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

Abstract

Abstract. This paper deals with continuous-domain quantization, which aims to create the illusion of a gray-value image by appropriately distributing black dots. For lack of notation, we refer to the process as halftoning, which is usually associated with the quantization on a discrete grid. Recently a framework for this task was proposed by minimizing an attraction-repulsion functional consisting of the difference of two continuous, convex functions. The first one of these functions describes attracting forces caused by the image gray values, the second one enforces repulsion between the dots. In this paper, we generalize this approach by considering quadrature error functionals on reproducing kernel Hilbert spaces (RKHSs) with respect to the quadrature nodes, where we ask for optimal distributions of these nodes. For special reproducing kernels these quadrature error functionals coincide with discrepancy functionals, which leads to a geometric interpretation. It turns out that the original attraction-repulsion functional appears for a special RKHS of functions on R 2 . Moreover, our more general framework enables us to consider optimal point distributions not only in R 2 but also on the torus T 2 and the sphere S 2 . For a large number of points the computation of such point distributions is a serious challenge and requires fast algorithms. To this end, we work in RKHSs of bandlimited functions on T 2 and S 2 . Then the quadrature error functional can be rewritten as a least squares functional. We use a nonlinear conjugate gradient method to compute a minimizer of this functional and show that each iteration step can be computed in an efficient way by fast Fourier transforms at nonequispaced nodes on the torus and the sphere. Numerical examples demonstrate the good quantization results obtained by our method. 1. Introduction. Halftoning is a method for creating the illusion of a continuous tone image having only a small number of tones available. It is usually associated with the quantization on a discrete grid. In this paper, we use the name halftoning for continuous-domain quantization; cf. [48,28]. We focus just on two tones, black and white, and ask for appropriate distributions of the black "dots." Applications of halftoning include printing and geometry processing [54] as well as sampling problems occurring in rendering [60], relighting [36], and artistic nonphotorealistic image visualization [4,49]. Halftoning has been an active field of research for many years.Dithering methods which place the black dots at image grid points include, e.g., ordered dithering [7,47], error diffusion [22,32,52,53], global or direct binary search [1,6], and structure-aware halftoning [45]. Ostromoukhov [44] extended the error

Quadrature Nodes Meet Stippling Dots

“…Interestingly, this is also related to 'capacity constraints' in stippling techniques, see [2,3]. Note that a relation between the distance kernels K(x, y) = x − y 2 on T 2 and S 2 and the corresponding discrepancy kernels was shown numerically in [9].…”

“…Note that for X = S 2 and w ≡ 1 there is a close relation between the minimizers E N (p) and spherical designs, see [19,8,9].…”

“…7 concludes the paper. Due to the page limitation proofs of certain theorems of this paper are given in the preprint [9].…”

“…To start with, the following theorem shows a relation between the quadrature error functional and the attraction-repulsion functional (2). For a proof we refer to [9].…”

“…The corresponding discrepancy kernel is K B (d) = d , S 1 and T 2 and their relations to distance kernels Φ(x, y) = − x − y 2 , we refer to [15,9].…”