. For

such instance, our algorithm unwrap phase very faster than

PUMA. As listed in the table III, G-PUMA unwrap phase

twice faster than PUMA even if the image is noisy and big. As

size of the image increases, the difference between the G-

PUMA and PUMA also increases. For an image of 512 X 512

size, G-PUMA unwrap image within 6.5 Sec where as PUMA

unwrap phase in 13.7 Sec, i.e. (2X faster).

All of the above interferogram’s are grid like structures and

have no irregular topologies. But, as per [12], the performance

gain will be degraded if the profiles have topological

irregularities. We would like to test and validate the above

scenario for our algorithm as well. So we have consider the

profiles like Shear Ramp and High phase rate Gaussian Hill

with a non-vertical and non-horizontal aligned sector set to

zero. Shear Ramp as shown in Figure 7(a), have no left and up

pixel interactions whereas the other one belongs to 2D grid 8

Connectivity as shown in Figure 6(b) have all the 6

connectivity available except the up and down information.

Fig.11. Elapsed time of PUMA and G PUMA for different profiles

When this profiles with missing edges, are provided as input

to the G-PUMA, it unwraps phase correctly as PUMA but

elapsed time is delayed. Finally, G-PUMA performance gain

is degraded if profiles have topological irregularities. This is

the only dis-advantage of the G-PUMA method.

E. Memory

Memory consumption is another critical parameter which

often times limits the effective use of phase unwrapping

algorithms on workstations. For larger images the memory

consumed by these algorithms are extremely high. By using

the memory optimization techniques, we can reduce the

memory consumption. Compact data structure with cache-

efficient memory layout is the technique we have followed

and showed that significant improvement in utilizing the

memory bandwidth as in table IV.

We have tested our new algorithm on different profiles and

find out that memory consumption is reduced by half for most

of the profiles. As size of the image/profile increases, more

memory is consumed by the PUMA or any other Phase

unwrapping algorithm. But G-PUMA consumes very less

memory even if the profile size increases.

PUMA Algorithm consumes more memory (nearly twice) for

64-bit system than 32-bit system, but G-PUMA consumes

same memory even if the system is 32-bit or 64-bit. For

instance, on a 64-bit system, the memory consumed by

PUMA algorithm to unwrap phase of 512 X 512 size is 27.24

MB and on a 32-bit system is 15.71 MB, where as G-PUMA

algorithm unwrap phase by consuming half of the memory

than PUMA (12.05 MB) for both 32-bit and 64-bit systems.

Finally, the G-PUMA is a memory saving algorithm and

consumes very less memory, even if the profile is big or noisy

or 64 bit system.

F. Phase Unwrapping to Real Image

1) Area corresponds to Long’s Peak, CO

Fig 10(a) shows a phase image associated with noise (152 X

458 pixels) to be unwrapped. It was obtained from an original

absolute phase surface that corresponds to a (simulated)

InSAR acquisition for a real steep-relief mountainous area



34131

TABLE IV. MEMORY CONSUMPTION

Interferogram 32-BIT 64-BIT

G-

PUMA

(in

MB)

PUMA

(in

MB)

G-

PUMA

(in

MB)

PUMA

(in

MB)

High phase rate

Gaussian Hill

0.4603 0.4835 0.4603 0.8388

High phase rate

Gaussian Hill

with a quarter set

to zero.

0.6908 0.8958 0.6908 1.5540

Shear Ramp 0.6908 1.2745 0.6908 1.5540

High phase rate

Gaussian Hill

with a non-

vertical and

non-horizontal

aligned sector set

to zero.

0.6908 0.8958 0.6908 1.5540

Gaussian Hill

(128 X 128)

0.7544 0.9794 0.7544 1.6988

Gaussian Hill

(256X256)

3.0154 3.9250 3.0154 6.8055

Gaussian Hill

(512 X 512)

12.0594 15.7143 12.0594 27.2425

inducing, therefore, many discontinuities and posing a very

tough PU problem. This area corresponds to Long’s Peak,

CO, and the data is distributed with book [1].

The wrapped image is generated according to an InSAR

observation statistics, producing an interferometric pair; by

computing the product of one image of the pair by the

complex conjugate of the other and finally taking the

argument, the wrapped phase image is then obtained as in Fig

10(b). Fig. 10(c) shows a quality map (also distributed with

book [1]) computed from the InSAR coherence estimate (see

[1, Ch.3]) for further details). The unwrapping was obtained

using the approximate version of PUMA, with m=2. The

resulting phase unwrapped is “3-D” rendered for G-PUMA in

Fig 10(d) and for PUMA in Fig 10(e).True phase of the area

corresponds to Long’s Peak, CO is in Fig 10(f).

Both PUMA and G-PUMA have the same view as shown in

Fig.10 (d) & 10(e).Elapsed time of PUMA is 9.25 seconds

whereas G-PUMA unwraps the real image within 4.9 seconds.

G-PUMA outperforms PUMA with a speed increase of about

45% on a real image and the error norm is same for both

algorithms.

In order to unwrap the phase, the memory consumed by the

PUMA Algorithm is 4.17 MB of memory for 32-bit system

and 7.23 MB of memory for 64-bit system whereas G-PUMA

unwraps phase by consuming very less memory of 3.20 MB.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 12(a) Wrapped phase image obtained from ENVISAT, area

corresponds to Mexicali, Baja California, USA (b) Extracted image

(1000x1000) of (a).(c) Image in (b) unwrapped by G-PUMA. (d)

Image in (b) unwrapped by PUMA.(e) 3-D image of (c). (f) 3-D

image of (d). (g) Correlation map of (a). (h) Google image of (a).

So finally, G-PUMA shows a graceful performance

improvement in both time (45%) and memory (23 % for 32-

bit and 55 % for 64-bit systems) for real image.

2) ENVISAT DATA (Area corresponds to Mexicali,

Baja California )

In this experiment, the original phase data of Interferometric

Synthetic Aperture Radar provided by ENVISAT are

unwrapped by both the algorithms and results are noted. The

ENVISAT interferogram of size (6000 X 12000) is shown in

Figure12(a).Its area corresponds to Mexicali, Baja California,

USA. The mid region of the interferogram is the earthquake

occurred area and its associated fringes. To process the

interferogram on PC workstation we have extracted some part

Total ENVISAT Interferogram

1000 2000 3000 4000 5000 6000

2000

4000

6000

8000

10000

12000

Extracted 1000x1000 ENVISAT Interferogram

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

G-Puma solution

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000

Puma solution

100 200 300 400 500 600 700 800 900 1000

100

200

300

400

500

600

700

800

900

1000



34132

of the interferogram (1000 X 1000) and processed. As shown

in Figure 12(b), the extracted area corresponds to the tip area

of the earthquake and processed by both the algorithms.

Figure 12(c) is the unwrapped phase by the G –PUMA

Algorithm and Figure 12(d) is the unwrapped phase by the

PUMA algorithm. The 3-D rendered image of Figure12(c) and

12(d) are shown in Figure 12(e) and 12(f) respectively.

Elapsed time of PUMA is 246 seconds whereas the G-PUMA

unwraps the real phase within 63 seconds. The Correlation

map of the corresponding area is shown in the Figure 12(g)

and Google image of that area is shown in Fig 12(h).

Memory consumed by PUMA is 60MB for 32-bit and 104

MB for 64-bit systems, where as the memory consumed by G-

PUMA is just 46 MB for both 32 and 64 bit systems. G-

PUMA out performs PUMA with a speed increase of about

74% and memory of about 55 % for 64 bit and 30 % for 32-bit

systems.

VI. CONCLUSION

An unwrapping via Gridcut method abbreviated as G-PUMA

is proposed in this letter. This method performs very well for

both the Convex and Non Convex potentials. The proposed

method does the unwrapping using Gridcut Graph Cut as an

Optimization step. The advantage is significant performance

gain in speed and memory consumption, which renders our

method particularly useful for interferogram’s of large size.

Our experimental evaluation shows that the proposed method

can achieve good result to the simulation phase map and the

real interferometric synthetic aperture radar phase image.

Acknowledgment

The authors would like to express great thanks for the help

from Daniel Sýkora and Ondřej Jamriška[12], Professor J. M.

Bioucas-Dias [8] in the Instituto de Telecomunicações,

Instituto Superior Técnico, Lisbon and Gonçalo Valadão[8]

PhD student in Electrical and Computer Engineering at

ISTand special thanks to ENVISAT for providing the data of

El Major-Cupacah earthquake.

References

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dimensional phase unwrapping: theory, algorithms,

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[2] Graham, Leroy C. "Synthetic interferometer radar for

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[3] Ferraioli, Giampaolo, AymenShabou, Florence

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[13] Syakrani, Nurjannah, Tati LR Mengko,

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