Grid-Based Techniques for Semi-Stochastic Sampling

Abstract

Stochastic point sampling, mostly blue-noise sampling, is a fundamental process in computer graphics, appearing in applications like image reconstruction, halftoning and stippling, (quasi)-Monte Carlo integration, and distributing various kinds of objects (people, plants, etc). Many of these applications require real-time performance, and such speeds are not met by common generation algorithms, which lead to the idea of tabulating and reusing pre-optimized sets of sampling points. Being "noise", however, such point sets inherently bear high entropy, which makes it challenging to store and manipulate them. Many ideas have been proposed over the past three decades to address this problem of storing and redistributing noise sets. The current state of the art encompasses two categories: methods based on sets of regular tiles, with multiple sample points per tile, and methods based on complex-shaped recursive tiles, with a single sample point per tile. These methods offer different trade-offs in terms of quality, speed, memory usage, flexibility, and coding complexity, but no method is considered optimal in all aspects. In this thesis we study a new approach for distributing pre-optimized sample points. The main idea is to morph a regular grid into the desired distribution. Thanks to the simple underlying structure, this approach is readily simpler than exisiting methods, and we also demonstrate its advantages in terms of both quality and performance. More importantly, we demonstrate that a grid-based approach enables the development of analytic and objective solutions to replace the heuristic treatments used earlier. We present three solutions aimed at three sampling scenarios. For uniform sampling, e.g. for distributing objects, we employ ornamental subsets of the regular grid known as AA Patterns, and demonstrate that, using a finite lookup table of any affordable size, the patterns can distribute non-periodic sets of optimized samples. For low-discrepancy sampling, mainly needed in quasi-Monte Carlo integration, we develop a new grid-based low-discrepancy point set, and demonstrate that such a set can be rearranged to acquire a blue-noise spectrum without loosing its low-discrepancy property. Finally, for adaptive sampling, e.g. for stippling or importance sampling, we develope a new indexing scheme, based on the Thue-Morse word, to identify individual points in a multi-resolution grid, and use it to build an adaptive sampler on a regular lattice. We demonstrate the superiority of our solutions over respective state-of-the-art methods. In addition, we present a new blue-noise optimization algorithm that is well-suited for offline optimization in our solutions and other lookup methods.