A projection algorithm on measures sets
Document type :
Pré-publication ou Document de travail
Title :
A projection algorithm on measures sets
Author(s) :
Chauffert, Nicolas [Auteur]
Modelling brain structure, function and variability based on high-field MRI data [PARIETAL]
Ciuciu, Philippe [Auteur]
Service NEUROSPIN [NEUROSPIN]
Modelling brain structure, function and variability based on high-field MRI data [PARIETAL]
Kahn, Jonas [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Weiss, Pierre [Auteur]
Institut des Technologies Avancées en sciences du Vivant [ITAV]
Institut de Mathématiques de Toulouse UMR5219 [IMT]
Modelling brain structure, function and variability based on high-field MRI data [PARIETAL]
Ciuciu, Philippe [Auteur]
Service NEUROSPIN [NEUROSPIN]
Modelling brain structure, function and variability based on high-field MRI data [PARIETAL]
Kahn, Jonas [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Weiss, Pierre [Auteur]
Institut des Technologies Avancées en sciences du Vivant [ITAV]
Institut de Mathématiques de Toulouse UMR5219 [IMT]
English keyword(s) :
Thomson problem
attraction-repulsion
optimization
measure projection
continuous line drawing
attraction-repulsion
optimization
measure projection
continuous line drawing
HAL domain(s) :
Mathématiques [math]/Analyse numérique [math.NA]
Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Optimisation et contrôle [math.OC]
Sciences de l'ingénieur [physics]/Traitement du signal et de l'image [eess.SP]
Sciences du Vivant [q-bio]/Ingénierie biomédicale/Imagerie
Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Optimisation et contrôle [math.OC]
Sciences de l'ingénieur [physics]/Traitement du signal et de l'image [eess.SP]
Sciences du Vivant [q-bio]/Ingénierie biomédicale/Imagerie
English abstract : [en]
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 ...
Show more >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.Show less >
Show more >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.Show less >
Language :
Anglais
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