### Finite-differences discretizations of the Mumford-Shah functional

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About two years ago, Gobbino [21] gave a proof of a De Giorgi's conjecture on the approximation of the Mumford-Shah energy by means of finite-differences based non-local functionals. In this work, we introduce a discretized version of De Giorgi's approximation, that may be seen as a generalization of Blake and Zisserman's “weak membrane” energy (first introduced in the image segmentation framework). A simple adaptation of Gobbino's results allows us to compute the -limit of this discrete functional...

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell 12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell 14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior...

We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset $S$ of a reference domain, and the complement of $S$ is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure $S$, which is the total work of the pressure and...

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of...

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [ (1990) 629–639], which has been proposed by Nitzberg and Shiota [ (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset of a reference domain, and the complement of is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure , which is the total work of the pressure...

In this paper we study the limit as → ∞ of minimizers of the fractional -norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results....

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of vergence by a sequence of integral functionals defined on piecewise affine functions.

We consider, in an open subset of ${\mathbb{R}}^{N}$, energies depending on the perimeter of a subset $E\subset \Omega $ (or some equivalent surface integral) and on a function which is defined only on $\Omega \setminus E$. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” may collapse into a discontinuity of , whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application,...

We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge to some...

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