A combined first and second order variational approach for image reconstruction

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.Comment: 34 pages, 89 figure

kk-MLE: A fast algorithm for learning statistical mixture models

We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expected complete) likelihood. Given prescribed mixture weights, the hard clustering $k$-MLE algorithm iteratively assigns data to the most likely weighted component and update the component models using Maximum Likelihood Estimators (MLEs). Using the duality between exponential families and Bregman divergences, we prove that the local convergence of the complete likelihood of $k$-MLE follows directly from the convergence of a dual additively weighted Bregman hard clustering. The inner loop of $k$-MLE can be implemented using any $k$-means heuristic like the celebrated Lloyd's batched or Hartigan's greedy swap updates. We then show how to update the mixture weights by minimizing a cross-entropy criterion that implies to update weights by taking the relative proportion of cluster points, and reiterate the mixture parameter update and mixture weight update processes until convergence. Hard EM is interpreted as a special case of $k$-MLE when both the component update and the weight update are performed successively in the inner loop. To initialize $k$-MLE, we propose $k$-MLE++, a careful initialization of $k$-MLE guaranteeing probabilistically a global bound on the best possible complete likelihood.Comment: 31 pages, Extend preliminary paper presented at IEEE ICASSP 201

Iterative algorithms for total variation-like reconstructions in seismic tomography

A qualitative comparison of total variation like penalties (total variation, Huber variant of total variation, total generalized variation, ...) is made in the context of global seismic tomography. Both penalized and constrained formulations of seismic recovery problems are treated. A number of simple iterative recovery algorithms applicable to these problems are described. The convergence speed of these algorithms is compared numerically in this setting. For the constrained formulation a new algorithm is proposed and its convergence is proven.Comment: 28 pages, 8 figures. Corrected sign errors in formula (25

Joint reconstruction-segmentation on graphs

Practical image segmentation tasks concern images which must be reconstructed from noisy, distorted, and/or incomplete observations. A recent approach for solving such tasks is to perform this reconstruction jointly with the segmentation, using each to guide the other. However, this work has so far employed relatively simple segmentation methods, such as the Chan--Vese algorithm. In this paper, we present a method for joint reconstruction-segmentation using graph-based segmentation methods, which have been seeing increasing recent interest. Complications arise due to the large size of the matrices involved, and we show how these complications can be managed. We then analyse the convergence properties of our scheme. Finally, we apply this scheme to distorted versions of ``two cows'' images familiar from previous graph-based segmentation literature, first to a highly noised version and second to a blurred version, achieving highly accurate segmentations in both cases. We compare these results to those obtained by sequential reconstruction-segmentation approaches, finding that our method competes with, or even outperforms, those approaches in terms of reconstruction and segmentation accuracy.Comment: 33 pages, 20 figure

Diffusion equations and inverse problems regularization.

The present thesis can be split into two dfferent parts: The first part mainly deals with the porous and fast diffusion equations. Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators. The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation