Algorithms for Comparing and Analyzing 3D Geometry

ˆe world around us consists of objects of vastly varying shapes, sizes, and geometric complexity. A natural question is how to capture such geometric variations? We propose algorithms for capturing, comparing, and analyzing such D geometry. In the rst part of this thesis, we propose algorithms to automate the various stages of a standard shape acquisition pipeline. Typically, a D scanner captures object geometry from multiple directions. From each viewpoint we get a partial geometric model or a scan. Scan registration involves stitching these scans together to form one consolidatedmodel. We present an algorithm for the automatic global rigid alignment of two D shapes, without any assumptions about their initial poses. We implicitly evaluate all the possible correspondence assignments between a set of selected feature points without having to explicitly enumerate each of the correspondences. ˆe rough initial alignment is further re ned using local alignment which is posed as a minimization of the squared distance between the underlying surfaces. We locally approximate the squared distance eld using quadratic functions and then develop a linear system whose solution gives the local aligning rigid transform. A er model registration, we o en get an incomplete representation of the scanned object with areas of missing data. We present a novel approach for plausible Dmodel completion using geometric priors. Our method retrieves suitable context models from a model database, warps the retrievedmodels to conformwith the input data, and consistently blends the warpedmodels to obtain the nal consolidated D shape. In the second part, we introduce two shape analysis tools. We present a new algorithm that processes geometricmodels and e ciently discovers and extracts a compact representation of their Euclidean symmetries. ˆese symmetries can be partial, approximate, or both. ˆe extracted symmetry graph representation captures important high-level information