Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

Abstract

Abstract. We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies {Jε}, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each Jε is nonsmooth and is expressed as the sum of an 1 regularization term and a smooth nonconvex function. Furthermore, the local minimization of each Jε is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.

Reconstruction of noisy signals by minimization of non-convex functionals

“…The high quality of the minimization results for some functionals of the class F(·) is asserted in the works of Nikolova [17] and Bar et al [15], which successfully deals with both PWC or PWS signals corrupted with Impulsive and/or White noise by using edge-preserving regularization methods with non-smooth fidelity terms such as L 1 , Student's t-distribution, and Mestimators.…”

“…Besides the trend-setting works of Geman and McClure [4] and Chipot et al [6], the practical effectiveness of non-convex functionals have also been shown in other studies. In signal/image denoising, the researches of Aubert et al [1], Nikolova [17] and Selesnick et al [18] illustrate by concrete examples the better performance of some non-convex functionals as compared to convex ones. Vese [19] shows the ability of several non-convex potentials to reconstruct signals, being particularly accurate at restoring edges and linear regions.…”

“…Numerical solutions to the minimization of functionals of the class (2) could produce apparently discontinuous discrete functions [17], which have proven to be satisfactory, computationally feasible, and good visual quality results. In spite of their suitability, these functionals are defined on a domain W 1,p (Ω), p > 1, which does not include functions with discontinuities.…”

Sparse Signal Estimation by Maximally Sparse Convex Optimization

“…Some of these methods proceed first by selecting a non-convex penalty function that induces sparsity more strongly than the 1 norm, and second by developing non-convex optimization algorithms for the minimization of F ; for example, iterative reweighted least squares (IRLS) [38], [69], FOCUSS [37], [58], extensions thereof [47], [65], half-quadratic minimization [12], [34], graduated nonconvexity (GNC) [6], and its extensions [50]- [52], [54].…”

“…We note that in [6], the proposed family of penalty functions are quadratic around the origin and that all a n are equal. On the other hand, the penalty functions we utilize in this work are non-differentiable at the origin as in [52], [54] (so as to promote sparsity) and the a n are not constrained to be equal.…”