# Hybrid Genetic Algorithm using a parametric method to solve

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"Hybrid Genetic Algorithm using a parametric method to solve"

```					      Hybrid genetic algorithm using a parametric method to solve the two-
dimensional phase unwrapping problem

Salah Karout, Munther A. Gdeisat, David R. Burton, Michael J. Lalor
CEORG, General Engineering Research Institute,
Liverpool John Moores University, UK,
s.a.karout@2004.ljmu.ac.uk

Abstract                                                   range – π       to + π. Thus, in order to retrieve the
continuous form of the phase map, an unwrapping step
A hybrid genetic algorithm is proposed to solve the
has to be added onto the phase retrieval techniques
two-dimensional phase unwrapping problem by
[Cusack et al., 1995]. This unwrapping step is not a
minimizing the Lp-norm between the unwrapped and
straightforward technique because of the presence of
wrapped phase gradients. The unwrapped phase is
noise, object discontinuity and the violation of
approximated by estimating the parameters of a
Shannon’s law due to undersampling in real wrapped
function chosen by the genetic algorithm that can
phase maps. As a result, many phase unwrapping
achieve the global minimum of the Lp-norm. Different
algorithms have been developed to solve this problem.
predefined functions are used on the basis of the object
However, the variety of forms, shapes and densities of
characteristics being unwrapped whether continuous
noise that might be found in real wrapped phase maps
and discontinuous object. The nth-grade polynomial is
makes the problem of phase unwrapping complex and
one of the functions used to approximate the
difficult to solve, even given the signifficant amount of
unwrapped phase. The hybrid genetic algorithm uses a
research effort expended to date and a large number of
polynomial weighted least-squares phase unwrapping
exiting phase unwrapping algorithms.
solution as an initial approximation of the unwrapped
phase map to speed up convergence to global optima.
1.1 Phase Unwrapping
The algorithm is robust in unwrapping very noisy
Phase Unwrapping (PU) is a technique used on
wrapped phase maps. It is computationally efficient
wrapped phase images to remove the 2π discontinuities
and very fast. The performance of the algorithm is
embedded within the phase map. It detects a 2π phase
tested using simulated and real wrapped phase maps
jump and adds or subtracts an integer offset of 2π to
and results are compared to that of the existing two-
successive pixels following that phase jump based on a
dimensional Lp-norm phase unwrapping algorithm.
threshold mechanism. The threshold mechanism states
that if the phase difference between two successive
Key words
pixels in a path {P} as in Eq. (1):
Phase unwrapping, Genetic Algorithm, weighted
least-squares and polynomial regression.
∆Φ ( p i ) = Φ ( p i ) − Φ ( pi −1 )        (1)
1 Introduction
is greater than +π, subtract a 2π offset to all successive
Many digital image processing techniques may be         pixels in the path. However, if the phase difference is
used to extract phase distributions from images (e.g.,     less than –π, add a 2π offset to all successive pixels in
fringe pattern or magnetic resonance scan images or        the path.     Then, by locating all discontinuities in the
Synthetic Radar Interferometry images) in order to         wrapped phase map, the phase at every pixel will
obtain the information embedded within the phase map       change by an integer k multiples of 2π depending on
(e.g., height, magnetic field, velocity and                the pixel position in the unwrapping path. This can be
displacement). Such techniques include Fourier fringe      summarised by the wrapping operator in Eq. (2):
analysis, phase stepping and the wavelet transform
method [Huntley et al., 2001]. These methods of                     W [Φ ( pi )] = Φ ( p i ) + 2πk ( pi )
calculating the phase distribution suffer from one
k ( pi ) ∈ Ζ                                (2)
disadvantage, which is the use of the arctangent
operator to extract the phase distribution. The
arctangent operator produces results wrapped onto the
where    − π ≤ W [Φ ( pi )] ≤ +π . The wrapping
operator could be modified to specify the gradient of
the corrected phase difference between two successive                   have the advantage that it they are more noise tolerant
pixels in the unwrapping path and it is stated in Eq. (3):              and they achieve the global smoothness of the
unwrapped solution.
