A projection algorithm on measures sets

We consider the problem of projecting a probability measure $\pi$ on a set $\mathcal{M}\mathrm{\_}N$ 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 $h\in L^2(\mathrm{\Omega })$ is a kernel, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ 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 $(\mathcal{M}\mathrm{\_}N)\mathrm{\_}N\in \mathbb{N}$ that ensures weak convergence of the projections $({\mu }^{\ast }\mathrm{\_}N)\mathrm{\_}N\in \mathbb{N}$ 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.