A Projection Method on Measures Sets

Abstract

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:

$$\begin{aligned} \inf _{\mu \in {\mathcal {M}}_N} \Vert h\star (\mu - \pi )\Vert _2^2, \end{aligned}$$

where $h\in L^2(\Omega )$ is a kernel, $\Omega \subset {\mathbb {R}}^d$, 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 $({\mathcal {M}}_N)_{N\in {\mathbb {N}}}$ that ensures weak convergence of the projections $(\mu ^*_N)_{N\in {\mathbb {N}}}$ 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 and drawings.