The unweighted and the weighted least-squares
∇Φ ( pi ) = W [Φ ( pi ) − Φ ( pi −1 )]              (3)   methods are well defined mathematicly and equivalent
to the solution of Poisson’s partial differential equation
A simple global method of phase unwrapping is to                        which leads to a spare linear equation that can be
minimize the distance between the phase gradient                        solved by any iterative method. However, these
estimate (unwrapped phase) and the true gradient as                     methods generate a very large number of linear
presented in Eq. (4) [Ghiglia et al., 1998]:                            equations to be solved equivelant to the total number of
M − 2 N −1                                        p         pixels in the phase map. They have a very large
ε = p
∑∑ψ
i=0 j =0
i + 1, j   − ψ i , j − ∇φ    x
i, j
computational time and their convergence rate is very
critical to a successful solution. Moreover, weighted
(4)   least-squares requires weights to achieve better results
M −1 N − 2                                         p
than the unweighted counter part. These weights are
+   ∑∑ψ i=0 j =0
i , j +1   − ψ i , j − ∇φ    y
i, j                    user defined weights generated from quality-maps used
to isolate corrupted areas with residues by masking
them out of the wrapped phase data to dimenish their
where i and j are indices of the pixel location in the                  effect on the unwrapped solution. A drawback to these
image respectively,                                                     two methods is if some residues where not masked out,
φi, j is the given wrapped phase,                              they will cause the unwrapped phase to be severely
corrupted depending on the density of the unmasked
ψ i, j is the approximated unwrapped phase,               residues. A more advance method developed by
∇φix j = W [φi +1, j − φi , j ]                           Ghiglia et al. [Ghiglia et al., 1998] is Lp-norm which
,
uses similar methods like the two previous least square
and   ∇φiy j = W [φi , j +1 − φi , j ]
,
methods to solve the phase unwrapping problem.
However, this method does not compute the minimum
are the wrapped phase gradients in x and y
L2-norm but the general minimum Lp-norm. In essence,
directions respectively,
by computing the minimum Lp-norm where p≠2; this
M and N are the size of the phase map image.
method can generate data dependent weight unlike the
Two major classes of phase unwrapping algorithms
weighted least-square method. The data-dependent
are path-following and least-square methods. The path-
weights can eliminate iteratively the presence of the
following methods deal with the problem of residues
residues in the unwrapped solution. Unfortunetly, this
directly by identifying the residues and eliminating
method is more robust than the previous mentioned
their presence in the phase map by balancing their
least-squares method but it is more computationally
polarities and branch-cutting them from the phase map.
intensive.
Once the residue’s effect eliminated, phase unwrapping
In this paper, a global phase unwrapping algorithm is
can take any path through the phase map and branch-
proposed that uses a genetic algorithm to estimate the
cut act as barriers to unwrapping corrupted areas in the
parameter coefficients of an nth-order polynomial used
phase map. There have been many path-following
to create the unwrapped phase solution that minimizes
algorithms developed which vary in the way this
the Lp-norm error between the gradient of the solution
method is implemented.
and the gradient of the wrapped phase map. This
On the other hand, least-squares methods are
method is similar in concept to least-square and Lp-
completely different than path-following methods.
norm phase unwrapping methods developed by Ghiglia
They are divided into three different types: unweighted
et al. [Ghiglia et al., 1998] except it does not rely on
least-squares, weighted least-squares and Lp-norm
the wrapped phase data to construct the unwrapped
methods. These methods in general minimize up to a
solution. However, it uses a polynomial to construct the
certain degree (least-square to the 2 degree order and
unwrapped surface solution. In essence, it has a major
Lp-norm raised to the p degree order) the difference
advantage on least-squares and Lp-norm methods by
being free from any residue disturbances generated
of the unwrapped solution in both x and y direction.
from the wrapped phase map. In other words, the
However, these methods do still indirectly deal with the
residues embedded in the wrapped phase cannot affect
residue problem because their solution is obtained by
the unwrapped phase solution. The reason is because
integrating over the residues to minimize the gradient
the wrapped and the unwrapped phase maps are
differences [Ghiglia et al., 1998]. Least-square methods
completely independent from each other. The only
relation between the wrapped and the unwrapped phase          them more possibility of moving on to the next
maps is the Lp-norm error minimization. The other             generation,
advantage of the proposed algorithm is that it generates      • Crossover, which is responsible for producing new
noise-free unwrapped phase map and achieves a global          chromosomes (offsprings) from the original
smoothness constraint. No matter how much noise and           chromosomes (parents),
residues are embedded in the wrapped phase map, this          • Mutation, applies deliberate changes to a gene at
method is capable of unwrapping with a certain                random, to keep variation in genes and increasing the
minimum degree of error which cannot be achieved by           probability of not falling into a local minimum
other mentioned algorithms as long good data are still        solution. It involves exploring the search space for new
present in the wrapped phase map.                             better solutions.
