The issue of a “mean shape” of a random set X often arises, in particular in image analysis and pattern detection. There is no canonical definition but one possible approach is the so-called Vorob’ev expectation EV (X), which is closely linked to quantile sets. In this paper, we propose a consistent and ready to use estimator of EV (X) built from independent copies of X with spatial discretizat...

Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduc...

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also anal...

Following in a vein similar to [Zhao et al., 2001], the goal of this project is to develop fast and accurate schemes to reconstruct shapes from unorganised data-sets using variational level set methods. The approach followed is based primarily on the gradient-flow scheme derived in class from variational considerations, and uses implementation details borrowed from [Zhao, 2004, Osher and Fedkiw...

Abstract The use of the fractional Laplacian in image denoising and regularization inverse problems has enjoyed a recent surge popularity, since for discontinuous functions it can behave less aggressively than methods based on H 1 norms, while being linear computable with fast spectral numerical methods. In this work, we examine regularized vanishing noise parameter regime. clean data is assume...