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

Structured sparsity with convex penalty functions

We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. This problem is relevant in Machine Learning, Statistics and Signal Processing. It is well known that a linear regression can benefit from knowledge that the underlying regression vector is sparse. The combinatorial problem of selecting the nonzero components of this vector can be “relaxed” by regularising the squared error with a convex penalty function like the ℓ1 norm. However, in many applications, additional conditions on the structure of the regression vector and its sparsity pattern are available. Incorporating this information into the learning method may lead to a significant decrease of the estimation error. In this thesis, we present a family of convex penalty functions, which encode prior knowledge on the structure of the vector formed by the absolute values of the regression coefficients. This family subsumes the ℓ1 norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We establish several properties of these penalty functions and discuss some examples where they can be computed explicitly. Moreover, for solving the regularised least squares problem with these penalty functions, we present a convergent optimisation algorithm and proximal method. Both algorithms are useful numerical techniques taylored for different kinds of penalties. Extensive numerical simulations highlight the benefit of structured sparsity and the advantage offered by our approach over the Lasso method and other related methods, such as using other convex optimisation penalties or greedy methods

Data-driven reconstruction methods for photoacoustic tomography:Learning structures by structured learning

Photoacoustic tomography (PAT) is an imaging technique with potential applications in various fields of biomedicine. By visualising vascular structures, PAT could help in the detection and diagnosis of diseases related to their dysregulation. In PAT, tissue is illuminated by light. After entering the tissue, the light undergoes scattering and absorption. The absorbed energy is transformed into an initial pressure by the photoacoustic effect, which travels to ultrasound detectors outside the tissue.This thesis is concerned with the inverse problem of the described physical process: what was the initial pressure in the tissue that gave rise to the detected pressure outside? The answer to this question is difficult to obtain when light penetration in tissue is not sufficient, the measurements are corrupted, or only a small number of detectors can be used in a limited geometry. For decades, the field of variational methods has come up with new approaches to solve these kind of problems. these kind of problems: the combination of new theory and clever algorithms has led to improved numerical results in many image reconstruction problems. In the past five years, previously state-of-the-art results were greatly surpassed by combining variational methods with artificial neural networks, a form of artificial intelligence.In this thesis we investigate several ways of combining data-driven artificial neural networks with model-driven variational methods. We combine the topics of photoacoustic tomography, inverse problems and artificial neural networks.Chapter 3 treats the variational problem in PAT and provides a framework in which hand-crafted regularisers can easily be compared. Both directional and higher-order total variation methods show improved results over direct methods for PAT with structures resembling vasculature.Chapter 4 provides a method to jointly solve the PAT reconstruction and segmentation problem for absorbing structures resembling vasculature. Artificial neural networks are embodied in the algorithmic structure of primal-dual methods, which are a popular way to solve variational problems. It is shown that a diverse training set is of utmost importance to solve multiple problems with one learned algorithm.Chapter 5 provides a convergence analysis for data-consistent networks, which combine classical regularisation methods with artificial neural networks. Numerical results are shown for an inverse problem that couples the Radon transform with a saturation problem for biomedical images.Chapter 6 explores the idea of fully-learned reconstruction by connecting two nonlinear autoencoders. By enforcing a dimensionality reduction in the artificial neural network, a joint manifold for measurements and images is learned. The method, coined learned SVD, provides advantages over other fully-learned methods in terms of interpretability and generalisation. Numerical results show high-quality reconstructions, even in the case where no information on the forward process is used.In this thesis, several ways of combining model-based methods with data-driven artificial neural networks were investigated. The resulting hybrid methods showed improved tomography reconstructions. By allowing data to improve a structured method, deeper vascular structures could be imaged with photoacoustic tomography.<br/