On the definition of solution to the total variation flow

We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals.

Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

We consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

The convergence of a numerical method for total variation flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.