The paper is organized as follows: a general
description of a simple genetic algorithm is given in          Randomly generate initial population, size Pop,
section 2. Then, the proposed algorithm is explained           Evaluate the fitness of every chromosome,
thoroughly in section 3. The proposed algorithm is then
tested on both computer-simulated and real wrapped             Do
phase data results. Its performance compared to two            For chromosome = 1 to Pop/2
well known global phase unwrapping algorithms; a               Use selection operator to select to
modified weighted multi-grid and synthesis algorithm           chromosomes,
and Lp-norm two-dimensional phase unwrapping                   If rand_no. < Px
algorithm is given in section 4. Finally, in section 5 a                Apply the crossover operator,
conclusion is presented of the work done in this paper.        If rand _no. < Pm
Apply the mutation operator,
2 Genetic Algorithm                                            End
A Genetic Algorithm (GA) is an artificial                   Replace the new population with the old
intelligence method that mimics the evolution of genes         population,
throughout human generations, leading to better ones.          Evaluate the fitness of all new chromosomes,
A Genetic Algorithm is a stochastic search technique           Preserve the most-fit chromosomes so far and
that uses a global random or problem-specific search           copy into the new population,
controlled by a set of different operators.6 It relies on a
probabilistic search mechanism and it uses self-                Repeat while not converged or not at maximum
adapting strategies for searching based on random             Figure 3: shows a simple genetic algorithm,
iteration.
exploration of the solution space coupled with a
memory component. This enables the algorithm to               Figure 1: A pseudo-code of a general genetic
learn the optimal search path from experience [Willi-         algorithm; where Px and Pm are the probability of
Hans et al., 2002]. Genetic algorithms are specifically       performing crossover and mutation respectively.
designed to solve non-deterministic polynomial
problems called NP-hard problems which involves               3 HGA for Parameter Estimation
large search spaces containing multiple local minima            The proposed algorithm as mentioned previously
[Willi-Hans et al., 2002]. It has also been applied to        relies on estimating the parameters of an nth order-
many combinational optimisation problems and has              polynomial to approximate the unwrapped surface
proved its robustness and speed.                              solution from the wrapped phase data. The proposed
Genetic algorithm iterations consist of a set of           algorithm uses a genetic algorithm to obtain the
chromosomes where every chromosome represents a               coefficients of the polynomial that best unwrap the
candidate solution for the problem to be solved.              wrapped phase map. However, by providing the genetic
Chromosomes consist of a number of genes that can be          algorithm with an initial population created by
considered as variables to the problem. Manipulating          randomly choosing a number between two limits will
these genes (variables) will result in creating new           cause the algorithm to take a very long time to
solutions. To evaluate how much the solution has been         converge to the global optimum solution. A faster way
improved, a fitness function (evaluation function) is         was achieved by obtaining an initial solution that is
tailored to the problem.                                      used to create the initial population.
Such a method uses three natural techniques:                 In this proposed algorithm, the complexity of the
• Natural selection, which allows the best genes (e.g.,       problem relies on the order of the polynomial used to
healthy in human terms) to be selected thus giving            reconstruct the unwrapped surface solution. By
increasing the order of the polynomial, more precision
and a lower minimum Lp-norm error are achieved. The                    stochastic search to achieve the global optimum
number of coefficients of the polynomial also increases                solution. The method that is used to create the initial
with the order of the polynomial.                                      population will determine the speed of convergence to
The proposed algorithm is summarised in Fig. 2:                        an optimum solution, as well as the size of the
population (the number of chromosomes in the
•     Calculate the quality map of the                                population). In essence, the size of the population
wrapped phase map using maximum                                 depends greatly on the method used to create the initial
phase gradient quality map [Ghiglia et                          population.
al., 1998],                                                       Moreover, as the size of the population increases, the
•     Unwrap the wrapped phase using quality                          complexity and memory usage increases, but on the
guided algorithm [Ghiglia et al., 1998],                        other hand, the tendency to converge to a global
•     Surface-fit the unwrapped solution with                         optimum solution also increases.
a polynomial using weighted least-                                Thus, it is required to have an initial population that
square multiple regression controlled by                        has the necessary information and gene possibilities for
the quality map weights,                                        the GA to converge, without the huge amount of
chromosomes in a population.
