Understanding the Structure of Large, Diverse Collections of Shapes

Due to recent developments in modeling software and advances in acquisition techniques for 3D geometry, large numbers of shapes have been digitized. Existing datasets include millions of real-world objects, cultural heritage artifacts, scientific and engineering models, all of which capture the world around us at nanoto planetary scales. As large repositories of 3D shape collections continue to grow, understanding the data, especially encoding the inter-model similarity and their variations, is of the utmost importance. In this dissertation we address the challenge of deriving structure from a large, unorganized, and diverse collection of 3D polygonal models. By structure we refer to how objects correspond to each other, how they are segmented into semantic parts, and how the parts deform and change across the models. While previous work has generally dealt with small and relatively homogeneous datasets, in this dissertation we concentrate on diverse and large collections. Our contribution is three-fold. First, we present an algorithm for establishing correspondences between pairs of shapes related by a non-uniform deformation. Second, we develop a robust and efficient algorithm for computing per-point similarities between all shapes in a collection of 3D models using only a small subset of all pairwise alignments. And third, we describe an algorithm for finding structure in an unorganized, unlabeled collection of diverse 3D shapes, which is achieved by jointly optimizing for point-to-point correspondences, part segmentations and an explicit model of part deformations. These algorithms enable finding correspondences in large diverse datasets where models are related by non-uniform deformations and model parts have different multiplicity and geometry. These methods also make it possible to segment large collections into consistent sets of parts and to represent most prominent geometric variations in the entire collection.