Building 2D Low-Discrepancy Sequences for Hierarchical Importance Sampling Using Dodecagonal Aperiodic Tiling

This paper introduces a new method for building 2D lowdiscrepancy sequences and fast hierarchical importance sampling. Our approach is based on self-similar tiling of the plane with a set of aperiodic tiles having twelve-fold (dedecagonal) rotational symmetry. Sampling points of our low-discrepancy sequence are associated with tiles, one point per tiles. Each tile is recursively subdivided until the desired local density of samples is reached. A numerical code generated during the subdivision process is used for thresholding to accept or reject the sample. A special number system is specially tailored in order to allow linear numbering of the tiles. The resulting point distribution is more even, compared with that of popular Halton and Hammersley 2D low-discrepancy sequences. It can be successfully applied in a large variety of graphical applications, where fast sampling with good spectral and visual properties is required. Typical applications application are digital halftoning, rendering, geometry processing etc.