•     Coefficients of the surface-fitted
The initial population is generated by creating an
polynomial are given to the genetic
initial solution using one of the simple phase
algorithm as an initial solution to lower
unwrapping algorithms such as ‘Quality guided phase
execution time,
unwrapping algorithm’. The initial solution is
•     Genetic Algorithm minimizes the Lp-
approximated using a ‘polynomial Surface-fitting
norm error between the gradient of the
weighted least-square multiple regression’ method. By
polynomial unwrapped surface solution
multiple regression method, the initial coeficients of
and the gradient of the original wrapped
the polynomial will be generated. These coeficients are,
phase map.
then, inserted into the first chromosome of the genetic
Figure 2: A summary of the proposed parameter                          algorithm initial population.
estimation genetic algorithm.                                            This method for creating initial solution is quite
powerful and gives the GA a good start to reach
3.1 Coding the Phase Unwrapping problem in GA                          convergence. It is more intelligent and problem-
Syntax Form                                                            specific than random initialization of polynomial
Any optimisation problem using a GA requires the                     coeficients. It also gives the GA the option of fewer
problem to be coded into GA syntax form, which is the                  chromosomes in a population and speeds up the
chromosome form. The chromosome can be used to                         convergence of the GA to an optimal solution.
represent a graph or an equation or a system. In this
problem, the chromosome consists of a number of                        3.3 Polynomial Surface-fitting Weighted Least-
genes where every gene correspond to a coefficient in                  Square Multiple Regression
the nth-order surface fitting polynomial as described int                Surface-fitting using polynomials is a very well
Eq. (5) and Fig. (3).                                                  established subject used to best fit a polynomial to set
of data. One way to surface fit a polynomial to a set of
f ( i , j ) = a 0 + a 1i + a 2 j + a 3 i 2 +                          data is by weighted least-square multiple regression.
(5)   This method of regression minimzes the sum of
a 4 ij + a 5 j 2 + K + a ( n +1 )( n + 2 ) j n                         residuals (least square error or L2-norm) controlled by a
2                                   set of weights. It involves smoothing of the data or
identifying an apparent trend in the data. The weights
used define how good the data to be fitted is and how
a0      a1       a2       a3                   a(n+1)(n+2)/2         much they can contribute to the fitting of the
polynomial to the data. The number of coefficients of
Figure 3: Coding scheme of the coefficients of the nth-                the polynomial specifies the size of the matrices used to
order surface fitting polynomial into the chromosome                   solve the least-square problem.
syntax form.                                                             The best fitted polynomial surface has the minimum
weighted least square error defined in Eq. (6):
3.2 Initial Population                                                                NM                                2

A GA requires an initial population of chromosomes                            S =   ∑      w i [z i − f ( x i , y i ) ]   (6)
where each chromosome represents a possible solution.                                 i =1

From this initial population, the GA starts using a                    where wi is the weight at at pixel i,
z i is the data pixel to be fitted,                      This can be solved by Gaussian Elimination to
calculated the values of the coeficient of the best fit
f ( xi , y i ) is the pixel value evaluated using        polynomial surface that can minimises the weighted
the polynomial in Eq. (5) at coordinate                   least-square error.
xi & y i in the x and y direction respectively.
3.4 Generating Initial Population Based on the
The coeficients of the nth-order polynomial are the              Initial Solution
unknowns need to be evaluated to construct the surface.                Once the initial solution coeficients are calculated,
However, the known parameters are the data to be                    the coeficients of the initial solution are inserted in the
fitted z i and the two control points defined by the                first chromosome in the initial population. The rest of
x i & y i coordinates of the polynomial surface in the x           the population is generated using the following method:
and y directions. To obtain the weighted least-square                  For every gene in the chromosome a random number
error, the unknown polynomial coeficients must yeild                is added to the value of the gene as in Eq. (11):
zero first derivatives.
ai = ai + δ(rand-0.5)          (11)
where     ai is the coefficient parameter stored in gene
⎧ dS        NM

⎪ da = 2 ∑ w i [z i − f ( x i , y i ) ] = 0                                   ‘i’,
⎪ 0               i =1                                                        δ is the fraction number which limit the
⎪ dS            NM                                                            generated random number to be smaller than
⎪        = 2 ∑ w i [z i − f ( x i , y i ) ] = 0               (7)             the value ai,
⎪ da 1
⎨                i =1
Rand is a random number generated between
⎪M                                                                            the values [0, 1].
