Colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given. We propose a couple of variational models using chromaticity color components to colorize black and white images. We first consider total variation minimizing (TV) colorization which is an extension from TV inpainting to color using chromaticity model. Second, we further modify our model to weighted harmonic maps for colorization. This model adds edge information from the brightness data, while it reconstructs smooth color values for each homogeneous region. We introduce penalized versions of the variational models, we analyze their convergence properties, and we present numerical results including extension to texture colorization.

We analyze a variational problem for the recovery of vector valued functions and compute its numerical solution. The data of the problem are a small set of complete samples of the vector valued function and some significant incomplete information where the former are missing. The incomplete information is assumed as the result of a distortion, with values in a lower dimensional manifold. For the recovery of the function we minimize a functional which is formed by the discrepancy with respect to the data and total variation regularization constraints. We show the existence of minimizers in the space of vector valued bounded variation functions. For the computation of minimizers we provide a stable and efficient method. First, we approximate the functional by coercive functionals on W-1,W-2 in terms of Gamma- convergence. Then we realize approximations of minimizers of the latter functionals by an iterative procedure to solve the PDE system of the corresponding Euler-Lagrange equations. The numerical implementation comes naturally by finite element discretization. We apply the algorithm to the restoration of color images from limited color information and gray levels where the colors are missing. The numerical experiments show that this scheme is very fast and robust. The reconstruction capabilities of the model are shown, also from very limited ( randomly distributed) color data. Several examples are included from the real restoration problem of A. Mantegna's art frescoes in Italy.

In recent years a number of variational models related to reconstruction problems in computer vision have been proposed.

We prove a semi-continuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends a well-known result by Ball to the BV framework.

We show the Gamma-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.

Progetto basato sullo sviluppo di modelli e metodi matematico-numerici per l'analisi di processi di degrado chimico (solfatazione, corrosione del ferro) e meccanico (danneggiamento, fratture). La finalita' e' di produrre gli strumenti per: a) simulare tali processi per poterli prevedere ed intervenire preventivamente, b)effettuare un monitoraggio a distanza, ovviamente non invasivo.

Progetto basato sullo sviluppo di modelli e metodi matematico-numerici per l'analisi di processi di degrado chimico (solfatazione, corrosione del ferro) e meccanico (danneggiamento, fratture). La finalita' e' di produrre gli strumenti per: a) simulare tali processi per poterli prevedere ed intervenire preventivamente, b)effettuare un monitoraggio a distanza, ovviamente non invasivo.

We introduce a functional for image segmentation which takes into account the transparencies (or shadowing) and the occlusions between objects located at different depths in space. By minimizing the functional, we try to reconstruct a piecewise smooth approximation of the input image, the contours due to transparencies, and the contours of the objects together with their hidden portions. The functional includes a Mumford-Shah type energy and a term involving the curvature of the contours. The variational properties of the functional are studied, as well as its approximation by Gamma-convergence. The comparison with the Nitzberg-Mumford variational model for segmentation with depth is also discussed.

The purpose of this paper is to show the theoretical soundness of a variational method proposed in image processing for supervised classification. Based on works developed for phase transitions in fluid mechanics, the classification is obtained by minimizing a sequence of functionals. The method provides an image composed of homogeneous regions with regular boundaries, a region being defined as a set of pixels belonging to the same class. In this paper, we show the Gamma-convergence of the sequence of functionals which differ from the ones proposed in fluid mechanics in the sense that the perturbation term is not quadratic but has a finite asymptote at infinity, corresponding to an edge-preserving regularization term in image processing.

Progetto basato sullo sviluppo di modelli e metodi matematico-numerici per l'analisi di processi di degrado chimico (solfatazione, corrosione del ferro) e meccanico (danneggiamento, fratture). La finalita' e' di produrre gli strumenti per: a) simulare tali processi per poterli prevedere ed intervenire preventivamente, b)effettuare un monitoraggio a distanza, ovviamente non invasivo.