a combination of particle filter, matrix pencil and ... - Semantic Scholar
A COMBINATION OF PARTICLE FILTER, MATRIX PENCIL AND REGION GROWING TECHNIQUES FOR PHASE UNWRAPPING IN SAR INTERFEROMETRY Juan J. Martinez-Espla, Tomas Martinez-Marin, Juan M. Lopez-Sanchez and J. David Ballester-Berman Signals, Systems and Telecommunication Group, EPS, University of Alicante, P.O. Box 99, E-03080 Alicante, Spain Tel/Fax: +34 96590 9811/9750. E-mail: [email protected], [email protected], [email protected], [email protected]
ABSTRACT This work presents an improved InSAR phase unwrapping (PU) method based on a combination of a particle filter, a region growing technique and a matrix pencil based local slope estimator. The better performance of this new solution when compared against some representative traditional methods and the previous particle filter PU approaches is justified and illustrated with results obtained from synthetic and real data. Index Terms— Phase unwrapping, SAR interferometry, state space, particle filter, matrix pencil, region growing. 1. INTRODUCTION In previous works, the authors presented some phase unwrapping (PU) algorithms based on the use of state space techniques, as the grid-based filter [1] and the particle filter (PF) [2]. Among all of them, the path following version of the Particle Filter Phase Unwrapping algorithm (2PFPU) was demonstrated in [2] to be a robust phase unwrapping tool. This algorithm combined a particle filter, a path following search strategy [3] and a local slope estimator based on the mode of the power spectral density (PSD) [4]. In the present work, an improved phase unwrapping solution is proposed. Like the 2PFPU algorithm, this solution also makes use of a particle filter. However, it incorporates two important differences. First, an optimized region growing strategy has replaced the path following one present in the 2PFPU. As described later, the region growing approach used here is efficient, since the unwrapping of one pixel per iteration is guaranteed and optimum, since only the pixel whose cost function is the best among all regions will be unwrapped. Second, the local slope estimator based on the matrix pencil (MP) has replaced the one based on the mode of the PSD, since more accurate local phase slope predictions are expected. The new algorithm is more robust than previous ones because it can unwrap zones that other PU algorithms simply mask or fail to unwrap. It will be illustrated in this work with some examples. The text is organized as follows. The optimized region
978-1-4244-3395-7/09/$25.00 ©2009 IEEE
growing technique is presented in Section 2. Section 3 justifies the use of the matrix pencil based local slope estimator. Then, Section 4 illustrates the performance of the new method with some examples. Finally, conclusions are drawn in Section 5. 2. OPTIMIZED REGION GROWING STRATEGY 2.1. Introduction The path following method used in [2] can be considered a mono-seed region growing technique since it grows from a single seed of the interferogram. As a consequence, sometimes there will be no other chance than crossing low quality areas (i.e. containing high density of residues), hence increasing the risk of introducing unwrapping errors and its propagation. On the contrary, the region growing technique introduced in [5] starts from different seeds and the corresponding areas grow simultaneously by following a pattern of thresholds. At every step, growth rings are defined and every candidate following the threshold criteria will be unwrapped. When no pixel is unwrapped at a step, the thresholds are relaxed and the process continues. As a result of the threshold scheme, this method is affected by two important drawbacks. First, it is computationally inefficient. Second, it does not guarantee an optimal solution. The new multi-seed algorithm presented in this work is based on optimized search strategies [3]. The algorithm starts selecting an arbitrary number of seeds and creating their corresponding regions. When a higher number of seeds is used, good quality areas can be unwrapped first without having to cross bad quality ones until the end, thus obtaining more accurate results than a mono-seed strategy. Independently of the number of regions, pixels of the interferogram are classified in only two different sets. First, the set of wrapped pixels with unwrapped neighbors. Second, the set of unwrapped pixels with neighbors without unwrap, whose final weight distributions must be saved in order to be used to unwrap subsequent pixels [2]. The pixels of this former set, which represent the candidates to be unwrapped, will be ordered according to their associated cost function fc. This cost function can be expressed as:
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fc
c1 COH c 2 UNW
(1)
where c1 and c2 are adjustable weights, COH represents the coherence of the pixel, and UNW the number of unwrapped neighbors. The higher fc is, the better the associated pixel will be. The slope estimation will be calculated as the average of the single slope estimations from the surrounding unwrapped neighbors [2]. The proposed multi-seed PU algorithm is an optimal solution since only the pixel whose cost function is the best will be unwrapped at every step. On the other hand, it is computationally efficient because it guarantees the unwrapping of one pixel per step. Since pixels of the interferogram are classified in only two different sets, the computational cost increase when compared to the mono-seed strategy is insignificant. It will be illustrated with an example in Section 4. 2.2. Merging growing regions procedure Procedure described in [5] for solving the region collision problem overlaps a group of pixels at once due to the strategy used to grow. Unlike this, the multi-seed PU algorithm proposed here will simplify the solution, since there will not be overlap between regions but a collision between adjacent pixels, one per growing region. For instance, let us suppose there exists collision between regions a and b. In such a case, a new region as the result of merging regions a and b will be created (region ab). To do so, the discontinuity between both regions (a multiple of 2ʌ) has to be removed. The unwrapped phase xuak of pixel k in region a after removing the discontinuity can be obtained as: u x ak
x ak 2S m
(2)
where xak is the unwrapped phase at pixel k in region a still containing the discontinuity, and m is the ambiguity factor given by: m
x x ak ½ round ® bn ¾ ¯ 2S ¿
(3)
being round(·) the nearest integer function, and xbn the unwrapped phase at pixel n in region b (collision pixel at region b).
2.3. Pseudocode of the proposed multi-seed PU algorithm The following steps define the proposed PU algorithm: Step 1: A group of seeds is selected. Step 2: Cost function fc per seed is obtained. REPEAT: Step 3: The pixel contained in the set of wrapped pixels with unwrapped neighbors whose fc is the highest will
be unwrapped making use of the slope estimations from the surrounding unwrapped neighbors and their weight distributions [2]. Step 4: IF region collision appears THEN solve the collision problem according to the method described in Section 2.2. Step5: Update lists of cost functions fc, weights and flags. WHILE wrapped pixels remain.
3. MATRIX PENCIL BASED LOCAL SLOPE ESTIMATOR The precision of a local phase slope estimator based on the mode of the PSD depends on the size of the window of data employed for the spectral (FFT) estimation, which defines the sampling in the slope spectrum. This precision is limited and, although a zero padding technique can be used to provide the slope estimations range with a higher granularity, it cannot improve the precision of the estimator. On the contrary, the matrix pencil technique does not suffer from this limitation since it belongs to the group of superresolution methods. Reference [6] shows the basic calculus of some useful parameters for one-dimensional sinusoids. However, since we are interested in the calculus of bidimensional frequencies, a complete explanation on how to obtain bidimensional frequencies and the phase slope estimates can be found in reference [7], which is the basis for the PU algorithm proposed in this work. 4. RESULTS This section shows the results obtained when using the MP local slope estimator and the proposed multi-seed PU algorithm. The first example illustrates a comparison between the MP local slope estimator and the local slope estimator based on the mode of the PSD. The more accurate predictions of the former are shown. The second example illustrates the better performance of the proposed multi-seed solution against a mono-seed or path following algorithm when dealing with low quality areas (high residue concentrations). This feature can be attributed to the diversity of growth paths used by the algorithm, which allows tackling difficult areas from different directions. Finally, the third example shows the good results when unwrapping a real interferogram by means of the proposed multi-seed PU algorithm. 4.1. Local phase slope estimates ǻĭ for a synthetic cone This example illustrates a comparison between slope estimates provided by the mode of the PSD and the MP local slope estimators. Fig. 1 shows a cone interferogram of size 100x100 pixels and 50 radians of height.
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Fig. 1: Synthetic cone interferogram: (a) input wrapped phase, and (b) expected unwrapped phase.