⎪                          NM
dS
⎪                      = 2 ∑ w i [z i − f ( x i , y i ) ] = 0       3.5 Fitness Evaluation
⎪ da ( n +1)( n + 2 )      i =1
⎪
⎩           2
To find the global optimum solution to the parameter
By expanding Eq. (7) will result in the set of linear               estimation Lp-norm phase unwrapping problem, the
equations presented in Eq. (8):                                     quality of the solution must be evaluated at every
generation in order to inform the genetic algorithm of
how good its current solution is at each stage. The
⎧ NM       NM                   NM           NM

⎪a0∑ i +a1∑ i xi L +a(n+1)(n+2) ∑ i yin =∑ i zi
w       w L                  w            w                  evaluation will increase the knowledge of the GA of
how good the quality level of the solution. This can be
⎪  i=1     i=1            2     i=1          i=1
achieved by using a problem-specific fitness function
⎪ NM         NM                    NM            NM
⎪a0∑ i xi +a1∑ i xi L +a(n+1)(n+2) ∑ i xi yi =∑ i xi zi
specified in Eq. (1). The genes of a chosen
⎪      w         w2 L                  w n            w
(8)    chromosomes are substituted as coeficients in Eq. (2) to
⎨ i=1        i=1             2     i=1            i=1
⎪                                                                   evaluate the approximated phase value at coordinate (i,
M                                                                  j). The obtained phase is subtracted from the adjacent
⎪
⎪ NM n NM n                           NM            NM              pixel approximated phase value to calculate the
⎪a0∑ i yi +a1∑ i xi yi L +a(n+1)(n+2) ∑ i yi2n =∑ i yinzi
w          w     L                  w            w           approximated unwrapped phase solution gradient. It is
⎪ i=1
⎩             i=1               2      i=1          i=1             then subtracted from the gradient of the wrapped phase
in the x and y direction. Then, the error is evaluated in
Eq. (8) can be simplified to the following matrix form:             the Lp-norm sense.

XA = Z                         (9)             ψi, j = f (i, j) = a0 + a1i + a2 jK+ a(n+1)(n+2) jn   (12)
where A is the matrix representing the polynomial                                                              2
coefficients,                                              where ak are the parameter coefficient that will be
is the matrix representing the left side of Eq.                   estimated by the genetic algorithm to
(8),                                                              approximated the unwrapped phase that can
Z is the matrix representing the right side of                    achieve the minimum Lp-norm, i and j are
Eq. (8).                                                          indices of the pixel location in the image
By reordering the matrices, the coeficient matrix can be                   respectively.
evaluated from Eq. (10):

A = X −1 Z                             (10)
3.6 Selection Operator                                                                                   Fitness
The selection operator is an important step in a
genetic algorithm. This reproduction operator selects                                                    100
Parent
the fittest chromosomes for the current population and           Chromosomes                             400
copies them to the new chromosomes in the next
generation. It applies the natural concept of evolution,                  Crossover
which states that: “the most fit individual survives to                   point
70
the next generation” [Willi-Hans et al., 2002]. Selected
parent chromosomes must be suitable for crossover                                                        200
(mating) to generate new child chromosomes, i.e., new
Child                                   100
solutions that have a high tendency to be better (more           Chromosomes
fit) than their parent chromosomes. The selection                                                        70
operator is required to be intelligent and problem-
specific in order to speed convergence and to avoid          Figure 4: The basic steps in the greedy 2-point
trapping the solution in local minima. It is required that   crossover.
the selection operator avoids causing a loss of
population diversity and also avoids the ineffective         3.8 Mutation Operator
execution of crossover operation[Willi-Hans et al.,            Mutation operator applies deliberate changes to a
2002].                                                       gene at random, to keep variation in genes and
increasing the probability of not falling into a local
3.7 Greedy 2-point Crossover                                 minimum solution. It involves exploring the search
Crossover is a natural operator used to generate new       space for new better solutions. This proposed operator
chromosomes from original chromosomes. It is an              uses a greedy technique which ensures only the best fit
important operator in genetic algorithm because it           chromosome is allowed to propagate to the next
introduces diversity into chromosomes. It is capable of      generation.
combining schemas (important genes) located in the             This operator can be summarized as follows:
original chromosomes. It avoids destroying the                    • Choose a chromosome from the current
schemas in the original chromosomes, in essence, good                 population at a probability; Pm,
schemas are conserved and propagated into the new                 • For every gene in the chromosome a random
chromosomes. Thus, it has more tendencies to create a                 number is added to the value of the gene using
better chromosome than weak one. A greedy method                      Eq. (11).
was also used in this crossover operator to ensure only
best fit chromosomes are allowed to propagate into the       3.9 Phase Matching
new generation.                                                This method matches the phase of the wrapped phase
The greedy 2-point crossover is summarized as              with approximated unwrapped phase to establish the
follows:                                                     best representation of the unwrapped phase. The phase
•     Choose two chromosomes from the initial           matching step extracts the small details embedded in
population for mating with a crossover             the wrapped phase data which was lost in the global
probability; Px,                                   phase unwrapping. This step is performed using Eq.
