Point sampling with general noise spectrum

Abstract

Point samples with different spectral noise properties (often defined using color names such as white, blue, green, and red) are important for many science and engineering disciplines including computer graphics. While existing techniques can easily produce white and blue noise samples, relatively little is known for generating other noise patterns. In particular, no single algorithm is available to generate different noise patterns according to user-defined spectra.In this paper, we describe an algorithm for generating point samples that match a user-defined Fourier spectrum function. Such a spectrum function can be either obtained from a known sampling method, or completely constructed by the user. Our key idea is to convert the Fourier spectrum function into a differential distribution function that describes the samples' local spatial statistics; we then use a gradient descent solver to iteratively compute a sample set that matches the target differential distribution function. Our algorithm can be easily modified to achieve adaptive sampling, and we provide a GPU-based implementation. Finally, we present a variety of different sample patterns obtained using our algorithm, and demonstrate suitable applications.

An adaptive point sampler on a regular lattice

“…This approach is well-suited for adaptive sampling, and aims at substantially higher spectral qualities than Wang tiles; but this comes at a considerable cost in memory, since the whole tiling structure has to be stored. Through two subsequent steps of development [Ostromoukhov 2007;Wachtel et al 2014], this approach reached a quality that enables almost full control over the spectral properties of the conveyed point sets, using sophisticated optimization techniques [Heck et al 2013;Öztireli and Gross 2012;Zhou et al 2012] that were developed concurrently. Unfortunately, to that end the required memory footprint grows beyond the practical limits of many applications: gigabyte-sized lookup tables for a single spectral profile.…”

“…11 Spectral Control. Target matching algorithms [Heck et al 2013;Öztireli and Gross 2012;Zhou et al 2012] can easily be adapted to our framework thanks to the toroidal domain optimization environment we have. The only change needed is to average the displacements of the points holding the same ID, as advocated by all of [Ahmed et al 2015;Ostromoukhov 2007;Ostromoukhov et al 2004;Wachtel et al 2014].…”

“…Please note that the radial distribution function of some noise profiles, e.g. pink noise [Zhou et al 2012], makes them unrealizable under the regular lattice constraint, and we have to allow the sample points to leave the tile boundaries, or use more samples per tile. The distribution of a point set (top) before and (bottom) after snapping, demonstrating that the net effect of snapping is a slight jitter.…”

AA patterns for point sets with controlled spectral properties

“…Direct production algorithms to generate such point sets include stratified jittering, dart throwing [Dippé and Wold 1985;Cook 1986;Mitchell 1987], and their variants. There are also iterative optimization techniques to modify a given point set, including Lloyd's relaxation algorithm [McCool and Fiume 1992] and its variants [Balzer et al 2009;Xu et al 2011;Chen et al 2012;de Goes et al 2012], other iterative methods [Schmaltz et al 2010;Fattal 2011;Schlömer et al 2011], and the recently invented target-matching algorithms [Zhou et al 2012;Öztireli and Gross 2012;Heck et al 2013]. …”

A Shape‐Aware Model for Discrete Texture Synthesis

“…More recently, several contributions focused on simulating point processes with more varied spectra or correlation functions [LWSF10, ZHWW12,OG12]. Such approaches show that pairwise point distances are the key characteristics of point processes.…”