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We continue the analysis of some modifications of the total variation imageinpainting method formulated on the space $BV(\Omega)^M$ in the sense that wegeneralize the main results of [32] to the case that a more general datafitting term is involved. As in [32] we deal with vector-valued images, we donot impose any structure condition on our density $F$ and the dimension of thedomain $\Omega$ is arbitrary. Precisely we discuss existence of generalizedsolutions of the corresponding variational problem and we will also pass to theassociated dual variational problem for which we show unique solvability. Amongother things, our results are the uniqueness of the absolutely continuous part$\nabla^a u$ of the gradient of $BV$-solutions $u$ on the entire domain$\Omega$, where outside of the damaged region $D$ we even get uniqueness of$BV$-solutions. Imposing stronger assumptions on our density $F$ and an$L^{\infty}$-condition on our partial observation $f$ we are going to prove amaximum principle for each generalized minimizer and deduce partial$C^{1,\beta}$-regularity of solutions on the entire domain $\Omega$ for all$0<\beta\leq\frac{1}{2}$.

Existence and almost everywhere regularity of generalized minimizers for  a class of variational problems with linear growth related to image  inpainting