Antonin Chambolle

A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging

In this paper we study a first-order primal-dual algorithm for non-smooth convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions for the complete class of problems. We further show accelerations of the proposed algorithm to yield improved rates on problems with some degree of smoothness. In particular we show that we can achieve O(1/N2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(ωN) for some ω∈(0,1), on smooth problems. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and multi-label image segmentation.

Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage

This paper examines the relationship between wavelet-based image processing algorithms and variational problems. Algorithms are derived as exact or approximate minimizers of variational problems; in particular, we show that wavelet shrinkage can be considered the exact minimizer of the following problem. Given an image F defined on a square I, minimize over all g in the Besov space B(1)(1)(L (1)(I)) the functional |F-g|(L2)(I)(2)+lambda|g|(B(1)(1 )(L(1(I)))). We use the theory of nonlinear wavelet image compression in L(2)(I) to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i.i.d., mean zero, Gaussian noise. A new signal-to-noise ratio (SNR), which we claim more accurately reflects the visual perception of noise in images, arises in this derivation. We present extensive computations that support the hypothesis that near-optimal shrinkage parameters can be derived if one knows (or can estimate) only two parameters about an image F: the largest alpha for which FinEpsilon(q)(alpha )(L(q)(I)),1/q=alpha/2+1/2, and the norm |F|B(q)(alpha)(L(q)(I)). Both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than Donoho and Johnstones VisuShrink procedure; an example suggests, however, that Donoho and Johnstones SureShrink method, which uses a different shrinkage parameter for each dyadic level, achieves a lower error than our procedure.

Practical, Unified, Motion and Missing Data Treatment in Degraded Video

We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.

Total variation minimization and a class of binary MRF models

We observe that there is a strong connection between a whole class of simple binary MRF energies and the Rudin-Osher-Fatemi (ROF) Total Variation minimization approach to image denoising. We show, more precisely, that solutions to binary MRFs can be found by minimizing an appropriate ROF problem, and vice-versa. This leads to new algorithms. We then compare the efficiency of various algorithms.

Image Decomposition into a Bounded Variation Component and an Oscillating Component

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.

Dual Norms and Image Decomposition Models

Following a recent work by Y. Meyer, decomposition models into a geometrical component and a textured component have recently been proposed in image processing. In such approaches, negative Sobolev norms have seemed to be useful to modelize oscillating patterns. In this paper, we compare the properties of various norms that are dual of Sobolev or Besov norms. We then propose a decomposition model which splits an image into three components: a first one containing the structure of the image, a second one the texture of the image, and a third one the noise. Our decomposition model relies on the use of three different semi-norms: the total variation for the geometrical component, a negative Sobolev norm for the texture, and a negative Besov norm for the noise. We illustrate our study with numerical examples.

Diagonal preconditioning for first order primal-dual algorithms in convex optimization

In this paper we study preconditioning techniques for the first-order primal-dual algorithm proposed in [5]. In particular, we propose simple and easy to compute diagonal preconditioners for which convergence of the algorithm is guaranteed without the need to compute any step size parameters. As a by-product, we show that for a certain instance of the preconditioning, the proposed algorithm is equivalent to the old and widely unknown alternating step method for monotropic programming [7]. We show numerical results on general linear programming problems and a few standard computer vision problems. In all examples, the preconditioned algorithm significantly outperforms the algorithm of [5].

A convex relaxation approach for computing minimal partitions

In this work we propose a convex relaxation approach for computing minimal partitions. Our approach is based on rewriting the minimal partition problem (also known as Potts model) in terms of a primal dual Total Variation functional. We show that the Potts prior can be incorporated by means of convex constraints on the dual variables. For minimization we propose an efficient primal dual projected gradient algorithm which also allows a fast implementation on parallel hardware. Although our approach does not guarantee to find global minimizers of the Potts model we can give a tight bound on the energy between the computed solution and the true minimizer. Furthermore we show that our relaxation approach dominates recently proposed relaxations. As a consequence, our approach allows to compute solutions closer to the true minimizer. For many practical problems we even find the global minimizer. We demonstrate the excellent performance of our approach on several multi-label image segmentation and stereo problems.

An algorithm for minimizing the Mumford-Shah functional

In this work we revisit the Mumford-Shah functional, one of the most studied variational approaches to image segmentation. The contribution of this paper is to propose an algorithm which allows to minimize a convex relaxation of the Mumford-Shah functional obtained by functional lifting. The algorithm is an efficient primal-dual projection algorithm for which we prove convergence. In contrast to existing algorithms for minimizing the full Mumford-Shah this is the first one which is based on a convex relaxation. As a consequence the computed solutions are independent of the initialization. Experimental results confirm that the proposed algorithm determines smooth approximations while preserving discontinuities of the underlying signal.

A l1-Unified Variational Framework for Image Restoration

Among image restoration literature, there are mainly two kinds of approach. One is based on a process over image wavelet coefficients, as wavelet shrinkage for denoising. The other one is based on a process over image gradient. In order to get an edge-preserving regularization, one usually assume that the image belongs to the space of functions of Bounded Variation (BV). An energy is minimized, composed of an observation term and the Total Variation (TV) of the image.