Dithering by Differences of Convex Functions

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 (DC). One of them describes attracting forces caused by the image 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 like its coercivity and propose to compute a minimizer by a forwardbackward splitting algorithm. We suggest to compute the special sums occurring in each iteration step by a fast summation technique based on the fast Fourier transform at non-equispaced knots which requires only O(m log m) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our dithering method.

Quadrature Errors, Discrepancies, and Their Relations to Halftoning on the Torus and the Sphere

“…In particular, his strategy generates a high-quality stochastic point distribution in a time that is linear in the number of points. In [55], a novel halftoning framework was proposed, where the vector p :…”

“…The black points are considered as small particles of equal size moving in an environment, e.g., under a glass pane above the image w. The particles are attracted by the image forces w(x) at the points x ∈ G. On the other hand, there is a force of repulsion between the particles modeled by the negative sign of the second sum which becomes minimal if the sum of distances between the particles are maximized. In [55] a circumstantial numerical comparison to other methods was performed which showed that the method achieves unsurpassed quality, has a blue noise spectrum which can keep up with state-of-the-art techniques, and performs superior with respect to Gaussian scale space properties. Furthermore, the use of the nonequispaced fast Fourier transform lowers the complexity of O(M 2 ) per iteration achieved in [48] to O(M log M ).…”

“…Very recently, a new parallel implementation for GPUs was proposed in [28] which reduces the runtime again substantially. [55].…”

“…In [48] the function ϕ(r) = − log(r) with a modification near zero was applied. Further, the authors in [55] also mentioned ϕ(r) = −r τ , 0 < τ < 2, and ϕ(r) = r −τ , τ > 0, with a modification near zero as possible choices. For monotone decreasingly, convex ϕ with ϕ(0) < ∞ it is easily shown that the minimizers of (1.3) exist; cf.…”

“…For monotone decreasingly, convex ϕ with ϕ(0) < ∞ it is easily shown that the minimizers of (1.3) exist; cf. [55].…”