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

https://u.math.biu.ac.il/~katzmik/goldman05.pdf

Curvature formulas for implicit curves and surfaces

A method and apparatus for obtaining an image to determine a three dimensional shape of a stationary or moving object using a bi dimensional coded light pattern having a plurality of distinct identifiable feature types. The coded light pattern is projected on the object such that each of the identifiable feature types appears at most once on predefined sections of distinguishable epipolar lines. An image of the object is captured and the reflected feature types are extracted along with their location on known epipolar lines in the captured image. Displacements of the reflected feature types along their epipolar lines from reference coordinates thereupon determine corresponding three dimensional coordinates in space and thus a 3D mapping or model of the shape of the object at any point in time.

3d geometric modeling and motion capture using both single and dual imaging

A method and apparatus for obtaining an image to determine a three dimensional shape of a stationary or moving object using a bi dimensional coded light pattern having a plurality of distinct identifiable feature types. The coded light pattern is projected on the object such that each of the identifiable feature types appears at most once on predefined sections of distinguishable epipolar lines. An image of the object is captured and the reflected feature types are extracted along with their location on known epipolar lines in the captured image. The locations of identified feature types in the 2D image are corresponded to 3D coordinates in space in accordance to triangulation mappings. Thus a 3D mapping or model of imaged objects at an point in time is obtained.

3d geometric modeling and 3d video content creation

Techniques are provided for searching on three dimensional (3D) objects across large, distributed repositories of 3D models. 3D shapes are created for input to a search system; optionally user-defined similarity criterion is used, and search results are interactively navigated and feedback received for modifying the accuracy of the search results. Search input can also be given by picking 3D models from a cluster map or by providing the orthographic views for the 3D model. Feedback can be given by a searcher as to which models are similar and which are not. Various techniques adjust the search results according to the feedback given by the searcher and present the new search results to the searcher.

Methods, systems, and data structures for performing searches on three dimensional objects

Shape skeletons are powerful geometric abstractions that provide useful intermediate representations for a number of geometric operations on solid models including feature recognition, shape decomposition, finite element mesh generation, and shape design As a result there has been significant interest in the development of effective methods for skeleton generation of general free-form solids In this paper we describe a method that combines Delaunay triangulation with local numerical optimization schemes for the generation of accurate skeletons of 3D implicit solid models The proposed method accelerates the slow convergence of Voronoi diagrams to the skeleton, which, without optimization, would require impraelically large sample point sets and resulting meshes to attain acceptable accuracy The Delaunay triangulation forms the basis for generating the topological structure of the skeleton The optimization step of the process generates the geometry of the skeleton patches by moving the vertices of Delaunay tetrahedra and relocating their centres to form maximally inscribed spheres The computational advantage of the optimization scheme is that it involves the solution of one small optimization problem per tetrahedron and its complexity is therefore only linear (O(n)) in the number of points used for the skeleton approximation We demonstrate the effectiveness of the method on a number of representative solid models

An accelerated triangulation method for computing the skeletons of free-form solid models

Starting with a solid model of a 3D object, an accurate skeleton is produced by minimizing internal and edge errors of an initial approximate skeleton. To produce the initial approximate skeleton, the boundary of the solid is densely sampled to obtain a set of surface data points. Delaunay triangulation is performed on the surface data points and exterior and spurious tetrahedra are removed, leaving substantially interior tetrahedra. Circumspheres are constructed that enclose the tetrahedra, and the centers of the circumspheres are connected based on tetrahedra adjacency, to form a Voronoi diagram. Closed Voronoi cells are identified comprising the initial approximate skeleton. The position of skeleton interior vertices are then adjusted to minimize their error, producing a refined polygonal approximation of the skeleton interior geometry. Positions of the skeleton edge vertices are adjusted to minimize the edge error, yielding an accurate polygonal approximation of the skeleton.

Hao Chen

Papers

An accelerated triangulation method for computing the skeletons of free-form solid models

Constructing shape skeletons of 3D objects using generalized Voronoi diagrams