Introduction of a Grid-based Filter Approach for InSAR Phase Filtering and Unwrapping

Juan J. Martinez-Espla, Tomas Martinez-Marin, and Juan M. Lopez-Sanchez Department of Physics, System Engineering and Signal Theory (DFISTS), University of Alicante

P.O. Box 99, E-03080 Alicante, Spain Tel/Fax: +34 96590 9811/9750 E-mail: jjme1@alu.ua.es, tomas@dfists.ua.es, juanma-lopez@ieee.org

Abstract— This work presents a phase unwrapping (PU) algorithm for SAR interferometry based on an approximate grid-based filter (GbF). This PU algorithm, which makes use of state space techniques, performs simultaneously noise filtering and phase unwrapping. The formulation of this technique provides independence from noise statistics and is not constrained by the non-linearity of the problem. Results obtained with synthetic and real data show a significant improvement with respect to conventional PU algorithms in some situations.

Keywords-component: phase unwrapping, SAR interferometry, extended Kalman filter, grid filter.

I. INTRODUCTION Phase unwrapping (PU) is one of the key processing steps

in all InSAR applications. We refer to the book by Ghiglia and Pritt [1] for an excellent overview. There are two general types of conventional PU methods. The algorithms of the first group, generally named path following or region growing algorithms, isolate and/or mask problematic zones containing residues and unwrap the interferogram by avoiding these zones. The techniques of the second group provide a global solution that minimizes a cost function over the whole interferogram. Some of these techniques, and other alternative approaches that do not correspond to these groups, make use of a pre-filtering stage before starting the unwrapping procedure with the filtered phase. Consequences of these strategies are the following: phase information in noisy pixels is not recovered by the path following algorithms; noisy pixels distort the solution in global approaches, thus affecting the noise-free areas; and the information contained in noisy pixels is lost if a pre-filtering stage is carried out. We are interested in recovering as much information as possible from the interferogram, including noisy pixels. Therefore, an ideal method would consider every pixel to be simultaneously unwrapped and filtered.

There exists an algorithm that shares the same objective and was already published in [2]-[3]. That algorithm combines a slope estimator with an extended Kalman filter (EKF), showing the mentioned feature of integrating a slope/frequency estimator and an EKF in an elegant fashion. Unfortunately, that solution is founded on the following two assumptions. First, both evolution and measurement models are available and correspond to linear functions. Second, the noises affecting both evolution and measurement stages are Gaussian. When the Gaussian noise constraint holds but the models are not

linear functions, the Extended Kalman filter (EKF) has been used in many cases as a good solution that approximates the nonlinearities by local linearizations. The main problem arises when the noise present in the interferogram is not Gaussian. The EKF cannot guarantee a successful PU, as we will show later with two examples.

In this work we propose to substitute the EKF by a grid-based filter (GbF), which is not subject to any linear or Gaussian constraints. A detailed tutorial about different methods to broach nonlinear/non-Gaussian problems can be consulted in [4]. This solution combines a GbF with a phase slope estimator to simultaneously unwrap and filter the interferometric phase. This approach could be combined with path following strategies to enhance their performance. However, at this moment the basic algorithm introduced here runs the interferogram row by row, passing through zones of low coherence and containing residues, unwrapping and filtering the phase at the same time. We refer to [1] for the definition of coherence and residues.

The text is organized as follows. The grid-based filter formulation for solving phase unwrapping is presented in Section II. Then, Section III illustrates the performance of the new method with synthetic and real data. A comparison with the cited EKF approach and the well-known branch-cut algorithm is also discussed. Finally, conclusions are drawn in Section IV.

II. APPROXIMATE GRID-BASED FILTER

A. Introduction to the grid-based filter Assuming a one-dimensional notation for the complex SAR

interferogram, the unwrapped phase ( )kϕ can be obtained through the recursive expression:

( ) ( ) ( )kkk ϕϕϕ ∆+=+ ~1 (1)

where ( )kϕ∆~ represent a slope estimation. From now on, for the sake of coherency with the notation of state space methods, the unwrapped phase at pixel k, )(kϕ , will be referred as the corresponding state kx at pixel k. The concept of state, also known as cell, is introduced below.

For a grid-based approximation, it is assumed that some necessary information is available:

- Initial state: ox

- Initial weight distribution: s

io Niw ,,1; …= where sN is the

number of cells (see cell definition below).

- Observations: To zz ,,… where Tkzk ,,0; …= is the observed wrapped noisy phase at pixel k, formerly denoted as

( )kϕ~ , and T is the total number of pixels.

- Evolution Model: ( ) ( )11 | −− ↔= kkkkk xxpxfx

- Observation Model: ( ) ( )kkkkk xzpxhz |↔=

The continuous state space of the interferometric problem is divided into

sN cells or states, { }sik Nix ,...,1; = , which

correspond to all possible phase values. The grid for this new discrete state space must be sufficiently dense to get a good approximation to the continuous state space.

In such a case, as it can be consulted in [4], the prediction and update equations are, respectively, given by:

( ) ( )∑=

−− −≈sN

i

ikk

ikkkk xxwzxp

11|1:1| δ (2)

( ) ( )∑=

−≈sN

i

ikk

ikkkk xxwzxp

1|:1| δ (3)

where )(δ represents the Dirac Delta and:

( )jk

ik

N

j

jkk

ikk xxpww

s

11

1|11| | −=

−−− ∑≈ (4)

( )( )∑

=−

−≈sN

j

jkk

jkk

ikk

ikki

kk

xzpw

xzpww

11|

1||

|

| (5)

Here jkx 1−

represents the center of the jth cell at pixel 1−k , whereas the probabilities ( )j

kik xxp 1| −

and ( )ikk xzp | are defined

by the evolution and observation models at pixel k , given by:

111 ˆ −−− +∆+= kkj

kik nxx ϕ (6)

kikk vxz += (7)

where 1ˆ −∆ kϕ represents a phase slope estimation, while kn and

kv are supposed to be, respectively, approximate models for the noise present in both evolution and measurement stages. Several techniques can be used to obtain an estimation of the phase slope. For instance, the mode of the power spectral density was used in [2] and [3] due to its supposed insensitiveness and robustness to superimposed white noise. We will use the same one here for the sake of comparison. The models for the noises present in both evolution and measurement stages, kn and kv , are respectively Gaussian and Wishart distributions.

Unlike a Grid-based solution, the EKF cannot be used in its pure conception. The pure EKF makes use of the following update equation:

( )pkk

pkk xzKxx −⋅+= (8)

where kz is the measurement, pkx is the predicted unwrapped

phase, K is the Kalman gain, and kx is the unwrapped solution at pixel k obtained by the update equation.

As described in [5], as the measurement error covariance kR approaches zero, the Kalman gain K approaches one, and

the measurement kz is trusted more and more whereas the predicted unwrapped phase p

kx is trusted less and less. As a consequence, the following result is obtained:

kk zx ≈ (9)

But kz is the input measure, i.e., the wrapped phase and the EKF fails to unwrap the interferogram, as we will show below.

Therefore, the EKF solution presented in [2]-[3] employed two dimensions per pixel (inphase and quadrature components of phase contained in the InSAR image), thus increasing complexity.

In addition, it can be demonstrated that:

( )[ ] radxzK pkk 1max =−⋅ (10)

which means that a bad prediction pkx only can be corrected up

to rad1± by the update equation, in the best case. In contrast, the application of a grid-based filter solution has the great advantage of using only one dimension per pixel (directly the phase), bringing down complexity. Moreover, unlike the EKF, a Grid-based Filter solution is able to reach a correction of up to radπ2± .

A priori, one drawback of the GbF could be the finite state space constraint. However, the grid-based solution we propose here makes use of a π2 wide sliding window S

kW to cover the complete state space. This window is centered in the previous unwrapped phase given by mxGbF , introduced below. This sliding state space is based on the assumption that the phase difference between two contiguous pixels must be contained in the interval ( ]ππ +− , . Computational cost increases linearly with sN , the number of cells.

B. 2D Phase unwrapping algorithm To extend the use of GbF to 2D interferograms, any

prediction estimate will be calculated depending on two neighbours so that only the evolution model (6) needs to be modified:

2/)(2/)ˆˆ(~1 VHVH

ik

ik mnxx ++∆+∆+= − ϕϕ (11)

where Hϕ̂∆ and

Vϕ̂∆ are the horizontal and vertical slope estimates calculated on the previously unwrapped pixels, placed in the same row and adjacent to current pixel k , and in the same column and adjacent to pixel k , respectively. Likewise,

Hn and Vm are Gaussian distributions that model the

noise present in Hϕ̂∆ and

Vϕ̂∆ slope estimations.

A pseudocode of the 2D PU algorithm is included in Fig. 1.

Figure 1. Pseudocode of the PU algorithm using a grid-based filter.

III. RESULTS

A. 1D Results The results in this section illustrate the performance of the

different phase unwrapping solutions: the EKF algorithm proposed in [2] and [3] and two approaches of the Grid-based Filter. mxGbF consists in selecting the state whose weight is the maximum and avGbF by selecting the average of the distribution. They have been applied to both a 1D synthetic interferogram and a 1D-cut extracted from a 2D real one.

