A Projection Method on Measures Sets - Archive ouverte HAL Access content directly
Journal Articles Constructive Approximation Year : 2017

A Projection Method on Measures Sets

(1, 2) , (1, 2) , (3) , (3, 4)


We consider the problem of projecting a probability measure π on a set MN of Radon measures. The projection is defined as a solution of the following variational problem: inf µ∈M N h (µ − π) 2 2 , where h ∈ L 2 (Ω) is a kernel, Ω ⊂ R d and 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 (MN) N ∈N that ensures weak convergence of the projections (µ * N) N ∈N to π. 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
Measure_Projection.pdf (18.75 Mo) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-01432720 , version 1 (12-01-2017)



Nicolas Chauffert, Philippe Ciuciu, Jonas Kahn, Pierre Weiss. A Projection Method on Measures Sets. Constructive Approximation, 2017, 45 (1), pp.83 - 111. ⟨10.1007/s00365-016-9346-2⟩. ⟨hal-01432720⟩
788 View
324 Download



Gmail Facebook Twitter LinkedIn More