Table I shows a comparison between the slope estimators for a single pixel of the same synthetic cone interferogram, for instance pixel (85,80). Slope estimates in horizontal (0 radians, ǻĭH), vertical (S/2 radians, ǻĭV), obliquehorizontal (S/4 radians, ǻĭOH) and oblique-vertical (-S/4 radians, ǻĭOV) directions have been calculated from pixel (85,80) to the surrounding ones with a 3x3 window. Values in this table clearly demonstrate the better precision of the MP estimator.
involving an erroneous solution. On the other hand, Figs. 3.d and 3.e show intermediate results when the proposed multi-seed algorithm is used. The proposed multi-seed PU algorithm can unwrap high quality regions independently. Therefore, success chances increase since the low quality strips can be approached from both high quality sides at once. Finally, the correct unwrapped phase when the proposed multi-seed algorithm is used can be observed in Fig. 3.f.
Fig. 2: Conical interferogram containing two strips of Wishart noise: (a) wrapped input signal, and (b) coherence.
TABLE I SLOPE ESTIMATES CALCULATED AT PIXEL (85,80): EXPECTED VALUES AND RESULTS OBTAINED BY MEANS OF BOTH THE MODE OF THE PDS AND THE MP LOCAL SLOPE ESTIMATORS Expected
PSDmode
MP
ǻĭH (rad/pixel)
-0.4645
-0.4909
-0.4602
ǻĭV (rad/pixel)
-0.5401
-0.4909
-0.5368
ǻĭOH (rad/pixel)
-0.9971
-0.9817
-0.9968
ǻĭOV (rad/pixel)
0.0614
0.0982
0.0767
4.2. Synthetic cone containing two very noisy strips The data contained in Figs. 2.a and 2.b correspond to the input wrapped phase and the coherence of a conical synthetic interferogram contaminated with two longitudinal strips of Wishart noise. It will be used to illustrate the better performance of the proposed multi-seed PU algorithm introduced in Section 2.3, when compared against a monoseed region growing or a conventional path following PU algorithm. When the former is used, good quality areas are unwrapped first, and low quality ones can be approached from different sides at once, thus increasing the chance of success. On the contrary, if a conventional or mono-seed strategy is used, sometimes there will be no other chance than crossing low quality areas with the risk of introducing erroneous phase unwrapping and error propagation. Figs. 3.a and 3.b illustrate intermediate results after 18% and 62% of the unwrapping process when a mono-seed version of the proposed method is used. Finally, Fig. 3.c shows the final solution for this method. It can be observed how the mono-seed algorithm has no other chance that crossing both noisy strips from one side to the other. Consequences of the low quality of the data inside the strips are a low reliability of the slope predictions and the phases themselves,
Fig. 3: Conical interferogram containing two strips of Wishart noise: (a) mono-seed algorithm: result after 18% unwrapped pixels, (b) mono-seed algorithm: result after 62% unwrapped pixels, (c) final solution of the mono-seed algorithm, (d) multiseed algorithm: collision of regions a and b, (e) multi-seed algorithm: collision of regions b and c, and (f) final solution of the multi-seed algorithm.
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Finally, computational cost increase of the multi-seed method when compared to the path following method was insignificant (only 1% higher).
4.3. Small crop extracted from a real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration The data shown in Figs. 4.a and 4.b are the input wrapped phase and the coherence, respectively, extracted from a small crop from a real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration over Alicante province in Spain. Fig. 4.c presents the unwrapped phase by means of the proposed multi-seed PU algorithm. Note that the continuous space contained inside the 2ʌ sliding window has been translated into a discrete state space composed of N=100 cells and a maximum of Ns=50 particles will be used (see [2] for sliding window, cell and particle definitions). Finally, a quality test of this solution can be observed in Fig. 4.d, where the wrapped phase difference between the solution and the original interferogram exhibits a noisy aspect, due to the filtering capability exhibited by this solution. On the other hand, as expected, no bias has appeared.
[5] and a local slope estimator based on the matrix pencil technique introduced in [7]. The better precision of the MP local slope estimator when compared to the one based on the mode of the PSD has been shown by means of example 4.1. On the other hand, example 4.2 has shown a comparison between the proposed multi-seed PU algorithm and its mono-seed version. It has illustrated the better performance of the former when unwrapping low quality areas. Finally, the proposed PU algorithm has been applied to a real interferogram (example 4.3), showing a correct performance. Our research line at present and in the near future is focused on the application of this state space unwrapping strategy in advanced differential SAR interferometry for ground deformation monitoring, where the time coordinate has to be incorporated in the formulation.