• Choose randomly two crossover points,                 (13).
• Swap the genes lying between these two                                         ⎡ 1
points of one the chromosomes with the genes
ξ ( p ) = Φ ( p ) + 2 πρ ⎢    (Φ ( p ) − Ψ ( p ) )⎤
⎥
(13)
⎣ 2π                     ⎦
lying between the same points in the other           Where ξ ( p ) is the phase matched unwrapped
chromosome; and vise versa,
phase,
• Evaluate the fitness of both new generated                      p is the pixel position in the phase map,
chromosomes,
Φ ( p ) is the given wrapped phase,
• Compare the fitness of the new generated
chromosomes with that of the original                       Ψ ( p ) is the approximated unwrapped phase,
chromosomes,                                                ρ [] is a rounding function defined by
.
• Choose the two best fit chromosomes from the
original and the new chromosomes to be                               ⎣      ⎦
ρ [t ] = t + 1 2 for t ≥ 0 and
added into the new population.                                       ⎣      ⎦
ρ [t ] = t − 1 2 for t < 0 .
4 Results and Discussion
The proposed algorithm is tested on two kinds of
wrapped phase maps; simulated and real phase maps, to
verify the performance of the proposed algorithm. The
results were also compared with a very well known
global phase unwrapping algorithms developed by
Ghiglia et al. called ‘Lp-norm two-dimensional phase
unwrapping algorithm’. The results of all these stated            (a)                       (b)
algorithms were executed on a Pentium IV- 3.0 GHz
computer. The order of the polynomial fitted was
chosen experimentally depending on the object surface
complexity.

4.1 Computer Simulated Results
The proposed algorithm was tested on computer
simulated object with high noise and the result was
compared with that of the Lp-norm algorithm and the               (c)                       (d)
weighted least squares algorithm. The wrapped phase      Figure 6: Simulated noisy object 128x128 (a) original
map and the rewrapped result of all the algorithms are   3d-suface, unwrapped phase map using (b) Lp-norm
presented in Fig. 5. The proposed algorithm best         algorithm, (c) weighted least-square algorithm, (d) the
matches the original wrapped phase with an advantage     proposed algorithm.
of smoothing the noise embedded in the wrapped phase
map. The unwrapped surface is presented in Fig. 6.       4.2 Experimental Results
The proposed algorithm was also implemented on a
real wrapped phase map generated from Interferometric
Synthetic Aperture Radar (IFSAR) data [Ghiglia et al.,
1998]. The IFSAR wrapped phase map and the
rewrapped result of the Lp-norm and the proposed
algorithm are presented in Fig. 7.

(a)                      (b)

(a)                       (b)

(c)                      (d)

Figure 5: The simulated noisy object 128x128 (a)
wrapped phase map, rewrapped phase map using (b)
Lp-norm algorithm, (c) weighted least-square
algorithm, (d) the proposed algorithm.
(c)                    (d)
Figure 7: (a) a 512x512 noisy IFSAR wrapped phase,
rewrapped phase map using (b) Lp-norm algorithm, (c)
weighted least-square algorithm, (d) the propsoed
algorithm.
The initial solution is achieved by unwrapping using a
simple unwrapping algorithm and estimating the
parameters of the polynomial using weighted least
squares multiple regression. The parameters of the
initial solution are then given to the genetic algorithm
as an initial solution.
The algorithm was then tested on simulated and
(a)                   (b)                         experimental data and it proved to be efficient and
robust. The comparison of performance of this
algorithm was made with powerful established phase
unwrapping algorithms such as the Lp-norm. Based on
the rewrapping of the solution, the proposed gave better
result best matched the original wrapped phase map.

Reference
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map which minimizes the difference between the                      (23), 3901-3908.
unwrapped phase gradient and the wrapped phase             Willi-Hans, S. (2002). The Nonlinear Workbook.
gradient. The genetic algorithm in this proposed                    (London: World       Scientific    Publishing).
method uses an initial solution to speed convergence.

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