integration Goldstein brunch cut algorithm [2, 3, 4 and 5],. Quality guided ..... New York:: Wiley,. 1998. [2] Goldstein, Richard M., Howard A. Zebker, and Charles L. Werner. "Satellite ... [21] Ghiglia, Dennis C., and Louis A. Romero. "Robust two- ...
International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 3 Issue 4, April - 2014
Two-Dimentional Phase Unwrapping Algorithm Based on Markov Random Field Mohammed Safy Moussa,Guang Ming Shi
Ahmed Saleh Amein
Xidian University Xian, China
Military technical college Cairo, Egypt
Abstract—Phase unwrapping is considered as the heart of the digital elevation model reconstruction. In this paper a new phase unwrapping algorithm is proposed based on Markov Random Field. Morkov Random Field is used to model the relation between the true phase and the wrapped phase. Efficient objective function is proposed to satisfy the smoothness of the true phase and the closeness between the true phase and wrapped one. The congruence function is used to achieve the closeness. Real and simulated images are used to prove the efficiency of the algorithm. The results show that the algorithm is efficient and gave a good result with high noise.
I.
INTRODUCTION
Random
Field;
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Keywords—phase unwrapping; Markov closeness; smoothness;standard deviation
Phase unwrapping is the critical step in the digital elevation model (DEM) generation and its accuracy depends on this step. Phase wrapping is the restoring of the true phase from the observed phase (modulo 2-π). In case of Phase unwrapping in one-dimensional, the problem can be avoided by calculating the phase difference between the two neighboring pixels and comparing them with π. If the phase difference between two adjacent pixels is less than π, the ideal solution would be to add 2 π and if the phase difference between two adjacent pixels is more than π, the ideal solution would be to subtract -2 π.
n n n 1, n n n 1 Where
n is the true phase and n
(1)
is the observed phase at
location n. By the same basic idea the two dimensional phase unwrapping can be solved, but it’s more complicated. The complications arise from the noise in the observed image, under sampling problem and its accumulative nature.
Phase unwrapping with filtering
Phase unwrapping without filtering
Fig.1. The noise effect on the phase unwrapping process
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When the phase unwrapping algorithm fails to remove the wrapping between two neighboring pixels this wrapping will propagate in the rest of the image in the range and azimuth direction. This ambiguity in the phase unwrapping process will affect the final result of the DEM. So the accumulative nature is essential in the phase unwrapping algorithm. There are two key solution of the phase unwrapping problem: 1) Phase model to express the true phase and 2) an efficient algorithm to recover the phase based on the model. There are two main categories of phase unwrapping algorithms; one of them is minimum norm algorithms which are based on the gradient between the unwrapped phases and the wrapped one. The other one is the path following algorithms which detect the good brunch cut between the residues and follow paths through the wrapped phases. D. Ghiglia and M. Pritt [1] presented most of the phase unwrapping methods and the problems associated with this problem. The path following algorithms works in a pixel base which computes the unwrapping phase from pixel to pixel. Due to the presence of some residues somewhere between the pixels inconsistencies in the phase unwrapping solution will occur. If the sum of the wrapped phase difference in a loop of four pixels is positive the residue takes positive polarity and vice versa. Once the residues are detected the brunch cuts are used to balance between the residues with different polarities. In this case while having different integral paths we will get the same unwrapping result. So the main goal is how to choose a good brunch set. There are four main approaches for path following integration Goldstein brunch cut algorithm [2, 3, 4 and 5], Quality guided path-following [6, 7, 8, 9, 10and 11], Mask cut algorithm [12, 13and 14] and Flynn’s minimum discontinuity approach [15, 16and 17] The minimum norm algorithms are global solution algorithms using the mathematical form of general minimum norm [18, 19, 20, 21 and 22]. It can be formulated as an unweighted or weighted phase unwrapping problem. These algorithms are based on minimizing the gradients between the wrapped and unwrapped phase. It’s effective in the case of noise but in general they need a very high memory.
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There are three main approaches for path following integration un-weighted Least Square phase unwrapping [23], weighted Least Square phase unwrapping [24] and minimum norm phase unwrapping [25]
likelihood function equals to the kronecker delta function of the observed phase. In [27] Hammersly-Clifford proved that MRF is equivalent to Gibbs distribution, so that: (4) p( ) exp( V ( , )
N
N
II.
