Low-discrepancy sequences for volume properties in solid modelling

This paper investigates the use of low-discrepancy sequences for computing volume integrals in geometric modelling. An introduction to low-discrepancy point sequences is presented which explains how they can be used to replace random points in Monte Carlo methods. The relative advantages of using low-discrepancy methods compared to random point sequences are discussed theoretically, and then practical results are given for a series of test objects which clearly demonstrate the superiority of the low-discrepancy method. Low-Discrepancy Sequences Monte Carlo methods of integration are used widely for calculating volume integrals in solid modelling. The Monte Carlo method uses randomly generated points inside a box enclosing an object of interest to calculate volume integrals. For example, the volume of the object can be estimated as the ratio of number of points that are contained within the object to the total number of points generated, multiplied by the volume of the box. Naturally, such a method is subject to errors because of the random nature of the sampling, and in particular we cannot guarantee that all parts of space will be sampled equally well. Quasi-Monte Carlo methods [4] use pseudorandom sequences of numbers, called low-discrepancy sequences, for computing multi-dimensional integrals, where here pseudo-random indicates that the sampling is to be done in a rather more structured manner.