Optimizing dyadic nets

Abstract

We explore the space of (0, m, 2)-nets in base 2 commonly used for sampling. We present a novel constructive algorithm that can exhaustively generate all nets - - up to m-bit resolution - - and thereby compute the exact number of distinct nets. We observe that the construction algorithm holds the key to defining a transformation operation that lets us transform one valid net into another one. This enables the optimization of digital nets using arbitrary objective functions. For example, we define an analytic energy function for blue noise, and use it to generate nets with high-quality blue-noise frequency power spectra. We also show that the space of (0, 2)-sequences is significantly smaller than nets with the same number of points, which drastically limits the optimizability of sequences.