Section: New Results

A projection method on measures sets.

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:

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

Projection of a lion image onto $P_N^{1,\infty }$ with $N=8,000$. The figure depicts the resulting line with several values of the iterates of our Algorithm.

More information can be found in [55] .