Density-controlled sampling of parametric surfaces using adaptive space-filling curves

Abstract. Low-discrepancy point distributions exhibit excellent unifor-mity properties for sampling in applications such as rendering and mea-surement. We present an algorithm for generating low-discrepancy point distributions on arbitrary parametric surfaces using the idea of convert-ing the 2D sampling problem into a 1D problem by adaptively mapping a space-filling curve onto the surface. The 1D distribution takes into ac-count the parametric mapping by employing a corrective approach simi-lar to histogram equalisation to ensure that it gives a 2D low-discrepancy point distribution on the surface. This also allows for control over the lo-cal density of the distribution, e.g. to place points more densely in regions of higher curvature. To allow for parametric distortion, the space-filling curve is generated adaptively to cover the surface evenly. Experiments show that this approach efficiently generates low-discrepancy distribu-tions on arbitrary parametric surfaces and creates nearly as good results as well-known low-discrepancy sampling methods designed for particular surfaces like planes and spheres. However, we also show that machine-precision limitations may require surface reparameterisation in addition to adaptive sampling.