A projection algorithm on measures sets

We consider the problem of projecting a probability measure $π$ on a set $\mathcal{M}\_N$ of Radon measures. The projection is defined as a solution of the following variational problem:\begin{equation*}\inf\_{μ\in \mathcal{M}\_N} \|h\star (μ- π)\|\_2^2,\end{equation*}where $h\in L^2(Ω)$ is a kernel, $Ω\subset \R^d$ 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 $(μ^*\_N)\_{N\in \N}$ 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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