On Solving SAR Imaging Inverse Problems Using Non-Convex Regularization with a Cauchy-based Penalty

Synthetic aperture radar (SAR) imagery can provide useful information in a multitude of applications, including climate change, environmental monitoring, meteorology, high dimensional mapping, ship monitoring, or planetary exploration. In this paper, we investigate solutions to a number of inverse problems encountered in SAR imaging. We propose a convex proximal splitting method for the optimization of a cost function that includes a non-convex Cauchy-based penalty. The convergence of the overall cost function optimization is ensured through careful selection of model parameters within a forward-backward (FB) algorithm. The performance of the proposed penalty function is evaluated by solving three standard SAR imaging inverse problems, including super-resolution, image formation, and despeckling, as well as ship wake detection for maritime applications. The proposed method is compared to several methods employing classical penalty functions such as total variation ($TV$) and $L_1$ norms, and to the generalized minimax-concave (GMC) penalty. We show that the proposed Cauchy-based penalty function leads to better image reconstruction results when compared to the reference penalty functions for all SAR imaging inverse problems in this paper.Comment: 18 pages, 7 figure

A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection

We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian distribution. The highly flexible variational structure of the corresponding regulariser encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM). A suitable preliminary variable splitting together with a novel result in multivariate non-convex proximal calculus yield a very efficient minimisation algorithm. Several numerical results showing significant quality-improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular in terms of texture and detail preservation

영상 잡음 제거와 수중 영상 복원을 위한 정규화 방법

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 강명주.In this thesis, we discuss regularization methods for denoising images corrupted by Gaussian or Cauchy noise and image dehazing in underwater. In image denoising, we introduce the second-order extension of structure tensor total variation and propose a hybrid method for additive Gaussian noise. Furthermore, we apply the weighted nuclear norm under nonlocal framework to remove additive Cauchy noise in images. We adopt the nonconvex alternating direction method of multiplier to solve the problem iteratively. Subsequently, based on the color ellipsoid prior which is effective for restoring hazy image in the atmosphere, we suggest novel dehazing method adapted for underwater condition. Because attenuation rate of light varies depending on wavelength of light in water, we apply the color ellipsoid prior only for green and blue channels and combine it with intensity map of red channel to refine the obtained depth map further. Numerical experiments show that our proposed methods show superior results compared with other methods both in quantitative and qualitative aspects.본 논문에서 우리는 가우시안 또는 코시 분포를 따르는 잡음으로 오염된 영상과 물 속에서 얻은 영상을 복원하기 위한 정규화 방법에 대해 논의한다. 영상 잡음 문제에서 우리는 덧셈 가우시안 잡음의 해결을 위해 구조 텐서 총변이의 이차 확장을 도입하고 이것을 이용한 혼합 방법을 제안한다. 나아가 덧셈 코시 잡음 문제를 해결하기 위해 우리는 가중 핵 노름을 비국소적인 틀에서 적용하고 비볼록 교차 승수법을 통해서 반복적으로 문제를 푼다. 이어서 대기 중의 안개 낀 영상을 복원하는데 효과적인 색 타원면 가정에 기초하여, 우리는 물 속의 상황에 알맞은 영상 복원 방법을 제시한다. 물 속에서 빛의 감쇠 정도는 빛의 파장에 따라 달라지기 때문에, 우리는 색 타원면 가정을 영상의 녹색과 청색 채널에 적용하고 그로부터 얻은 깊이 지도를 적색 채널의 강도 지도와 혼합하여 개선된 깊이 지도를 얻는다. 수치적 실험을 통해서 우리가 제시한 방법들을 다른 방법과 비교하고 질적인 측면과 평가 지표에 따른 양적인 측면 모두에서 우수함을 확인한다.1 Introduction 1 1.1 Image denoising for Gaussian and Cauchy noise 2 1.2 Underwater image dehazing 5 2 Preliminaries 9 2.1 Variational models for image denoising 9 2.1.1 Data-fidelity 9 2.1.2 Regularization 11 2.1.3 Optimization algorithm 14 2.2 Methods for image dehazing in the air 15 2.2.1 Dark channel prior 16 2.2.2 Color ellipsoid prior 19 3 Image denoising for Gaussian and Cauchy noise 23 3.1 Second-order structure tensor and hybrid STV 23 3.1.1 Structure tensor total variation 24 3.1.2 Proposed model 28 3.1.3 Discretization of the model 31 3.1.4 Numerical algorithm 35 3.1.5 Experimental results 37 3.2 Weighted nuclear norm minimization for Cauchy noise 46 3.2.1 Variational models for Cauchy noise 46 3.2.2 Low rank minimization by weighted nuclear norm 52 3.2.3 Proposed method 55 3.2.4 ADMM algorithm 56 3.2.5 Numerical method and experimental results 58 4 Image restoration in underwater 71 4.1 Scientific background 72 4.2 Proposed method 73 4.2.1 Color ellipsoid prior on underwater 74 4.2.2 Background light estimation 78 4.3 Experimental results 80 5 Conclusion 87 Appendices 89Docto

Nonlocal Myriad Filters for Cauchy Noise Removal

The contribution of this paper is two-fold. First, we introduce a generalized myriad filter, which is a method to compute the joint maximum likelihood estimator of the location and the scale parameter of the Cauchy distribution. Estimating only the location parameter is known as myriad filter. We propose an efficient algorithm to compute the generalized myriad filter and prove its convergence. Special cases of this algorithm result in the classical myriad filtering, respective an algorithm for estimating only the scale parameter. Based on an asymptotic analysis, we develop a second, even faster generalized myriad filtering technique. Second, we use our new approaches within a nonlocal, fully unsupervised method to denoise images corrupted by Cauchy noise. Special attention is paid to the determination of similar patches in noisy images. Numerical examples demonstrate the excellent performance of our algorithms which have moreover the advantage to be robust with respect to the parameter choice

Space adaptive and hierarchical Bayesian variational models for image restoration

The main contribution of this thesis is the proposal of novel space-variant regularization or penalty terms motivated by a strong statistical rational. In light of the connection between the classical variational framework and the Bayesian formulation, we will focus on the design of highly flexible priors characterized by a large number of unknown parameters. The latter will be automatically estimated by setting up a hierarchical modeling framework, i.e. introducing informative or non-informative hyperpriors depending on the information at hand on the parameters. More specifically, in the first part of the thesis we will focus on the restoration of natural images, by introducing highly parametrized distribution to model the local behavior of the gradients in the image. The resulting regularizers hold the potential to adapt to the local smoothness, directionality and sparsity in the data. The estimation of the unknown parameters will be addressed by means of non-informative hyperpriors, namely uniform distributions over the parameter domain, thus leading to the classical Maximum Likelihood approach. In the second part of the thesis, we will address the problem of designing suitable penalty terms for the recovery of sparse signals. The space-variance in the proposed penalties, corresponding to a family of informative hyperpriors, namely generalized gamma hyperpriors, will follow directly from the assumption of the independence of the components in the signal. The study of the properties of the resulting energy functionals will thus lead to the introduction of two hybrid algorithms, aimed at combining the strong sparsity promotion characterizing non-convex penalty terms with the desirable guarantees of convex optimization