The BBM formula revisited

In this paper, we revise the BBM formula due to J. Bourgain, H. Brezis, and P. Mironescu in [1]

On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

BIow-up solutions of some nonlinear elliptic problems

Sin resume

A sharp relative isoperimetric inequality for the square

We compute the exact value of the least “relative perimeter” of a shape $S$, with a given area, contained in a unit square; the relative perimeter of $S$ being the length of the boundary of $S$ that does not touch the border of the square

A sharp relative isoperimetric inequality for the square

We compute the exact value of the least “relative perimeter” of a shape $S$, with a given area, contained in a unit square; the relative perimeter of $S$ being the length of the boundary of $S$ that does not touch the border of the square

Non-convex, non-local functionals converging to the total variation

We present new results concerning the approximation of the total variation, integral(Omega)vertical bar del u vertical bar, of a function u by non-local, non-convex functionals of the form Lambda delta(u) = integral(Omega)integral(Omega)delta phi(vertical bar u(x) - u(y)vertical bar/delta)/vertical bar x-y vertical bar(d+1)dxdy, as delta -> 0, where Omega is a domain in R-d and phi : [0, +infinity) > [0, +infinity) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate, and numerous problems remain open. The original motivation of our work comes from Image Processing. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS

Two subtle convex nonlocal approximations of the BV-norm

Inspired by the BBM formula and by work of G. Leoni and D. Spector, we analyze the asymptotic behavior of two sequences of convex nonlocal functionals (Psi(n)(u)) and (Phi(n)(u)) which converge formally to the BV-norm of u. We show that pointwise convergence when u is not smooth can be delicate; by contrast, Gamma-convergence to the BV-norm is a robust and very useful mode of convergence. (c) 2016 Elsevier Ltd. All rights reserved