As we introduced before, a pure EKF (not the two dimensional method proposed in [2]-[3]) is unable to unwrap the phase in the ideal case in which the input signal is free of noise. On the contrary, a correct performance of the avGbF can be observed in Fig.2 because the phase is perfectly unwrapped.

To compare the performance of the Grid-based Filter against the EKF proposed in [2] and [3], a synthetic pyramidal signal has been generated, and Gaussian noise with variance

rad40,2 has been added. There exist zones where the coherence of data is very poor, for instance, in the surroundings of pixels 12 and 52, reaching values of 0.29 and 0.38 respectively. The results can be observed in Fig. 3.

In such a case, the assumption of high coherence made to approximate the estimation of

kR by expression (8) in [3] does not hold. The assumption of a power spectral density unimodal and nearly symmetric around the central mode does not hold

either, and the spectral slope estimator fails for some consecutive pixels. The result is that the EKF fails and will never recover.

An example with real data has been carried out with similar conclusions in Fig. 4. Note that a synthetic ramp of 20 pixels has been added at the beginning to avoid confusing real phase jumps with discontinuities due to the wrapping operation. As it is well known, noise present in a real interferogram is Wishart, not Gaussian. A local concentration of noise can be observed in the surroundings of pixel 30. The lowest values for coherence in this zone are around 0.5, and the approximation made by expression (8) in [3] holds. However, the non-Gaussian constraint still stands.

Figure 2. 1D wrapped synthetic Gaussian. The continuous space contained inside the π2 sliding window at every iteration, has been translated into a

discrete state space composed of 100=N cells. In this plot, the EKF solution is a pure one dimension algorithm, not the one proposed in [2]-[3].

Figure 3. 1D synthetic noisy pyramid. π2 sliding window divided into

100=N cells.

Figure 4. 1D real cut. π2 sliding window divided into 200=N cells.

As observed in Fig. 3 and Fig. 4, both Grid-based Filter solutions, mxGbF and avGbF , can deal with non Gaussian

noise distributions and recover from errors since they are not subject to any Gaussian and linear constraints.

Finally, as a part of the simultaneous filtering process, from the comparison of both Grid-based Filter solutions, it can be concluded that the avGbF approach is more likely to produce softer profiles than the mxGbF solution, as observed in Fig. 4.

B. 2D Results The following results correspond to a 2D crop extracted

from a real interferogram obtained by the ERS1&2 satellites in tandem configuration. As observed in Figs. 5(e-f), the EKF fails in two different areas. First, a concentration of residues around pixel (40,80) produces errors in the slope estimates and in the unwrapped phase in this area. Second, the conjunction of low coherence and high variance of the slope estimators in pixel (7,70) produces an error which propagates in a diagonal fashion, generating an erroneous phase discontinuity. In contrast, Figures 5.g-h show the expected performance of the GbF approach. A π2 sliding window divided into 100=N cells was used in this case. Finally, the branch-cut algorithm proposed by Goldstein [1] was also applied to this data set. Results from the branch-cut method (Figs. 5.i-j) exhibit a noisier solution because this technique does not incorporate any filtering, in contrast to the GbF solution. Some other illustrative 2D results can be consulted in [6].

Figure 5. Results with 2D real data: (a) input, (b) coherence, (c) zoom of residues, (d) zoom of rewrapped EKF, (e) rewrapped EKF, (f) unwrapped

EKF, (g) rewrapped avGbF , (h) unwrapped

avGbF , (i) rewrapped Goldstein, (j) unwrapped Goldstein.

IV. CONCLUSION An approximate grid-based filter (GbF) solution of the

phase unwrapping problem in SAR interferometry has been proposed. Two variants of this method have been compared against the EKF method, which shares the same philosophy of simultaneously filtering and unwrapping, and the brunch-cut algorithm proposed by Goldstein. We have demonstrated with examples some situations where the EKF fails and never recovers, whereas the grid-based solutions perform better and are also able to recover from errors. It must be noted that the GbF can deal with zones containing residues.

Our research lines at present are focused on two main issues. First, introduction of different slope estimators into the GbF approach. For instance, the Matrix Pencil algorithm proposed in [7] and the slope average-based method introduced in [8] are possible options. Second, the combination of the GbF approach with path following techniques (e.g. region growing), since the synergy between both strategies may overcome their particular limitations.

ACKNOWLEDGMENT This work was supported by the Spanish Ministry of

Education and Science, and EU FEDER, under Project TEC2005 06863-C02-02.

The SAR real images used in this work have been supplied by the European Spatial Agency (ESA) within the framework of the ESA EO project Cat.1-2494. The interferometric processing has been carried out with the DORIS software, provided by the Delft University of Technology.

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