A projection algorithm on measures sets

Overview & Synopsis

We consider the problem of projecting a probability measure $\pi$ on a set $\mathcal{M}_N$ 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 $h\in L^2(\Omega)$ is a kernel, $\Omega\subset \R^d$ 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 $(\mu^*_N)_{N\in \N}$ 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.