The Intrinsic Shape of Point Clouds

The Intrinsic Shape of Point Clouds Stefan Ohrhallinger, PhD Concordia University, 2012 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 R, 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.