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Multichannel Phase Unwrapping with Graph-Cuts Giampaolo Ferraioli, Aymen Shabou, Florence Tupin, and Vito Pascazio

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Multichannel Phase Unwrapping with Graph-Cuts Giampaolo Ferraioli, Aymen Shabou, Florence Tupin, and Vito Pascazio 1 Multichannel phase unwrapping with Graph-cuts Giampaolo Ferraioli, Aymen Shabou, Florence Tupin, and Vito Pascazio Abstract—Markovian approaches have proven to be effective minimum Lp norm unwrapping methods. Therefore, they do for solving the multichannel phase unwrapping problem, es- not optimally exploit statistical properties of the noise present pecially when dealing with noisy data and big discontinuities. on the data and they are not optimal from the information This paper presents a markovian approach to solve the phase unwrapping problem based on a new a priori model, the Total theoretical point of view. Moreover, differently from these Variation, and graph-cut based optimization algorithms. The approaches, we propose an algorithm that is able to unwrap proposed method turns to be fast, simple and robust. A set of and restore the solution at the same time. experimental results on both simulated and real data illustrate In the next section, we introduce the multichannel phase un- the effectiveness of our approach. wrapping (MCPU) technique with an inhomogeneous GMRF model. In section III, we present our new approach based on I. INTRODUCTION TV model and graph-cut optimization algorithms. Finally, we present some results showing the reconstruction obtained on In interferometric synthetic aperture radar systems, estima- simulated and real data. tion of the phase is a crucial point since there exists a known relation between InSAR phase and height values of the ground [1]. It is known that the measured phase (wrapped phase) is II. MULTICHANNEL PHASE UNWRAPPING given in the principal interval [−π, π], so phase unwrapping Multichannel phase unwrapping approach consists in com- (PU) problem has to be solved providing the absolute phase bining two or more independent interferograms. These inter- (unwrapped phase). This problem is known to be an ill- ferograms can be obtained in two different ways, multifre- posed problem if the so-called Itoh condition (the absolute quency and multibaseline configurations [7]. The interferomet- value of phase difference between neighboring pixels is less ric phase signals can be modeled as: than π) is not satisfied [2]. Usually, in InSAR systems, Itoh condition is violated due to the presence of discontinuities φp,c =< αchp + wp,c >2π; p ∈ {1, ..., N}; c ∈ {1, ..., M} (1) and/or interferometric noise. ⊥ 4πBc with αc = in case of multibaseline situation and One of the methods proposed recently to solve the PU λR0sin(θ) 4πB⊥ problem in case of non-Itoh condition is the multichannel αc = in case of multifrequency situation. The λcR0sin(θ) Maximum a posteriori (MAP) estimation method described index p refers to the pixel position inside the image of size in [3]. In this approach, the a priori statistical term is N, index c to the considered channel which is one of the M modeled by an inhomogeneous Gaussian Markov Random possible interferograms (frequencies or baselines), w is the Field (GMRF), i.e. GMRF with local hyperparameters [4]. The phase decorrelation noise, < . >2π represents the modulo−2π effectiveness of this method has been proved, even in presence operation, λ is the wavelength, B⊥ is the orthogonal baseline th of discontinuities, high sloped areas and low coherence areas. of the c SAR interferogram, θ is the SAR view angle and R0 Anyway, it suffers of some limits in particular concerning the is the distance of the first antenna to the center of the scene. computational time and the optimization step (no guaranty of Defining these notations, the height reconstruction problem finding the global optimum). consists in estimating the height values hp of the whole scene, In this work, we propose to improve this approach through using the N × M measured available wrapped phases φp,c. a new a priori Total Variation (TV) based model and using In the case of M statistically independent channels, the energy optimization algorithms based on graph-cut theory. multichannel likelihood function is given by [3]: This new approach presented in this paper, gives similar N M solutions to the work of [3] in 1/10th of the computation time. Y Y F (Φ|h) = f(φp,c|hp; αc, γp,c) (2) Furthermore, the global optimum for the considered energy p=1 c=1 function can be provided. The proposed method is validated both on simulated and real data, showing its effectiveness. where: −1 It is important to mention the work [5] that proposed graph- 1 1−|γp,c| dcos (−d) f(φp,c|hp; αc, γp,c) = 2 1 + 2 1/2 (3) cut based optimization algorithms to solve the PU problem 2π 1−d (1−d ) and [6] which is based on network programming optimization d = |γp,c| cos(φp,c − αchp) technique. We underline that these two approaches belong to is the single-channel likelihood function and γp,c is the co- G. Ferraioli and V. Pascazio are with Dipartimento per le Tecnolo- herence coefficient that depends on pixel p and on channel T T T T gie, Universita` degli Studi di Napoli Parthenope, Napoli, Italy. E-mail: c. In (2) Φ = [Φ0 Φ1 ...