2018/03/27 by Jan Mueller, Mueller, Jan, Christian Tietz +1
Computer Science · Engineering · Mathematics · #49J45 #49N15 #49Q20 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1803.09970
openalex publication_date 2018/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue the analysis of some modifications of the total variation image\ninpainting method formulated on the space BV(\Ω)M in the sense that we\ngeneralize the main results of [32] to the case that a more general data\nfitting term is involved. As in [32] we deal with vector-valued images, we do\nnot impose any structure condition on our density F and the dimension of the\ndomain \Ω is arbitrary. Precisely we discuss existence of generalized\nsolutions of the corresponding variational problem and we will also pass to the\nassociated dual variational problem for which we show unique solvability. Among\nother things, our results are the uniqueness of the absolutely continuous part\n\∇a u of the gradient of BV-solutions u on the entire domain\n\Ω, where outside of the damaged region D we even get uniqueness of\nBV-solutions. Imposing stronger assumptions on our density F and an\nL\∞-condition on our partial observation f we are going to prove a\nmaximum principle for each generalized minimizer and deduce partial\nC1,\β-regularity of solutions on the entire domain \Ω for all\n0<\β\≤\(1)/(2).\n