A projection algorithm on measures sets
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...
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| Main Authors | , , , |
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| Format | Journal Article |
| Language | English |
| Published |
01.09.2015
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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. |
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| DOI: | 10.48550/arxiv.1509.00229 |