Interferometric Phase Denoising Using Total Generalized Variation (TGV) Gang Liu, Robert Wang, YunKai Deng, Wensen Feng, Runpu Chen, YunFeng Shao Department of Space Microwave Remote Sensing System Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China e-mail:
[email protected] Abstract—The quality of Interferometric phase significantly affects the result of phase unwrapping and even the entire interferometric SAR application. Since traditional space borne interferometric products always contain much noise due to the influence of geometric decorrelation and temporal decorrelation phenomenon, it is necessary to remove the noise from the real interferometric phase. Because of the feature interferometric phase is modulo 2π, the wrapping phase will leads many steep edges in the phase image. Therefore the filter methods must satisfy that they should not break the edge information. This paper proposes an approach for interferometic phase denoising with Total Generalized Variation (TGV) regulation. TGV is developed recently and its mathematical framework is very suitable for interferometic phase noise model. It has favorable abilities of edge preservation and noise removal. In order to test the validity of our algorithm, both simulated and real SAR interferogrms data experiments are performed. The results demonstrate the effectiveness of the TGV denoising algorithm.
I. INTRODUCTION Synthetic aperture radar (SAR) is one of the essential tools of remote sensing for observation of the earth. Interferometric SAR, as an extend technique of SAR, utilizes phase different observation between two radar acquisition for estimation of surface height or deformation [1]. Its measurement precision has reached cm degree. Because of the advantage of high precision and widely mapping abilities, recent years, InSAR technique has been studied deeply by scientist around the world and has become a big family including DInSAR (Differential SAR Interferometry), PSInSAR (Permanent Scatterer SAR Interferometry), SBAS InSAR (Small Baseline Subset), tomography SAR and so on. In all these applications, phase unwrapping plays a crucial role to obtain the final results. However, phase unwrapping extremely depends on the quality of the phase. In reality, good quality phase is hard to obtain directly because of the deccorelation phenomenon. Spatial separations of the two antennas (baseline) lead to the speckle decorrelation pattern in the two images, introducing higher level noise in the interferometric phase image. In the case of repeat-pass interferometry, temporal changes also degrade the phase quality. The coherence between the two images is the reflection of the degree of deccorelation. In other words, any loss of accuracy of the interferometric phase means loss in coherence. Additionally, in the complex terrain area, layover and shadow phenomenon also reduce the coherence. As a
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result, before phase unwrapping step, interferometic phase denoising must be done to improve the phase quality. The research in [2] indicates that the statistics of interferometric phase images can be characterized by a probability density function (pdf) based on the circular Gaussian assumption. According to the characteristics of interferometric phase, a series of filter methods have been developed. Lee et al. developed an adaptive filtering algorithm based on noise model [2]. It emphasizes filtering noise adaptively according to the local noise level and filtering along fringes using directionally dependent windows and it is proved to be effective in [3]. Baran et al. presented an improvement on the Goldstein filter [4]. This method makes the Goldstein filter parameter alpha dependent on coherence, and makes the incoherent areas filtered more than coherent areas. Vasile et al. invented IDAN (Intensity-Driven AdaptiveNeighborhood) filter for denoising of interferometric phase images. This algorithm originally uses region-growth method on intensity images so as to define the adaptive neighborhood (AN) of a pixel to be filtered [5]. In this paper, we try to solve this problem with optimization theory. Total variation (TV) regularization has been developed for many years for additive white Gaussian noise image denoising [6]. However, there is a problem with TV regularization that is the “staircase artifact,” a tendency to produce small flat regions with steep edges. That is deadly restriction hindering it using in the interferometric phase filter as all the interferometric phase data is modulo 2π which leads to that it include so many steep edges. Another problem of TV is that it assumes the images consist of regions, which are piecewise constant. It is not in accordance with the actual situation of terrain that represented by InSAR phase. The new concept of Total Generalized Variation (TGV) can overcome these problems. TGV mathematical theory has recently been developed [7]. It performs favorable edge preservation and noise removal abilities; it also can be applied in imaging situations where the assumption that the image is piecewise constant is not valid. II. PHASE NOISE CHARACTERITICS The quality of the interferometric phase can be estimated from the interferometric data as a function of the complex coherence γ between two complex SAR images Um and Us. It is defined as
γ=
E [U m ∗ U s ] E[| U m |2 *E[| U s |2 ]
=| γ | eiθ
(1)
represents the mathematical expectation where operator, | | is the absolute coherence measuring interferometric phase quality, and is the phase of the complex correlation coefficient. Based on (1), Middleton derived the one-look phase distribution based on circular Gaussian statistics. Later the following multilook pdf (2) for multilook phase distributions were derived by Lee et al. [2].
