A Projection Method on Measures Sets

Nicolas Chauffert (1, 2) , Philippe Ciuciu (1, 2) , Jonas Kahn (3) , Pierre Weiss (3, 4)

1 PARIETAL - Modelling brain structure, function and variability based on high-field MRI data
2 NEUROSPIN - Service NEUROSPIN
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 ITAV - Institut des Technologies Avancées en sciences du Vivant

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.

Mots clés: Constructive quantization; measure theory; nonconvex opti- mization; halftoning; continuous line drawing

Domaines: Imagerie; Traitement du signal et de l'image [eess.SP]

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⟩