In this paper we study a first-order primal-dual algorithm for convex optimization problems with known saddle-point structure. We prove convergence to a saddle-point with rate O(1/N) in finite dimensions, which is optimal for the complete class of non-smooth problems we are considering in this paper. We further show accelerations of the proposed algorithm to yield optimal rates on easier problems. In particular we show that we can achieve O(1/N 2) convergence on problems, where the primal or the dual objective is uniformly convex, and we can show linear convergence, i.e. O(1/e N) on problems where both are uniformly convex. The wide applicability of the proposed algorithm is demonstrated on several imaging problems such as image denoising, image deconvolution, image inpainting, motion estimation and image segmentation. 1

Most variational models for multi-phase image segmentation are non-convex and possess multiple local minima, which makes solving for a global solution an extremely difficult task. In this work, we provide a method for computing a global solution for the (non-convex) multi-phase piecewise constant Mumford-Shah (spatially continuous Potts) image segmentation problem. Our approach is based on using a specific representation of the problem due to Lie et al. [27]. We then rewrite this representation using the dual formulation for total variation so that a variational convexification technique due to Pock et al. [30] may be employed. Unlike some recent methods in this direction, our method can guarantee that a global solution is obtained. We believe our method to be the first in the literature that can make this claim. Once we have the convex optimization problem, we give an algorithm to compute a global solution. We demonstrate our algorithm on several multi-phase image segmentation examples, including a medical imaging application.

In this paper, we address the problem of segmenting data defined on a manifold into a set of regions with uniform properties. In particular, we propose a numerical method when the manifold is represented by a triangular mesh. Based on recent image segmentation models, our method minimizes a convex energy and then enjoys significant favorable properties: it is robust to initialization and avoid the problem of the existence of local minima present in many variational models. The contributions of this paper are threefold: firstly we adapt the convex image labeling model to manifolds; in particular the total variation formulation. Secondly we show how to implement the proposed method on triangular meshes, and finally we show how to use and combine the method in other computer vision problems, such as 3D reconstruction. We demonstrate the efficiency of our method by testing it on various data. 1.

2 Brief review of the state of the art 2

My research project is to design variational models and fast optimization algorithms to solve efficiently problems arising in image processing, machine learning and other applications such as medical imaging and physics. An important part of my research project is to design convex variational models for basic problems in image processing such as image segmentation and image registration. Indeed, most models published in the last twenty years have used non-convex energy minimization models that capture non-optimal solutions. However, in the last few years, I have developed with my collaborators new convex minimization models along with fast algorithms that, I believe, will provide a new paradigm to solve more effectively basic problems in image processing, computer vision, medical imaging, machine learning and physics. Another part of my research project is to develop a unified framework for image processing. Since several image processing problems such as image denoising, image segmentation and image registration have been defined as variational models, I have developed with my collaborators a new variational model, based on the Polyakov energy, to unify these models. The Polyakov model from the physics of high-energy seems a very promising method to unify image processing models. Unlike standard models, ours can define denoising, segmentation, registration on any smooth and parameterized surface s.a. the sphere. The model is also purely geometric,

◮ Surfaces and edges detection, construction and analysis ◮ PET reconstruction–numerical algorithms ◮...Landmarks ◮ AOS scheme – one popular scheme for nonlinear filtering (proposed in Lu-Tai-Neittaanmaki (1991, 1992), popularized by Weickert-etal-1998). ◮ MG mesh independent convergence proof (Tai- Numer Math.

Abstract. We propose an algorithmic framework to compute global solutions of variational models with convex regularity terms that permit quite arbitrary data terms. While the minimization of variational problems with convex data and regularity terms is straight forward (using for example gradient descent), this is no longer trivial for functionals with non-convex data terms. Using the theoretical framework of calibrations the original variational problem can be written as the maximum flux of a particular vector field going through the boundary of the subgraph of the unknown function. Upon relaxation this formulation turns the problem into a convex problem, however, in higher dimension. In order to solve this problem, we propose a fast primal dual algorithm which significantly outperforms existing algorithms. In experimental results we show the application of our method to outlier filtering of range images and disparity estimation in stereo images using a variety of convex regularity terms. Key words. Variational methods, calibrations, total variation, convex optimization. AMS subject classifications. 49M20, 49M29, 65K15, 68U10. 1. Introduction. Energy

Maximum flow (and minimumcut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of maxflow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual mincut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow (CCMF) problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.