1 Γ(n + )(1− | γ |2 )n β (1− | γ |2 ) n 2 pψ (ψ ) = + 1 n+ 2π 2 π Γ(n)(1 − β 2 ) 2 1 ⋅ F (n;1; ; β 2 ) − π < (ψ − θ ) ≤ π 2
(2)
(3)
where ψ z denotes observed phase, ψ x is wrapped phase without noise. With some approximation, the noise term can be described as a zero-mean, Gaussian independent random variable. On the basis of (2), the goal of phase image denoising can be considered as an inverse problem in particular of mathematical imaging problems to recover ψ z from ψ x . III.
THE CONCEPT OF TOTAL GENERALIZED VARIATION (TGV) A. The defination of TGV The Total Generalized Variation is recently developed in [7]. It is capable to measure image characteristics up to a certain order of differentiation. Before introducing it, let us recall the
definition of the Total Variation [6], which is, for a given image u, usually expressed as
TV(u ) = ∫ ∇u dx
(4)
Ω
The Total Generalized Variation of second-order [8] which we will use is based on this definition. And it is generalized to represent a minimization problem itself: TGVα2 (u ) = min α1 ∫ | ∇u − v | dx + α 0 ∫ | ε (v) | dx v
Ω
Ω
f =u+v
(5)
Here, the minimum is taken over all complex vector fields |/2 which denotes the | v on Ω and ε symmetrized derivative. This definition provides a way of balancing between the first and second derivative of the function.
(6)
stand for ψ z , ψ x , and v respectively.
Where f, u,
Also begin with TV method, from a Bayesian point of view, (6) can be solved by a maximum a posteriori estimate using a TV prior. From the AWGN noise model, the conditional probability density p(f|u) is 1
p( f | u ) =
2πσ
2
exp(−
− log p( f | u ) = const +
| | cos where and F is a Gauss hypergeometric function. The multilook distributions, such as onelook, two-look, three-look, etc., can been derived through (2). In (2), the distribution ψ is symmetrical about θ , θ is the mean with modulus 2 , and the standard deviation is independent of θ . Consequently, ψ can be characterized by an additive noise model [3].
ψ z = ψ x +ν
B. TGV Image Denoising Denoising is to solve the problem of removing noise from an image. The most common studied case is with additive white Gaussian noise (AWGN), as showed in (3). We rewrite the equation (3) as following so as to unify the expression with usual mathematics form.
1
∫
2σ 2 1
2σ 2
Ω
∫
Ω
( f ( x ) − u ( x)) 2 dx) (7) 2
( f − u ) dx,
where σ is the noise variance. The maximum a posteriori estimate is
u = arg max p(u | f ) u
= arg max p(u ) p ( f | u ) u
(8)
= arg min − log p(u ) − log p( f | u ) u
= arg min − log p(u ) + u
1 2σ 2
∫
Ω
( f − u )2 dx.
The log term is the prior on u, an a priori assumption on the likelihood of a solution u. With total variation regularization, the selected prior is − log p(u ) = μ || u ||TV( Ω )
(9)
where μ is a positive parameter controlling the regularization strength. TV models have the main benefit that they are very well suitable to remove random noise. However, the assumption of TV is that the images consist of regions, which are piecewise constant. According to the characteristic of Interferometric phase, the real phase mostly keep piecewise linearity. In this situation, TV models may be not valid. Here, we use TGV replace the TV regulation because it can be applied in imaging situations where the assumption that the image is piecewise constant is not hold. So formulation (9) can be rewritten as follows − log p(u ) = μ || u ||TGV( Ω )
(10)
The problem is equivalent to the maximum a posteriori estimate
min u
2 1 (u − f ) dx + TGVα2 (u ) ∫ Ω 2λ
(11)
The discrete version of the denoising problem (11) is given by
min h h
u ∈U
1 || u h − f h ||22 +TGVα2 (u h ) 2λ
(12)
Where, the superscript h indicates the discrete setting. according to (5), used the corresponding discrete divergence operators and discretized, problem (12) becomes min
u h ∈U h , v h ∈V h
1 || u h − f h ||22 +α1 || ∇ h u h − ||1 +α 0 || ε h (v h ) ||1 (13) 2λ
TABLE I.
PERFORMANCE COMOPARISAN BETWEEN TGV FILTER AND THE OTHERS
Statistics
Filter Algorithm Goldstein filter
Lee filter
TGV filter
MSE. (rad2)
0.917
0.953
0.714
Perc. residues
80.8%
91.46%
98.9%
Exec. time(s)
0.078
19.81
0.651
We apply the primal-dual algorithm to get the solution of (13). Limited by the paper length, the algorithm is not discussed here; more details can be seen in [8]. EXPERIMENT RESULT IV. A. Synthetic data Experiment To test the performance of the proposed algorithm, a simulated interferogram with the DEM (458 157 pixels) of mountains around the Isolation Peak (Colorado, USA) is used. Starting from this height profile, we simulate the interferogram that generated with the height of ambiguity ha = 29.2 m. The noise added in the data is generated from the interferometic phase pdf with the coherence coefficient that equals to 0.89.
