On Some Perturbations of the total variation image inpainting method. Part II: Relaxation and Dual Variational Formulation

Abstract

We continue the analysis of some strongly elliptic modifications of the total variation image inpainting model formulated in the space BV(Ω) and investigate the corresponding dual variational problems. Remarkable features are the uniqueness of the dual solution and the uniqueness of the absolutely continuous part ∇ a u of the gradient of BV-solutions u on the whole domain. Additionally, any BV-minimizer u automatically satisfies the inequality 0 ≤ u ≤ 1, which means that u measures the intensity of the grey level. Outside of the damaged region we even have the uniqueness of BV-solutions, whereas on the damaged domain the L 2-deviation $$ {\left\Vert u-\upsilon \right\Vert}_{L^2} $$ of different solutions is governed by the total variation of the singular part ∇ s (u − υ) of the vector measure ∇(u − υ). Moreover, the dual solution is related to the BV-solutions through an equation of stress-strain type. Bibliography: 23 titles.