Where V ( , N ) is the potential function, there are different
PROPOSED METHOD
In this work, Markov Random field [26, 27,28and29] is used to model the true phase image. The ability to take contextual information into account is the major advantage of MRF [30].There are two presuppositions [31] that could be considered as a priori acknowledgment in phase unwrapping and make the data distribution suitable for MRF. One of them is the smoothness of local area topography and the other one is the observed phase is considered as the true phase but polluted by the noise, under these conditions the similarity between the images appears.
[32]
where V (m, n ,m, n 1 )
m, n ,m, n 1
p
,
1
2,
V (m, n ,m, n 1 ) log coshc(m, n ,m, n 1 ) where c is a
specified parameter [33], the standard deviation of all the adjacent pixels is used [34] and in [35] the potential function is performed by calculating the square of all the differences between the four adjacent pixels. B. Phase Unwrapping Algorithm In order to achieve an exact potential function formulation, two factors must be considered: the smoothness of the true phase and the closeness between true phase and observed phase. The smoothness condition is achieved when the following two inequalities are considered. (5) and
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A. Phase Model Markov Random field is used to model the true phase image, in which the observed phase Ψm,n and the true one at each pixel are considered as a random variables. The following figure presents the dependency of the true phase on the observed phase.
potential functions have been studied. In [26] the potential function is the square of the differences between previous two pixels and the origin pixel, generalized p-Gaussian model
xm, n
ym, n
In The path following algorithms, if the sum of the wrapped phase difference in a loop of four pixels is vanished that means there is no residue. Based on this idea our potential function is developed. If the sum of the unwrapped phase difference between the center pixel and the adjacent four pixels is vanishes, that means the phase unwrapping condition is satisfied.
Fig.2. The relation between the observed phase and true one [31]
In figure (2), the filled nodes are the observed phases (wrapped phase) while the empty one are the true phase (unwrapped phase). The links represent the phase’s dependency between the two sites. The following equation represents the links between the phases. (2) m,n m,n 2k the sum of the wrapped phase difference=0
Where k is the ambiguity factor In MRF maximum a posteriori (MAP) is used to estimate the true phase.
arg max p( / ) arg amxp( / ) p( )
(3)
Fig.3.The proposed idea
p( ) arg min ( (m, n m, n 1 ) m, n
(m, n m 1, n ) (m, n m, n 1 ) (m, n m 1, n ) )
Where p( / ) is the posterior probability, p( ) is the prior probability and ( / ) is the likelihood function [26].
(6 )
The previous equation is achieved under the following conditions:
According to the fact that the possibility of observed phase with the knowledge of the true phase is always one, the
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The structure of the first order neighborhood system
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m, n m, n 1
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International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 3 Issue 4, April - 2014
m, n m 1, n
(7)
m, n 1 m, n m 1, n m, n To achieve the closeness property between the true phase and the observed phase, the congruence is used. Congruence means if the true phase is rewrapped it will be the same as the observed one [36].
a tan 2( m, n ,m, n ) ( (m, n m, n 1 ) p( ) arg min( (m, n m 1, n ) ) m , n m , n (m, n m, n 1 ) (m, n m 1, n ) )
(8)
To overcome the disadvantages of the deterministic algorithms, the stochastic algorithms are used which generate and use random variables. Simulated annealing, Tabu search and genetic algorithm are the most common applicable algorithms. The proposed objective function is non linear, constrained and static. The Goldstein algorithm is used to estimate the initial solution, thus a deterministic algorithm is used to estimate the optimum solution. Matlab’s fmincon is used to minimize the nonlinear constrained proposed objective function.
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Taking equation (2) inconsideration the equation (8) will be: a tan 2( m, n , m, n 2km, n ) (9) ( m, n m , n 1 ) ( k m , n k m , n 1 ) 2 p( ) arg min m, n m 1, n ) (km, n km 1, n ) 2 m, n m, n m, n m, n 1 ) (km, n km, n 1 ) 2 m, n m 1, n ) (km, n km 1, n ) 2
The optimization algorithms are divided into two main categories, deterministic and stochastic. The deterministic algorithms depend on the linear algebra because they are based on the gradient computation or the Hessian computation of the variable’s response. Its solution convergences faster than that of the stochastic one, it is also precisely and replicable, on the other hand the deterministic optimizations have some disadvantages: A. The solution is a local optimum solution and not a global optimum solution. B. The local optimum solution needs an initial solution which is not available in many cases. C. It is normally not easy to get the upper limit of the computation time.
To solve the optimization function in (9), there are many solutions [37] but all of them are computational cost for realistic images [39]. One of the path following algorithms is used to estimate the initial phase unwrapping solution. This initial solution is used to estimate the final phase unwrapping solution through the equation (9)..Noise variance values from 0.1 to 1 are added to the input observed image. The standard deviation between the true phase and the observed one is used to evaluate the phase unwrapping results [39]. (10) 1 M 1 N 1
2
III.
RESULTS
The interferogram of Las Vegas in USA [78] and a simulated image are used in the analysis of the algorithm. Goldstein algorithm is used to calculate the initial solution for the phase unwrapping. The results are compared with R.chen algorithm [31], which based on the same basic idea.
A. Simulate interferogram Matlab program is used to perform a phase image to act as an interferogram, the phase of this image is between π and – π and its size is 128x128.
[ (m, n) (m, n) ]
2
MN
m0 n 0
Where:
1 M 1 N 1 [ (m, n) (m, n)] MN m 0 n 0
(11)
1) Objective function optimization Mathematical optimization detects the minimum or maximum of the objective function. To choose the best algorithm, the objective function should be well understood as the following. A. The objective function is linear or non linear. B. The variables are static or dynamic, in other words they do change with time or not. C. The objective function is constrained or unconstrained.
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Fig.4. The simulated interferogram
The following table presents the standard deviations between the original image and the unwrapped image.
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International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 3 Issue 4, April - 2014
TABLE I. The standard deviation results for the simulated image Noise variance 0 0.2 0.6 0.8 1
Goldstein 0.0491 0.0511 0.0605 0.0964 0.0991
variance R. Chen algorithm 0.0498 0.0495 0.0517 0.0965 0.0973
Proposed algorithm 0.0190 0.0197 0.0200 0.0450 0.0436 (c)Noise variance=0.6
In figure (5), the scattering points represent the efficiency of the propose algorithm. The standard deviation between the original image and the unwrapped one in the proposed algorithm is less than that of the Goldstein algorithm and also with respect to R.chen algorithm [31].
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(d)Noise variance=0.8
Fig .5. The phase unwrapping evaluation using simulated interferogram
The following figure presents the unwrapping phase results of the proposed algorithm and R.Chen [31] algorithm at different values of the noise variance.
(e)Noise variance=1
Fig.6.The phase unwrapping results at different values of the noise variance for the simulated interfergram. On the left the proposed algorithm and on the right R.Chen algorithm
B. Real interferogram The interferogram of Las Vegas in USA [40] is used in the analysis.
(a)Noise variance=0
Fig.7. Las Vegas interferogram (b)Noise variance=0.2
The following table presents the standard deviations between the original image and the unwrapped image.
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TABLE II. The standard deviation results for the Las Vegas interferogram
Noise variance
Goldstein
0 0.2 0.6 0.8 1
0.0240 0.0259 0.0282 0.0295 0.0315
variance R. Chen algorithm 0.0166 0.0143 0.0167 0.0169 0.0195
Proposed algorithm 0.0123 0.0135 0.0163 0.0167 0.0182
(c)Noise variance=0.6
In figure (8), the scattering points represent the efficiency of the propose algorithm. The standard deviation between the original image and the unwrapped one in the proposed algorithm is less than that of the Goldstein algorithm and also with respect to R.chen algorithm [31].
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(d)Noise variance=0.8
(e)Noise variance=1
Figure .9 The phase unwrapping results at different values of the noise variance for real interfergram. On the left the proposed algorithm and on the right R.Chen algorithm
Fig.8. The phase unwrapping evaluation using Las Vegas interferogram
The following figure presents the unwrapping phase at different values on the noise variance.
CONCLUSION
An efficient algorithm is proposed, by which the smoothness in the true phase and also the closeness between the unwrapped phase and wrapped one is achieved. The standard deviation between the true phase and the original phase is used to detect the efficiency of the proposed algorithm. The results show that the standard deviation between the original image and the unwrapped one in the proposed algorithm is less than that of the Goldstein algorithm and less than that of R.Ceh algorithm [31],which used the same basic idea.
(a)Noise variance=0
(b)Noise variance=0.2
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