Variational problems for maps of bounded variation with values inS 1

Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings

“…An essentially different manifold constrained relaxation problem is the one when the variational functional (0.1) is supposed to be finite only on smooth W 1,1 -maps in C 1 (B n , Y) rather than on the whole class of Sobolev maps W 1,1 (B n , Y). In this setting, as to functional with linear growth, the case Y = S 1 was studied by Demengel and Hadiji [6] in the case of dimension n = 2, and by Giaquinta, Modica and Souček [15] in the case of higher dimension n ≥ 2. Dealing with more general target manifolds Y, Giaquinta and Mucci [11] studied the relaxation problem in the case of the total variation integrand f = |Du|, and more recently [14] in the case of integrands satisfying a suitable isotropy condition of the type f = f (x, u, |Du 1 |, .…”

Variational Models for Image Colorization via Chromaticity and Brightness Decomposition

“…For this kind of constraints there are various numerical methods introduced to accurately compute non-flat features. Some mathematical analysis on S N constraints is studied in [26], and numerical methods are explored in [47]. In this paper, we introduce variational colorization models which use a penalty term to deal with S 2 constraint.…”

Total Variation Regularization for Functions with Values in a Manifold

“…For the special case of the circle M = S 1 , Giaquinta et al [7] prove the existence of minimizers for certain energies in the space of functions with bounded total cyclic variation, again using an embedding in the Euclidean plane. For the case M = S 1 , Cremers and Strekalovskiy [5] recently proposed an implementation of various models for cyclic data, including total variation, quadratic, Huber-TV and Mumford-Shah regularization.…”