[show abstract][hide abstract] ABSTRACT: We study the kinetic mean-field limits of the discrete systems of interacting
particles used for halftoning of images in the sense of continuous-domain
quantization. Under mild assumptions on the regularity of the interacting
kernels we provide a rigorous derivation of the mean-field kinetic equation.
Moreover, we study the energy of the system, show that it is a Lyapunov
functional and prove that in the long time limit the solution tends to an
equilibrium given by a local minimum of the energy. In a special case we prove
that the equilibrium is unique and is identical to the prescribed image
profile. This proves the consistency of the particle halftoning method when the
number of particles tends to infinity.
[show abstract][hide abstract] ABSTRACT: The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2×2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfeld’s algorithm for the computation of ℓ2 vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfeld’s algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.
J. Computational Applied Mathematics. 01/2012; 236:2200-2222.
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