A projection algorithm on measures sets - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2015

A projection algorithm on measures sets

Résumé

We consider the problem of projecting a probability measure $\pi$ on a set $\mathcal{M}_N$ of Radon measures. The projection is defined as a solution of the following variational problem: \begin{equation*} \inf_{\mu\in \mathcal{M}_N} \|h\star (\mu - \pi)\|_2^2, \end{equation*} where $h\in L^2(\Omega)$ is a kernel, $\Omega\subset \R^d$ and $\star$ denotes the convolution operator. To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with $N$ dots) or continuous line drawing (representing an image with a continuous line). We provide a necessary and sufficient condition on the sequence $(\mathcal{M}_N)_{N\in \N}$ that ensures weak convergence of the projections $(\mu^*_N)_{N\in \N}$ to $\pi$. We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings.
Fichier principal
Vignette du fichier
papier_projection.pdf (11.42 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01188069 , version 1 (28-08-2015)

Identifiants

Citer

Nicolas Chauffert, Philippe Ciuciu, Jonas Kahn, Pierre Weiss. A projection algorithm on measures sets. 2015. ⟨hal-01188069⟩
566 Consultations
98 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More