Average and variance of a quasi-parallel family of surfaces. (September 2018) - Sati, Mukul; Rossignac, Jarek
Abstract: We provide theoretical foundations and practical computational tools for the statistical analysis of the local disparity between a family of situated surfaces. We do not mean statistics on discrete measures, such as pairwise Hausdorff distance, of these surfaces, but instead local, shape-variability statistics for all points on these surfaces in a manner that generalizes the mean and variance of numbers. Given a family F of n input surfaces B i, we wish to compute a surface, B, that is the average of the surfaces in F and to associate, with each point p of B, a variance value, v ( p ), which is the average of its squared distances to the input surfaces and hence measures the local disparity between the surfaces of the family. We choose B k as any one of the input surfaces in F . We define B as union of all points p, each resulting from 'snapping' a different point p k of B k . Snapping followed by closest projections may be used to establish pointwise correspondence between all input surfaces. When this correspondence defines a homeomorphism between B and each B i, and hence between each pair of input surfaces, we say that F is a quasi-parallel family of surfaces and that B is their average. In such valid configurations, B exhibits properties that one would expect from an average of surfaces and the resulting variance field over B may be used for analysis, optimization, and visualization. A sufficient condition for this validity is to require that each pair of surfaces in F be projection-homeomorphic. In practice, we only snap the vertices of a triangulation T k that approximates B k . The snap produces a triangulation T that approximates B . We obtain a triangulation T i of an input surface B i by projecting the vertices of T onto it. We propose a practical, although partial validity test that compares T to each T i . Graphical abstract: Highlights: For certain configurations of inputs surfaces, our proposed Snap algorithm yields an intuitive average surface. Snapping involves iterative projections onto local 'average' manifolds – planes in our case – and displays fast convergence to the average surface for such quasi-parallel configurations. For a practical implementation on triangle meshes, we espouse a compute and test paradigm — run the Snap algorithm, and, then, test conformance of computed result with each input.