Fast tile-based adaptive sampling with user-specified Fourier spectra

Abstract

We introduce a fast tile-based method for adaptive two-dimensional sampling with user-specified spectral properties. At the core of our approach is a deterministic, hierarchical construction of self-similar, equi-area, tri-hex tiles whose centroids have a spatial distribution free of spurious spectral peaks. A lookup table of sample points, computed offline using any existing point set optimizer to shape the samples' Fourier spectrum, is then used to populate the tiles. The result is a linear-time, adaptive, and high-quality sampling of arbitrary density functions that conforms to the desired spectral distribution, achieving a speed improvement of several orders of magnitude over current spectrum-controlled sampling methods.

An adaptive point sampler on a regular lattice

“…In recursive mode our sampler is an order of magnitude slower, delivering 13M samples per second, but is still faster than the speeds reported for common adaptive samplers [Wachtel et al 2014]. Thus, our sampler meets the speed requirements of time-sensitive applications such are real-time rendering.…”

“…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.…”

“…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]. Confining the sample points to the tile boundary is not mandatory, though it is highly desirable.…”

“…[Kopf et al 2006;Ostromoukhov 2007;Wachtel et al 2014], but it simplifies greatly for our case that uses orthogonal coordinates. Each ID is assigned log 2 (1/s) fractional bits, on each axis, to indicate the row and column of the nominated target child; then Eq (7) reduces to concatenation of bits.…”

“…1. Please note that in contrast to Polyhexes [Wachtel et al 2014] and Polyominos [Ostromoukhov 2007], we do not need to store any geometric rules to define the tiles. …”