Vicent Pallardó Julià

We study a generalization of the manifold-valued Rudin-Osher-Fatemi (ROF) model, which involves an initial datum f mapping from a curved compact surface with smooth boundary to a complete, connected and smooth n-dimensional Riemannian manifold. We prove the existence and uniqueness of minimizers under curvature restrictions on the target and topological ones on the range of f. We obtain a series of regularity results on the associated PDE system of a relaxed functional with Neumann boundary condition. We apply these results to the ROF model to obtain Lipschitz regularity of minimizers without further requirements on the convexity of the boundary. Additionally, we provide variants of the regularity statement of independent interest: for 1-dimensional domains (related to signal denoising), local Lipschitz regularity (meaningful for image processing) and Lipschitz regularity for a version of the Mosolov problem coming from fluid mechanics.

We consider manifold constrained weak solutions of quasilinear uniformly elliptic systems of divergence type with a source term that grows at most quadratically with respect to the gradient of the solution. As we impose that the solution lies on a Riemannian manifold, the classical smallness condition for regularity can be relaxed to an inequality relating strict convexity of the squared distance and growth of the leading order term in the tangent component of the source. As a key tool for the proof of a partial regularity result, we derive a fully intrinsic Caccioppoli inequality which may be of independent interest. Finally we show how the systems under consideration have a variational nature and arise in the context of F- or V-harmonic maps.

We consider an abstract concept of perimeter measure space as a very general framework in which one can properly consider two of the most well-studied variational models in image processing: the Rudin–Osher–Fatemi model for image denoising (ROF) and the Mumford–Shah model for image segmentation (MS). We show the linkage between the ROF model and the two phases piecewise constant case of MS in perimeter measure spaces. We show applications of our results to nonlocal image segmentation, via discrete weighted graphs, and to multiclass classification on high dimensional spaces.

In this paper we study a variant to Chan–Vese (CV) segmentation model with rectilinear anisotropy. We show existence of minimizers in the 2-phases case and how they are related to the (anisotropic) Rudin–Osher–Fatemi (ROF) denoising model. Our analysis shows that in the natural case of a piecewise constant on rectangles image (PCR function in short), there exists a minimizer of the CV functional which is also piecewise constant on rectangles over the same grid that the one defined by the original image. In the multiphase case, we show that minimizers of the CV multiphase functional also share this property in the case that the initial image is a PCR function. We also investigate a multiphase and anisotropic version of the Truncated ROF algorithm, and we compare the solutions given by this algorithm with minimizers of the multiphase anisotropic CV functional.

In this paper, we provide a new insight to the two-phase signal segmentation problem. We propose an augmented Lagrangian variational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.