6. ACKNOWLEDGMENT This work was supported by the Spanish Ministry of Science and Innovation and EU FEDER under Project TEC2008-06764-C02-02. The ERS images used in this work have been provided by the European Space Agency (ESA) in the framework of the ESA EO project Cat.1-2494. 7. REFERENCES [1] J. J. Martinez-Espla, T. Martinez-Marin, and J. M. LopezSanchez. “Using a Grid-based Filter to Solve InSAR Phase Unwrapping”. IEEE Geoscience and Remote Sensing Letters. Vol. 5, No. 2, pp. 147-151. April 2008. [2] J. J. Martinez-Espla, T. Martinez-Marin, and J. M. LopezSanchez. “A particle filter approach for InSAR phase filtering and unwrapping”. IEEE Transactions on Geoscience and Remote Sensing. Vol. 47, No. 4, pp. 1197–1211. April 2009. [3] N. J. Nilsson, Artificial Intelligence: a new synthesis. Morgan Kaufmann, 1998. [4] O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase Unwrapping for SAR Interferometry – A Data Fusion Approach by Kalman Filtering”, IEEE Transactions on Geoscience and Remote Sensing. Vol. 46, No. 1, pp. 47-58. January 2008.
Fig. 4: Real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration: (a) input phase, (b) coherence, (c) unwrapped phase by the multi-seed algorithm, and (d) wrapped phase difference between the unwrapped phase (output of the proposed multi-seed algorithm) and the input wrapped phase.
[5] W. Xu and I. Cumming. “A Region-Growing Algorithm for InSAR Phase Unwrapping”. IEEE Transactions on Geoscience and Remote Sensing. Vol. 37, No. 1, pp. 124-134. January 1999. [6] G. Nico and J. Fortuny. “Using the Matrix Pencil Method to Solve Phase Unwrapping”. IEEE Transactions on Signal Processing. Vol. 51, No. 3, pp. 886-880. March 2003. [7] Y. Hua. “Estimating Two-Dimensional Frequencies by Matrix Enhancement and Matrix Pencil”. IEEE Transactions on Signal Processing. Vol. 40, No. 9, pp. 2267-2280. September 1992.
5. CONCLUSION This work presents a new method for simultaneous filtering and phase unwrapping in SAR interferometry. The new solution combines a particle filter, an optimized version of the region growing search strategy introduced in
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ABSTRACT This work presents an improved InSAR phase unwrapping (PU) method based on a combination of a particle filter, a region growing technique and a matrix pencil based local slope estimator. The better performance of this new solution when compared against some representative traditional methods and the previous particle filter PU approaches is justified and illustrated with results obtained from synthetic and real data. Index Terms— Phase unwrapping, SAR interferometry, state space, particle filter, matrix pencil, region growing. 1. INTRODUCTION In previous works, the authors presented some phase unwrapping (PU) algorithms based on the use of state space techniques, as the grid-based filter [1] and the particle filter (PF) [2]. Among all of them, the path following version of the Particle Filter Phase Unwrapping algorithm (2PFPU) was demonstrated in [2] to be a robust phase unwrapping tool. This algorithm combined a particle filter, a path following search strategy [3] and a local slope estimator based on the mode of the power spectral density (PSD) [4]. In the present work, an improved phase unwrapping solution is proposed. Like the 2PFPU algorithm, this solution also makes use of a particle filter. However, it incorporates two important differences. First, an optimized region growing strategy has replaced the path following one present in the 2PFPU. As described later, the region growing approach used here is efficient, since the unwrapping of one pixel per iteration is guaranteed and optimum, since only the pixel whose cost function is the best among all regions will be unwrapped. Second, the local slope estimator based on the matrix pencil (MP) has replaced the one based on the mode of the PSD, since more accurate local phase slope predictions are expected. The new algorithm is more robust than previous ones because it can unwrap zones that other PU algorithms simply mask or fail to unwrap. It will be illustrated in this work with some examples. The text is organized as follows. The optimized region
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growing technique is presented in Section 2. Section 3 justifies the use of the matrix pencil based local slope estimator. Then, Section 4 illustrates the performance of the new method with some examples. Finally, conclusions are drawn in Section 5. 2. OPTIMIZED REGION GROWING STRATEGY 2.1. Introduction The path following method used in [2] can be considered a mono-seed region growing technique since it grows from a single seed of the interferogram. As a consequence, sometimes there will be no other chance than crossing low quality areas (i.e. containing high density of residues), hence increasing the risk of introducing unwrapping errors and its propagation. On the contrary, the region growing technique introduced in [5] starts from different seeds and the corresponding areas grow simultaneously by following a pattern of thresholds. At every step, growth rings are defined and every candidate following the threshold criteria will be unwrapped. When no pixel is unwrapped at a step, the thresholds are relaxed and the process continues. As a result of the threshold scheme, this method is affected by two important drawbacks. First, it is computationally inefficient. Second, it does not guarantee an optimal solution. The new multi-seed algorithm presented in this work is based on optimized search strategies [3]. The algorithm starts selecting an arbitrary number of seeds and creating their corresponding regions. When a higher number of seeds is used, good quality areas can be unwrapped first without having to cross bad quality ones until the end, thus obtaining more accurate results than a mono-seed strategy. Independently of the number of regions, pixels of the interferogram are classified in only two different sets. First, the set of wrapped pixels with unwrapped neighbors. Second, the set of unwrapped pixels with neighbors without unwrap, whose final weight distributions must be saved in order to be used to unwrap subsequent pixels [2]. The pixels of this former set, which represent the candidates to be unwrapped, will be ordered according to their associated cost function fc. This cost function can be expressed as:
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fc
c1 COH c 2 UNW
(1)
where c1 and c2 are adjustable weights, COH represents the coherence of the pixel, and UNW the number of unwrapped neighbors. The higher fc is, the better the associated pixel will be. The slope estimation will be calculated as the average of the single slope estimations from the surrounding unwrapped neighbors [2]. The proposed multi-seed PU algorithm is an optimal solution since only the pixel whose cost function is the best will be unwrapped at every step. On the other hand, it is computationally efficient because it guarantees the unwrapping of one pixel per step. Since pixels of the interferogram are classified in only two different sets, the computational cost increase when compared to the mono-seed strategy is insignificant. It will be illustrated with an example in Section 4. 2.2. Merging growing regions procedure Procedure described in [5] for solving the region collision problem overlaps a group of pixels at once due to the strategy used to grow. Unlike this, the multi-seed PU algorithm proposed here will simplify the solution, since there will not be overlap between regions but a collision between adjacent pixels, one per growing region. For instance, let us suppose there exists collision between regions a and b. In such a case, a new region as the result of merging regions a and b will be created (region ab). To do so, the discontinuity between both regions (a multiple of 2ʌ) has to be removed. The unwrapped phase xuak of pixel k in region a after removing the discontinuity can be obtained as: u x ak
x ak 2S m
(2)
where xak is the unwrapped phase at pixel k in region a still containing the discontinuity, and m is the ambiguity factor given by: m
x x ak ½ round ® bn ¾ ¯ 2S ¿
(3)
being round(·) the nearest integer function, and xbn the unwrapped phase at pixel n in region b (collision pixel at region b).
2.3. Pseudocode of the proposed multi-seed PU algorithm The following steps define the proposed PU algorithm: Step 1: A group of seeds is selected. Step 2: Cost function fc per seed is obtained. REPEAT: Step 3: The pixel contained in the set of wrapped pixels with unwrapped neighbors whose fc is the highest will
be unwrapped making use of the slope estimations from the surrounding unwrapped neighbors and their weight distributions [2]. Step 4: IF region collision appears THEN solve the collision problem according to the method described in Section 2.2. Step5: Update lists of cost functions fc, weights and flags. WHILE wrapped pixels remain.
