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    • "This variational problem is a prototypical example that motivates our study. As explained later, it is intimately related to recent works on image halftoning by means of attraction-repulsion potentials proposed in [26] [28] [13]. In references [11] [9] [10] this principle is shown to have far reaching applications ranging from numerical integration, quantum physics, economics (optimal location of service centers) or biology (optimal population distributions). "
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    ABSTRACT: 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: inf µ∈M N h (µ − π) 2 2 , where h ∈ L 2 (Ω) is a kernel, Ω ⊂ 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 (MN) N ∈N that ensures weak convergence of the projections (µ * N) N ∈N to π. 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.
    Preview · Article · Mar 2015
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    • "in the context of discrete variational methods for image dithering [20] [21] [17]. In particular, in [21], for ψ a (·) = ψ r (·) = | · | being the Euclidean norm, the authors considered the discrete energy functional (1.4) E N [p] "
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    ABSTRACT: We study the long time behavior of the Wasserstein gradient flow for an energy functional consisting of two components: particles are attracted to a fixed profile $\omega$ by means of an interaction kernel $\psi_a(z)=|z|^{q_a}$,and they repel each other by means of another kernel $\psi_r(z)=|z|^{q_r}$. We focus on the case of one space dimension and assume that $1\le q_r\le q_a\le 2$. Our main result is that the flow converges to an equilibrium if either $q_r<q_a$ or $1\le q_r=q_a\le4/3$,and if the solution has the same (conserved) mass as the reference state $\omega$. In the cases $q_r=1$ and $q_r=2$, we are able to discuss the behavior for different masses as well, and we explicitly identify the equilibrium state, which is independent of the initial condition. Our proofs heavily use the inverse distribution function of the solution.
    Preview · Article · Jan 2014 · SIAM Journal on Mathematical Analysis
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    • "However, the generation of optimal quantization points by the minimization of functionals of the type (1.2) might also be subjected to criticism. First of all the functionals are in general nonconvex, rendering their global optimization, especially in high-dimension, a problem of high computational complexity, although, being the functional the difference of two convex terms, numerical methods based on the alternation of descent and ascent algorithms proved to be rather efficient in practice, see [32] for details. Especially one has to notice that for kernels generated by radial symmetric functions applied on the Euclidean distance of their arguments, the evaluation of the functional and of its subgradients may result in the computation of convolutions which can be rapidly implemented by non-equispaced fast Fourier transforms [18] [29]. "
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    ABSTRACT: This paper is concerned with the study of the consistency of a variational method for probability measure quantization, deterministically realized by means of a minimizing principle, balancing power repulsion and attraction potentials. The proof of consistency is based on the construction of a target energy functional whose unique minimizer is actually the given probability measure \omega to be quantized. Then we show that the discrete functionals, defining the discrete quantizers as their minimizers, actually \Gamma-converge to the target energy with respect to the narrow topology on the space of probability measures. A key ingredient is the reformulation of the target functional by means of a Fourier representation, which extends the characterization of conditionally positive semi-definite functions from points in generic position to probability measures. As a byproduct of the Fourier representation, we also obtain compactness of sublevels of the target energy in terms of uniform moment bounds, which already found applications in the asymptotic analysis of corresponding gradient flows. To model situations where the given probability is affected by noise, we additionally consider a modified energy, with the addition of a regularizing total variation term and we investigate again its point mass approximations in terms of \Gamma-convergence. We show that such a discrete measure representation of the total variation can be interpreted as an additional nonlinear potential, repulsive at a short range, attractive at a medium range, and at a long range not having effect, promoting a uniform distribution of the point masses.
    Preview · Article · Oct 2013 · Journal of Fourier Analysis and Applications
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