We aim at constructing solutions to the minimizing problem for the variant of RudinOsher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (P CR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a P CR function, then solutions to both considered problems also have this property. For P CR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.MSC: 68U10, 35K67, 35C05, 49N60, 35B65

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i. e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some assumptions.

Let u be the minimizer of vectorial total variation (V T V ) with L 2 data-fidelity term on an interval I. We show that the total variation of u over any subinterval of I is bounded by that of the datum over the same subinterval. We deduce analogous statement for the vectorial total variation flow on I.

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