Computer and Robot Vision Vol. 1, by R.M. Haralick and Linda G. Shapiro, Addison-Wesley, 1992, ISBN 0-201-10887-1.

Computer and Robot Vision

Author(s): Amenta, Nina; Bern, Marshall; Kamvysselis, Manolis | Abstract: We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in IR3. The algorthim is the first for this problem with provable guarantees. Given a ''good sample'' from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling sensity increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for thsi fundamental subroutine, the algorithm is quite easy to implement.

/pdf/a-new-voronoi-based-surface-reconstruction-algorithm-3z96n4eoxj.pdf

A new Voronoi-based surface reconstruction algorithm

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 Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations

Voronoi diagrams and Delaunay triangulations

We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs.

/pdf/surface-reconstruction-by-voronoi-filtering-44o7pe0kyy.pdf

Surface Reconstruction by Voronoi Filtering

Computing and Simplifying 2D and 3D Continuous Skeletons

The medial axis of a geometric shape captures its connectivity. In spite of its inherent instability, it has found applications in a number of areas that deal with shapes. In this survey paper, we focus on results that shed light on this instability and use the new insights to generate simplified and stable modifications of the medial axis.

/pdf/stability-and-computation-of-medial-axes-a-state-of-the-art-4dxv78m8wi.pdf

Stability and Computation of Medial Axes - a State-of-the-Art Report

In this paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has the property to be included inside the Delaunay graph. A criterion to detect faces of the normalized mesh inside the Delaunay graph is proposed. This criterion is proved to provide the exact solution in 2D for points sampling a r-regular shapes with a sampling path e sin ( π 8 )r . In 3D, this result cannot be extended and the criterion cannot retrieve every face. A heuristic is proposed in order to complete the surface.

r-regular shape reconstruction from unorganized points

It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).

/pdf/complexity-of-the-delaunay-triangulation-of-points-on-6bjgfm2jkh.pdf

Complexity of the delaunay triangulation of points on surfaces the smooth case

The skeleton of an object is the locus of the centers of maximal discs included in the shape. The skeleton provides a compact representation of objects, useful for shape description and recognition. A well-known drawback of the skeleton transformation is its lack of continuity. This paper is concerned with the modeling of noise that may affect objects and the consequence of this noise on the skeleton. A graph (called the parameter graph) is introduced, on which branches due to noise are characterized. We deduce from this preliminary study a method to simplify skeletons. It depends on thresholds that can be chosen directly on the parameter graph associated to each skeleton.

Dominique Attali

Papers

Computing and Simplifying 2D and 3D Continuous Skeletons

Stability and Computation of Medial Axes - a State-of-the-Art Report

r-regular shape reconstruction from unorganized points

Complexity of the delaunay triangulation of points on surfaces the smooth case

Modeling noise for a better simplification of skeletons