Article

# Dithering by Differences of Convex Functions.

• ##### Joachim Weickert
SIAM Journal on Imaging Sciences (Impact Factor: 2.97). 01/2011; 4:79-108. DOI: 10.1137/100790197
Source: DBLP

ABSTRACT Motivated by a recent halftoning method which is based on electrostatic principles, we analyze a halftoning framework where one minimizes a functional consisting of the difference of two convex functions. One describes attracting forces caused by the image’s gray values; the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: The points are pairwise distinct, lie within the image frame, and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional, such as its coercivity, and propose computing a minimizer by a forward-backward splitting algorithm. We suggest computing the special sums occurring in each iteration step of our dithering algorithm by a fast summation technique based on the fast Fourier transform at nonequispaced knots, which requires only O(mlogm) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our dithering method.

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