ΦN ] is the vector collecting all {giampaolo.ferraioli,vito.pascazio}@uniparthenope.it T available wrapped phase values, Φp = [φp,0φp,1...φp,M ] is A. Shabou and F. Tupin are with Institut TELECOM, TELECOM ParisTech, CNRS LTCI, France. E-mail: {aymen.shabou, florence.tupin}@telecom- the vector of the wrapped phases measured in pixels p for T paristech.fr the M different channels and h = [h0h1...hN ] is the vector containing the ground elevation values. Following the Bayes phase unwrapping algorithm. This new approach is a trade law, the Multichannel MAP estimation solution is given by: off between the computation complexity, optimum quality and prior model adaptation. hˆ = argmax F (Φ|h)g (h) (4) h β The TV model introduced in 1992 [10] is one of the most where the function g(.) is the prior probability of h.A MRF used prior model in image processing due to its adequation is used to model h, whose expression is given in this case by: to different contextual information. In SAR applications, TV model is mainly used for image restoration [11]. Our proposal 1 −Eprior (h) gβ(h) = e β (5) is to apply it to the PU problem. The prior energy correspond- Z(β) ing to the discretization of TV [12] can be written as follows where Z(β) is a normalization factor called the partition : function, Eprior(.) is the so-called energy function expressing X Eprior = β wp,q|hp − hq| (7) relationship between pixels and β = [β0β1...βN ] is the p∼q hyperparameter vector [8] that will be discussed later. The energy function is defined in such a way to impose some where wp,q depends on neighbor connexity (1 for 4 − connexity and √1 for the 4 diagonal ones in the case of constraint on the neighboring pixels. 2 Different prior energy functions can be used to model our 8 − connexity). Note that in this expression, β is a scalar and problem. An effective model is the one proposed in [3], where, not a vector of hyperparameters as in the GMRF model. This the energy Eprior is modeled by a local GMRF [9]: makes the TV model a non-local model. The choice of a non- 2 local prior energy is done in order to have a simplified model X (hp − hq) E (h) = (6) and a faster algorithm since it avoids the estimation of a vector priorβ 2β2 p∼q p,q of local hyperparameters. This choice is not as powerfull as p ∼ q denotes that pixels p and q are neighbors within the the local one proposed in [3], since it is not local. However, TV neighborhood system N of the MRF model. The hyperparam- between the existing non-local prior energy models, has eter vector β represents in this case the local spatial variations been chosen because its main advantage, is that it does not of the unwrapped heights. penalize discontinuities in the image while simultaneously not According to this approach, MAP estimation solution (4) penalizing smooth functions either [10]. As it is well adapted needs a previous estimation of the local hyperparameters when dealing with strong discontinuities, it can be used in InSAR β which is performed using the Expectation Maximization case of applications and particularly it well fits to urban algorithm (EM) [8]. Then, optimization step is carried out areas. using a semi-deterministic solution. The ICM (Iterated Con- We propose in the next section a fast MCPU algorithm ditional Modes) [8] algorithm is used for this purpose and based on graph-cut optimization method and TV prior. it is initialized with high-probability samples of the image generated in the hyperparameter estimation. This algorithm, B. Graph-cut based optimization although faster than simulated annealing, could not provide, in some cases, a global optimum. In the recent years, energy optimization with graph-cut has become very popular in computer vision [13]. Graph- III. TOTAL VARIATION MODEL AND GRAPH-CUT cut optimization is successful because the exact minimum or OPTIMIZATION an approximate minimum with certain guaranties of quality can be found in a polynomial time based on minimum- In this section, we introduce a new, simple and effective cut/maximum flow algorithms [14]. Compared to the classical energy function model, which combined with graph-cut based optimization algorithm, Simulated Annealing [4], it provides optimization algorithms allows to develop a fast and robust comparable results with much less computational time and MCPU approach. compared to the deterministic algorithm ICM [8], it avoids the risk of being trapped in local minima solution which can A. Total Variation based model be far from the global one.
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