A Class of Nonconvex Penalties Preserving Overall Convexity in Optimization-Based Mean Filtering

$\ell_1$ mean filtering is a conventional, optimization-based method to estimate the positions of jumps in a piecewise constant signal perturbed by additive noise. In this method, the $\ell_1$ norm penalizes sparsity of the first-order derivative of the signal. Theoretical results, however, show that in some situations, which can occur frequently in practice, even when the jump amplitudes tend to $\infty$, the conventional method identifies false change points. This issue is referred to as stair-casing problem and restricts practical importance of $\ell_1$ mean filtering. In this paper, sparsity is penalized more tightly than the $\ell_1$ norm by exploiting a certain class of nonconvex functions, while the strict convexity of the consequent optimization problem is preserved. This results in a higher performance in detecting change points. To theoretically justify the performance improvements over $\ell_1$ mean filtering, deterministic and stochastic sufficient conditions for exact change point recovery are derived. In particular, theoretical results show that in the stair-casing problem, our approach might be able to exclude the false change points, while $\ell_1$ mean filtering may fail. A number of numerical simulations assist to show superiority of our method over $\ell_1$ mean filtering and another state-of-the-art algorithm that promotes sparsity tighter than the $\ell_1$ norm. Specifically, it is shown that our approach can consistently detect change points when the jump amplitudes become sufficiently large, while the two other competitors cannot.Comment: Submitted to IEEE Transactions on Signal Processin

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

Non-Convex Methods for Compressed Sensing and Low-Rank Matrix Problems

In this thesis we study functionals of the type \mathcal{K}_{f,A,\b}(\x)= \mathcal{Q}(f)(\x) + \|A\x - \b \| ^2 , where $A$ is a linear map, \b a measurements vector and $\mathcal{Q}$ is a functional transform called \emph{quadratic envelope}; this object is a very close relative of the \emph{Lasry-Lions envelope} and its use is meant to regularize the functionals $f$. Carlsson and Olsson investigated in earlier works the connections between the functionals \mathcal{K}_{f,A,\b} and their unregularized counterparts f(\x) + \|A\x - \b \| ^2 . For certain choices of $f$ the penalty $\mathcal{Q}(f)(\cdot)$ acts as sparsifying agent and the minimization of \mathcal{K}_{f,A,\b}(\x) delivers sparse solutions to the linear system of equations A\x = \b . We prove existence and uniqueness results of the sparse (or low rank, since the functional $f$ can have any Hilbert space as domain) global minimizer of \mathcal{K}_{f,A,\b}(\x) for some instances of $f$, under Restricted Isometry Property conditions on $A$. The theory is complemented with robustness results and a wide range of numerical experiments, both synthetic and from real world problems