3. MATRIX PENCIL BASED LOCAL SLOPE ESTIMATOR The precision of a local phase slope estimator based on the mode of the PSD depends on the size of the window of data employed for the spectral (FFT) estimation, which defines the sampling in the slope spectrum. This precision is limited and, although a zero padding technique can be used to provide the slope estimations range with a higher granularity, it cannot improve the precision of the estimator. On the contrary, the matrix pencil technique does not suffer from this limitation since it belongs to the group of superresolution methods. Reference [6] shows the basic calculus of some useful parameters for one-dimensional sinusoids. However, since we are interested in the calculus of bidimensional frequencies, a complete explanation on how to obtain bidimensional frequencies and the phase slope estimates can be found in reference [7], which is the basis for the PU algorithm proposed in this work. 4. RESULTS This section shows the results obtained when using the MP local slope estimator and the proposed multi-seed PU algorithm. The first example illustrates a comparison between the MP local slope estimator and the local slope estimator based on the mode of the PSD. The more accurate predictions of the former are shown. The second example illustrates the better performance of the proposed multi-seed solution against a mono-seed or path following algorithm when dealing with low quality areas (high residue concentrations). This feature can be attributed to the diversity of growth paths used by the algorithm, which allows tackling difficult areas from different directions. Finally, the third example shows the good results when unwrapping a real interferogram by means of the proposed multi-seed PU algorithm. 4.1. Local phase slope estimates ǻĭ for a synthetic cone This example illustrates a comparison between slope estimates provided by the mode of the PSD and the MP local slope estimators. Fig. 1 shows a cone interferogram of size 100x100 pixels and 50 radians of height.
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Fig. 1: Synthetic cone interferogram: (a) input wrapped phase, and (b) expected unwrapped phase.
Table I shows a comparison between the slope estimators for a single pixel of the same synthetic cone interferogram, for instance pixel (85,80). Slope estimates in horizontal (0 radians, ǻĭH), vertical (S/2 radians, ǻĭV), obliquehorizontal (S/4 radians, ǻĭOH) and oblique-vertical (-S/4 radians, ǻĭOV) directions have been calculated from pixel (85,80) to the surrounding ones with a 3x3 window. Values in this table clearly demonstrate the better precision of the MP estimator.
involving an erroneous solution. On the other hand, Figs. 3.d and 3.e show intermediate results when the proposed multi-seed algorithm is used. The proposed multi-seed PU algorithm can unwrap high quality regions independently. Therefore, success chances increase since the low quality strips can be approached from both high quality sides at once. Finally, the correct unwrapped phase when the proposed multi-seed algorithm is used can be observed in Fig. 3.f.
Fig. 2: Conical interferogram containing two strips of Wishart noise: (a) wrapped input signal, and (b) coherence.
TABLE I SLOPE ESTIMATES CALCULATED AT PIXEL (85,80): EXPECTED VALUES AND RESULTS OBTAINED BY MEANS OF BOTH THE MODE OF THE PDS AND THE MP LOCAL SLOPE ESTIMATORS Expected
PSDmode
MP
ǻĭH (rad/pixel)
-0.4645
-0.4909
-0.4602
ǻĭV (rad/pixel)
-0.5401
-0.4909
-0.5368
ǻĭOH (rad/pixel)
-0.9971
-0.9817
-0.9968
ǻĭOV (rad/pixel)
0.0614
0.0982
0.0767
4.2. Synthetic cone containing two very noisy strips The data contained in Figs. 2.a and 2.b correspond to the input wrapped phase and the coherence of a conical synthetic interferogram contaminated with two longitudinal strips of Wishart noise. It will be used to illustrate the better performance of the proposed multi-seed PU algorithm introduced in Section 2.3, when compared against a monoseed region growing or a conventional path following PU algorithm. When the former is used, good quality areas are unwrapped first, and low quality ones can be approached from different sides at once, thus increasing the chance of success. On the contrary, if a conventional or mono-seed strategy is used, sometimes there will be no other chance than crossing low quality areas with the risk of introducing erroneous phase unwrapping and error propagation. Figs. 3.a and 3.b illustrate intermediate results after 18% and 62% of the unwrapping process when a mono-seed version of the proposed method is used. Finally, Fig. 3.c shows the final solution for this method. It can be observed how the mono-seed algorithm has no other chance that crossing both noisy strips from one side to the other. Consequences of the low quality of the data inside the strips are a low reliability of the slope predictions and the phases themselves,
Fig. 3: Conical interferogram containing two strips of Wishart noise: (a) mono-seed algorithm: result after 18% unwrapped pixels, (b) mono-seed algorithm: result after 62% unwrapped pixels, (c) final solution of the mono-seed algorithm, (d) multiseed algorithm: collision of regions a and b, (e) multi-seed algorithm: collision of regions b and c, and (f) final solution of the multi-seed algorithm.
