The Intrinsic Shape of Point Clouds

Abstract

Given a point cloud, in the form of unorganized points, the problem of automatically connecting the dots to obtain an aesthetically pleasing and piecewise-linear closed interpolating boundary shape has been extensively researched for over three decades. In R3 , it is even more complicated to find an aesthetic closed oriented surface. Most previous methods for shape reconstruction exclusively from coordinates work well only when the point spacing on the shape boundary is dense and locally uniform. The problem of shape construction from non-dense and locally non-uniformly spaced point sets is in our opinion not yet satisfactorily solved. Various extensions to earlier methods do not work that well and do not provide any performance guarantees either. Our main thesis in this research is that a point set, even with non-dense and locally non-uniform spacing, has an intrinsic shape which optimizes in some way the Gestalt principles of form perception. This shape can be formally defined as the minimum of an energy function over all possible closed linear piece-wise interpolations of this point set. Further, while finding this optimal shape is NP-hard, it is possible to heuristically search for an acceptable approximation within reasonable time. Our minimization objective is guided by Gestalt s laws of Proximity, Good Continuity and Closure. Minimizing curvature tends to satisfy proximity and good continuity. For computational simplification, we globally minimize the longest-edge-in-simplex, since it is intrinsic to a single facet and also a factor in mean curvature. And we require a closed shape. Using such an intrinsic criterion permits the extraction of an approximate shape with a linearithmic algorithm as a simplicial complex, which we have named the Minimum Boundary Complex. Experiments show that it seems to be a very close approximation to the desired boundary shape and that it retains its genus. Further it can be constructed locally and can also handle sensor data with significant noise. Its quick construction is due to not being restricted by the manifold property, required in the boundary shape. Therefore it has many applications where a manifold shape is not necessary, e.g. visualization, shape retrieval, shadow mapping, and topological data analysis in higher dimensions. The definition of the Minimum Boundary Complex is our first major contribution. Our next two contributions include new methods for constructing boundary shapes by transforming the boundary complex into a close approximation of the minimum boundary shape. These algorithms vary a topological constraint to first inflate the boundary complex to recover a manifold hull and then sculpture it to extract a Minimum Boundary approximation, which interpolates all the points. In the R3 method, we show how local minima can be avoided by covering holes in the hull. Finally, we apply a mesh fairing step to optimize mean curvature directly. We present results for shape construction in R2 and R3 , which clearly demonstrate that our methods work better than the best performing earlier methods for non-dense and locally non-uniformly spaced point sets, while maintaining competitive linearithmic complexity.