Ricerc@Sapienza

We introduce a novel computational framework for digital geometry processing, based upon the derivation of a nonlinear operator associated to the total variation functional. Such an operator admits a generalized notion of spectral decomposition, yielding a convenient multiscale representation akin to Laplacian-based methods, while at the same time avoiding undesirable over-smoothing effects typical of such techniques.

We consider the partial differential equation

u−f=div(u^m ∇u/|∇u|)
with f nonnegative and bounded and m∈R. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ^{N−1}-Hausdorff measure. Results and proofs extend to more general nonlinearities.

Let u be the minimizer of vectorial total variation with L2 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.