A duality based approach to the minimizing total variation flow in the space H−sH^{-s}

We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given

A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

Asymptotic Expansions for Higher Order Elliptic Equations with an Application to Quantitative Photoacoustic Tomography

In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically

Nonlinear diffusion filtering of images using the topological gradient approach to edges detection

In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain $\Omega \subset \mathbb{R}^2$. In order to generate a nonuniform grid, the adaptive coarsening technique is proposed

The forward-backward scheme for the minimizing total variation flow in H−sH^{-s}.

In the talk, we consider a gradient flow of the total variation in the negative Sobolev space $H^{-s}$, $s\in[0,1]$, under the periodic boundary condition. We derive a dual formulation of a convex variational problem associated with a semi-implicit time discretization of this flow. Based on the forward-backward scheme, we construct a minimizing sequence of a given functional and discuss issues concerning its convergence. We also show and compare results of numerical experiments for simple initial data and different values of the index s. This is joint work with Y. Giga (University of Tokyo) and P. Rybka (University of Warsaw).Non UBCUnreviewedAuthor affiliation: Wrocław University of Science and TechnologyResearche

Nonlinear diffusion filtering of images using the topological gradient approach to edges detection

In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain $\Omega \subset \mathbb{R}^2$. In order to generate a nonuniform grid, the adaptive coarsening technique is proposed

An approach to the minimization of the Mumford–Shah functional using Γ-convergence and topological asymptotic expansion

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius ε> 0 and use the number of balls, multiplied with 2ε, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of Γ-limits to the Mumford–Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli.

An approach to the minimization of the Mumford–Shah functional using Γ-convergence and topological asymptotic expansion

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius ϵ>0 and use the number of balls, multiplied with 2ϵ, as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of Γ-limits to the Mumford–Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli