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A Projection Method on Measures Sets

Abstract : 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.
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Contributor : Philippe Ciuciu Connect in order to contact the contributor
Submitted on : Thursday, January 12, 2017 - 8:23:35 AM
Last modification on : Tuesday, January 25, 2022 - 3:11:49 AM
Long-term archiving on: : Friday, April 14, 2017 - 2:35:44 PM


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



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