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,

Image segmentation is often performed via the minimization of an energy function over a domain of possible segmentations. The effectiveness and applicability of such methods depends greatly on the properties of the energy function and its domain, and on what information can be encoded by it. Here we propose an energy function that achieves several important goals. Specifically, our energy function is convex and incorporates shape prior information while simultaneously generating a probabilistic segmentation for multiple regions. Our energy function represents multi-region probabilistic segmentations as elements of a vector space using the isometric log-ratio (ILR) transformation. To our knowledge, these four goals (convex, with shape priors, multi-region, and probabilistic) do not exist together in any other method, and this is the first time ILR is used in an image segmentation method. We provide examples demonstrating the usefulness of these features. 1.