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Finally, computational cost increase of the multi-seed method when compared to the path following method was insignificant (only 1% higher).
4.3. Small crop extracted from a real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration The data shown in Figs. 4.a and 4.b are the input wrapped phase and the coherence, respectively, extracted from a small crop from a real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration over Alicante province in Spain. Fig. 4.c presents the unwrapped phase by means of the proposed multi-seed PU algorithm. Note that the continuous space contained inside the 2ʌ sliding window has been translated into a discrete state space composed of N=100 cells and a maximum of Ns=50 particles will be used (see [2] for sliding window, cell and particle definitions). Finally, a quality test of this solution can be observed in Fig. 4.d, where the wrapped phase difference between the solution and the original interferogram exhibits a noisy aspect, due to the filtering capability exhibited by this solution. On the other hand, as expected, no bias has appeared.
[5] and a local slope estimator based on the matrix pencil technique introduced in [7]. The better precision of the MP local slope estimator when compared to the one based on the mode of the PSD has been shown by means of example 4.1. On the other hand, example 4.2 has shown a comparison between the proposed multi-seed PU algorithm and its mono-seed version. It has illustrated the better performance of the former when unwrapping low quality areas. Finally, the proposed PU algorithm has been applied to a real interferogram (example 4.3), showing a correct performance. Our research line at present and in the near future is focused on the application of this state space unwrapping strategy in advanced differential SAR interferometry for ground deformation monitoring, where the time coordinate has to be incorporated in the formulation.
6. ACKNOWLEDGMENT This work was supported by the Spanish Ministry of Science and Innovation and EU FEDER under Project TEC2008-06764-C02-02. The ERS images used in this work have been provided by the European Space Agency (ESA) in the framework of the ESA EO project Cat.1-2494. 7. REFERENCES [1] J. J. Martinez-Espla, T. Martinez-Marin, and J. M. LopezSanchez. “Using a Grid-based Filter to Solve InSAR Phase Unwrapping”. IEEE Geoscience and Remote Sensing Letters. Vol. 5, No. 2, pp. 147-151. April 2008. [2] J. J. Martinez-Espla, T. Martinez-Marin, and J. M. LopezSanchez. “A particle filter approach for InSAR phase filtering and unwrapping”. IEEE Transactions on Geoscience and Remote Sensing. Vol. 47, No. 4, pp. 1197–1211. April 2009. [3] N. J. Nilsson, Artificial Intelligence: a new synthesis. Morgan Kaufmann, 1998. [4] O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase Unwrapping for SAR Interferometry – A Data Fusion Approach by Kalman Filtering”, IEEE Transactions on Geoscience and Remote Sensing. Vol. 46, No. 1, pp. 47-58. January 2008.
Fig. 4: Real interferogram obtained with images acquired by the ERS-1 and ERS-2 satellites in tandem configuration: (a) input phase, (b) coherence, (c) unwrapped phase by the multi-seed algorithm, and (d) wrapped phase difference between the unwrapped phase (output of the proposed multi-seed algorithm) and the input wrapped phase.
[5] W. Xu and I. Cumming. “A Region-Growing Algorithm for InSAR Phase Unwrapping”. IEEE Transactions on Geoscience and Remote Sensing. Vol. 37, No. 1, pp. 124-134. January 1999. [6] G. Nico and J. Fortuny. “Using the Matrix Pencil Method to Solve Phase Unwrapping”. IEEE Transactions on Signal Processing. Vol. 51, No. 3, pp. 886-880. March 2003. [7] Y. Hua. “Estimating Two-Dimensional Frequencies by Matrix Enhancement and Matrix Pencil”. IEEE Transactions on Signal Processing. Vol. 40, No. 9, pp. 2267-2280. September 1992.
5. CONCLUSION This work presents a new method for simultaneous filtering and phase unwrapping in SAR interferometry. The new solution combines a particle filter, an optimized version of the region growing search strategy introduced in
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