k-d Darts: Sampling by k-dimensional flat searches

We formalize sampling a function using k-d darts. A k-d Dart is a set of independent, mutually orthogonal, k-dimensional hyperplanes called k-d flats. A dart has d choose k flats, aligned with the coordinate axes for efficiency. We show k-d darts are useful for exploring a function's properties, such as estimating its integral, or finding an exemplar above a threshold. We describe a recipe for converting some algorithms from point sampling to k-d dart sampling, if the function can be evaluated along a k-d flat.

We demonstrate that k-d darts are more efficient than point-wise samples in high dimensions, depending on the characteristics of the domain: for example, the subregion of interest has small volume and evaluating the function along a flat is not too expensive. We present three concrete applications using line darts (1-d darts): relaxed maximal Poisson-disk sampling, high-quality rasterization of depth-of-field blur, and estimation of the probability of failure from a response surface for uncertainty quantification. Line darts achieve the same output fidelity as point sampling in less time. For Poisson-disk sampling, we use less memory, enabling the generation of larger point distributions in higher dimensions. Higher-dimensional darts provide greater accuracy for a particular volume estimation problem.