Golden Ratio Sequences for Low-Discrepancy Sampling

Most classical constructions of low-discrepancy point sets are based on generalizations of the one-dimensional binary van der Corput sequence whose implementation requires non-trivial bit-operations. As an alternative we introduce the quasi-regular golden ratio sequences which are based on the fractional part of successive integer multiples of the golden ratio. By leveraging results from number theory we show that point sets which evenly cover the unit square or disc can be computed by a simple incremental permutation of a generator golden ratio sequence. We compare ambient occlusion images generated with a Monte Carlo ray-tracer based on random, Hammersley, blue noise and golden ratio point sets. The source code of the ray-tracer used for our experiments is available online. 1. Low-discrepancy point sets Hammersley A classical approach for generating quasi-random sequences of point samples in the unit square [0, 1) relies on generalizations of the van der Corput low-discrepancy sequence based on small prime numbers [Niederreiter 92]. For instance, the Hammersley point set of size N is {(H2(i), i/N)}i=1 (1) © A K Peters, Ltd. 1 1086-7651/06 $0.50 per page