The power crust is a construction which takes a sample of points from the surface of a three-dimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.This idea leads to a simple algorithm with theoretical guarantees comparable to those of other surface reconstruction and medial axis approximation algorithms. It also comes with a guarantee that does not depend in any way on the quality of the input point sample. Any input gives an output surface which is the `watertight' boundary of a three-dimensional polyhedral solid: the solid described by the approximate MAT. This unconditional guarantee makes the algorithm quite robust and eliminates the polygonalization, hole-filling or manifold extraction post-processing steps required in previous surface reconstruction algorithms.In this paper, we use the theory to develop a power crust implementation which is indeed robust for realistic and even difficult samples. We describe the careful design of a key subroutine which labels parts of the MAT as inside or outside of the object, easy in theory but non-trivial in practice. We find that we can handle areas in which the input sampling is scanty or noisy by simply discarding the unreliable parts of the MAT approximation. We demonstrate good empirical results on inputs including models with sharp corners, sparse and unevenly distributed point samples, holes, and noise, both natural and synthetic.We also demonstrate some simple extensions: intentionally leaving holes where there is no data, producing approximate offset surfaces, and simplifying the approximate MAT in a principled way to preserve stable features.

The power crust

The medial axis transform (or MAT) is a representation of an object as an infinite union of balls. We consider approximating the MAT of a three-dimensional object, and its complement, with a finite union of balls. Using this approximate MAT we define a new piecewise-linear approximation to the object surface, which we call the power crust. We assume that we are given as input a sufficiently dense sample of points from the object surface. We select a subset of the Voronoi balls of the sample, the polar balls, as the union of balls representation. We bound the geometric error of the union, and of the corresponding power crust, and show that both representations are topologically correct as well. Thus, our results provide a new algorithm for surface reconstruction from sample points. By construction, the power crust is always the boundary of a polyhedral solid, so we avoid the polygonization, hole-filling or manifold extraction steps used in previous algorithms. The union of balls representation and the power crust have corresponding piecewise-linear dual representations, which in some sense approximate the medial axis. We show a geometric relationship between these duals and the medial axis by proving that, as the sampling density goes to infinity, the set of poles, the centers of the polar balls, converges to the medial axis.

/pdf/the-power-crust-unions-of-balls-and-the-medial-axis-mhmmyz0jdt.pdf

The power crust, unions of balls, and the medial axis transform

We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation. Additional

/pdf/fast-computation-of-generalized-voronoi-diagrams-using-35wpydj9nw.pdf

Fast computation of generalized Voronoi diagrams using graphics hardware

Skeletonization promises to become a powerful tool for compact shape description, path planning and other applications. However, current techniques can seldom efficiently process real, complicated 3D data sets, such as MRI and CT data of human organs. In this paper, we present an efficient voxel coding-based algorithm for the skeletonization of 3D voxelized objects. The skeletons are interpreted as connected center lines, consisting of sequences of medial points of consecutive clusters. These center lines are initially extracted as paths of voxels, followed by medial point replacement, refinement, smoothing and connection operations. The voxel-coding techniques have been proposed for each of these operations in a uniform and systematic fashion. In addition to preserving basic connectivity and centeredness, the algorithm is characterized by straightforward computation, no sensitivity to object boundary complexity, explicit extraction of ready-to-parameterize and branch-controlled skeletons, and efficient object hole detection. These issues are rarely discussed in traditional methods. A range of 3D medical MRI and CT data sets were used for testing the algorithm, demonstrating its utility.

Efficient skeletonization of volumetric objects

The medial surface is a skeletal abstraction of a solid that provides useful shape information, which compliments existing model representation schemes. The medial surface and its associated topological entities are defined, and an algorithm for computing the medial surface of a large class of B-rep solids is then presented. The algorithm is based on the domain Delaunay triangulation of a relatively sparse distribution of points, which are generated on the boundary of the object. This strategy is adaptive in that the boundary point set is refined to guarantee a correct topological representation of the medial surface.

D.J. Sheehy

Papers

Shape description by medial surface construction

Computing the medial surface of a solid from a domain Delaunay triangulation