Fast agglomerative clustering for rendering

Abstract

Hierarchical representations of large data sets, such as binary cluster trees, are a crucial component in many scalable algorithms used in various fields. Two major approaches for building these trees are agglomerative, or bottom-up, clustering and divisive, or top-down, clustering. The agglomerative approach offers some real advantages such as more flexible clustering and often produces higher quality trees, but has been little used in graphics because it is frequently assumed to be prohibitively expensive (O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) or worse). In this paper we show that agglomerative clustering can be done efficiently even for very large data sets. We introduce a novel locally-ordered algorithm that is faster than traditional heap-based agglomerative clustering and show that the complexity of the tree build time is much closer to linear than quadratic. We also evaluate the quality of the agglomerative clustering trees compared to the best known divisive clustering strategies in two sample applications: bounding volume hierarchies for ray tracing and light trees in the Lightcuts rendering algorithm. Tree quality is highly application, data set, and dissimilarity function specific. In our experiments the agglomerative-built tree quality is consistently higher by margins ranging from slight to significant, with up to 35% reduction in tree query times.