UDC 517.9We continue the analysis of modifications of the total variation image inpainting method formulated on the space BV (Ω) M and treat the case of vector-valued images where we do not impose any structure condition on the density F and the dimension of the domain Ω is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part ∇ a u of the gradient of BV -solutions u on the domain Ω and get the uniqueness of BV -solutions outside the damaged region D. We also prove new density results for functions of bounded variation and for Sobolev functions. Bibliography: 36 titles.
Dedicated to the memory of Stefan Hildebrandt AMS Subject Classification: 26A45, 49J05, 49J45, 49M29, 34B15 Keywords: total variation, signal denoising, variational problems in one independent variable, linear growth, existence and regularity of solutions.
AbstractWe consider one-dimensional variants of the classical first order total variation denoising model introduced by Rudin, Osher and Fatemi. This study is based on our previous work on various denoising and inpainting problems in image analysis, where we applied variational methods in arbitrary dimensions. More than being just a special case, the one-dimensional setting allows us to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, we obtain quite strong regularity results for a class of data functions that contains many of the standard examples from signal processing such as rectangle-or triangle signals as a special case. An analysis of the related Euler-Lagrange equation, which turns out to be a second order two-point boundary value problem with Neumann conditions, by ODE methods completes the picture of our investigation.Acknowledgement: The authors thank Michael Bildhauer for many stimulating discussions.
Keywords: variational problems of linear growth, TV-regularization, denoising and inpainting of multicolor images, existence of solutions in Sobolev spaces.
AbstractWe study classes of variational problems with energy densities of linear growth acting on vector-valued functions. Our energies are strictly convex variants of the TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerful tool in the field of image recovery. In contrast to our previous work we here try to figure out conditions under which we can solve these variational problems in classical spaces, e.g. in the Sobolev class W 1,1 .
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