(a)
(b)
(c)
Figure 2. Filter results (a) Goldstein filter (b) Lee filter (c) TGV filter
(a)
(b)
Figure 1. simulated InSAR phase (a) original phase (b) noised phase
In order to evaluate the performance of TGV denoing algorithm, two other classical filters, Goldstein filter and Lee filter, are used to process the simulated data. Fig 2 shows the filters results. Fig 3 displays the estimated noise term of the three filters. In the meanwhile, several statistical characteristics, including mean square error (MSE), residues decrease ratio and execution time, are given in TABLE I. From Fig 2, it can be validated that TGV filter has the ability of reducing noise and preserving edges. It can be seen that TGV result is better than Goldstein’s and similar to the original phase. In Fig 3, from the visual, the estimated noise term of Lee filter contains less textures information than the original phase. In Fig 3, from the visual, the estimated noise
(a)
(b)
(c)
Figure 3. The estimated noise term (a) Goldstein filter (b) Lee filter (c) TGV filter
term of Lee filter contains less textures information than the Goldstein filter’s and the TGV’s. Discussion above is based on the qualitative inorganic analysis. Table 1 illustrates its performance from quantities analysis. It can be figured out that the TGV filter has the best performance in the MSE and
residues decrease ratio. It has the minimum MSE and maximum residues decrease ratio. The Lee filter although can keep good textures but its executive time is a little longer than the other ones. B. Real data Experiment In order to test the performance of the proposed algorithm on real applications, two TerraSAR-X images covering Grand Canyon, USA are downloaded from infoterra website. Here we only pick a part of 351 501 pixels from the original imagesso that the detailed result can be seen. Fig 4 shows the interferogram generated from the two acquisitions. This phase data are very complicated because of the extremely steeply terrain changes. We compare the results of the TGV filter with those using other filters. Fig 5 exhibits the results of all the filters applied on this interferogram.
(c) Figure 5. Real InSAR Data filter result (a) Goldstein filter result (b) Lee filter result (c) TGV filter result
From the above results, we can see that the precision of the TGV filter is similar to the Lee filter; however, its advantage lies in the faster executive speed. Contract to the Goldstein filter, the TGV filter performs with a better precision. V.
Figure 4. Real InSAR Data
CONCLUSION
In this paper, we present an effective denoising approach through exploiting the Total Generalized Variation (TGV) for interferometric SAR phase data. Both the simulated and real data experiments prove that the proposed approach can obtain a rapid result while keeping a pretty good precision. Besides, it has the potential to improve the ability of preserving textures information through adding back the estimated noise term to the original image, and carrying out the second iteration. We will continue this work in the future. REFERENCES [1]
[2]
[3]
[4] (a) [5]
[6] [7] [8]
(b)
P. A. Rosen, S. Hensley, I. R. Joughin, F. K. Li, S. N. Madsen, E. Rodriguez, and R. M. Goldstein, “Synthetic aperture radar interferometry,” Proc. IEEE, vol.88, no.3, pp.333-382, March 2000. J. S. Lee, K. W. Hoppel, and S. A. Mango, “Intensity and phase statistics of mutilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci Remote Sensing, vol.32, pp. 1017-1027, Sept 1994 J. S. Lee, P. Papathanassiou, T. L. Ainsworth, R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci Remote Sensing, vol.36, no.5, pp.14561465, Sep 1998 I. Baran, M. P. Stewart, B. M. Kampes, Z. Perski, P. Lilly, "A modification to the Goldstein radar interferogram filter," IEEE Trans. Geosci Remote Sensing, vol.41, no.9, pp. 2114- 2118, Sept. 2003 G. Vasile, E. Trouvé, J. S. Lee, V. Buzuloiu, “IntensityDrivenAdaptive-Neighborhood Technique for Polarimetric and Interferometric SAR Parameters Estimation”, IEEE Trans. Geoscience and RemoteSensing, vol.44, no.6, June 2006 L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Phys. D, vol.60, pp.259–268,1992. K. Bredies, K. Kunisch, and T. Pock, “Total generalized variation”, SIAM Journal on Imaging Sciences 2010, vol.3, no.3, pp.492–526. F. Knoll, K. Bredies, and T. Pock, R. Stollberger, “Second order total generalized variation (TGV) for MRI”, Magn Reson Med 2010, vol.65, pp.480–491
