Patch-based Interferometric Phase Estimation via Mixture of Gaussian Density Modelling in the Complex Domain, Proc The European Space Agency Fringe, International Workshop on “Advances in the Science and Applications of SAR Interferometry and Sentinel-1 InSAR", Helsinki, Finland, Vol. , pp. - , June, 2017.
Phase imaging systems play a vital role in many present-day technologies, namely in the field of surveillance, remote sensing, medical diagnostic, weather forecasting and photography. Often, in such systems, a physical quantity of interest is coded in an image of the phase using a suitable coherent imaging technique (e.g., InSAR, InSAS). Since the phase is closely linked with the wave propagation phenomenon, the measured signals depend only on the principal (wrapped) values of the original phase (absolute phase), which we term as interferometric phase (InPhase), usually defined in the interval [-∏, ∏). The interferometric phase is thus a sinusoidal and nonlinear function of the absolute phase, which renders absolute phase estimation a hard inverse problem. In addition, the interferometric phase is usually corrupted by the noise introduced by the acquisition mechanism and electronic equipments, which further complicates the inverse problem which is the inference of the absolute phase from interferomertic measurements. This problem is often tackled in a two-step approach. In the first step, denoising of the noisy wrapped phase is taken care of and in the second step, the denoised phase image is unwrapped. InPhase image denoising should be addressed with special care since the wrapping discontinuities should be preserved carefully for the second stage of unwrapping.
In this paper, we propose a novel approach to address the problem of interferometric phase denoising by modelling the complex domain phase using Mixture of Gaussian densities. The recent state-of-the-art techniques in image denoising are based on non-local self-similarity and sparsity , which may be exploited via nonlocal patch-based techniques. The fact that the phase images are natural images motivates the application of patch-based approaches in InPhase image denoising. Following the standard procedures in patch-based image restoration, the phase image is decomposed into small overlapping patches. Vectors corresponding to these patches are modelled using Mixture of Gaussian (MoG) densities in the complex domain. The parameters, i.e.,
the covariance matrix, mean and mixing coefficients of the MoG are learned from complex domain patches of the training data. The learned MoG is used as a prior for estimating the interferometric phase images from the noisy images.
The main contribution of our work, which is inspired by the recent state-of-the-art image denoising techniques based on MoGs (see, e.g., ), can be summarized as follows: 1) an algorithm to learn the model; this is accomplished by formulating an Expectation Maximization (EM) algorithm for MoG densities in the complex domain; 2) computing the Minimum Mean Square Error (MMSE) estimates of the clean patches from the noisy ones using the learned model. The experiments conducted on simulated and real data of InSAR/InSAS shows results which are competitive with the state-of-the-art techniques , . Also, the entire process of InPhase estimation is illustrated by unwrapping the denoised interferogram using PUMA  algorithm.
One of the relevant contributions of our work is that it opens the door to the exploitation of "learned priors" from the specified classes of interferometric phase images, which can then be used in various phase inverse problems.
 J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman, "Non-local sparse models for image restoration," in 2009 IEEE 12th International Conference on Computer Vision, Sept 2009, pp. 2272-2279.
 T. Afonso, M. Almeida, and M. Figueiredo, "Single-frame image denoising and inpainting using Gaussian mixtures," in Proceedings of the International Conference on Pattern Recognition Applications and Methods, 2015, vol. 2, pp. 283-288.
 H. Hongxing, J. Bioucas-Dias, and V. Katkovnik, "Interferometric phase image estimation via sparse coding in the complex domain," IEEE Transactions on Geoscience and Remote Sensing, vol. 53, no. 5, pp. 2587- 2602, 2015.
 Q.Kemao, "Two-dimensional windowed fourier transform for fringe pattern analysis: Principles, applications and implementations," Optics and Lasers in Engineering, vol. 45, no. 2, pp. 304 - 317, 2007.
 J. Bioucas-Dias and G. Valadao, "Phase unwrapping via graph cuts," IEEE Transactions on Image Processing, vol. 16, no. 3, pp. 698-709, 2007.