We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.

Poisson surface reconstruction

Poisson surface reconstruction creates watertight surfaces from oriented point sets. In this work we extend the technique to explicitly incorporate the points as interpolation constraints. The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation. In contrast to other image and geometry processing techniques, the screening term is defined over a sparse set of points rather than over the full domain. We show that these sparse constraints can nonetheless be integrated efficiently. Because the modified linear system retains the same finite-element discretization, the sparsity structure is unchanged, and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.

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Screened poisson surface reconstruction

Computational geometry

TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

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

1 Introduction The problem of computing a piecewise linear approximation to a surface from a set of sam- ple points on the surface has been a focus of research in solid modeling and graphics due to its many applications. The input to this sur- face reconstruction problem consists of the three dimensional coordinates of the sampled points. The crust algorithm of [1] reconstructs a surface with topological and geometric guarantees using the Voronoi diagram of the input point set. We present new observations that simplify both the algorithm and the proofs for the crust, and we give for the first time a proof that the crust is homeomorphic to the input surface. *Dept. of Computer Science, U. of Texas, Austin TX 78712. e-mail: amenta@cs.utexas.edu tDept, of Computer Science, U. of Texas, Austin, TX 78712. e-mail: sunghee@cs.utexas.edu SDept. of Computer and Information Science, Ohio State U., Columbus, OH 43210. e-mail: t amaldey¢ci s. ohio-state, edu §Dept. of Computer and Information Science, Ohio State U., Columbus, OH 43210. e-mail: leekha@cis, ohio-state, edu

/pdf/a-simple-algorithm-for-homeomorphic-surface-reconstruction-tchngwvibf.pdf

A simple algorithm for homeomorphic surface reconstruction

The problem of computing a piecewise linear approximation to a surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent algorithm1 using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from the original sampled surface assuming that the sample was sufficiently dense. We give a similar algorithm, simplifying the computation and the proof of the geometric guarantee. In addition, we guarantee that our output surface is homeomorphic to the original surface; to our knowledge this is the first such topological guarantee for this problem.

/pdf/a-simple-algorithm-for-homeomorphic-surface-reconstruction-1oj9eybjlp.pdf

Randomized incremental constructions are widely used in computational geometry, but they perform very badly on large data because of their inherently random memory access patterns. We define a biased randomized insertion order which removes enough randomness to significantly improve performance, but leaves enough randomness so that the algorithms remain theoretically optimal.

/pdf/incremental-constructions-con-brio-w4gtqwj61l.pdf

Sunghee Choi

Papers

The power crust

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

A simple algorithm for homeomorphic surface reconstruction

A simple algorithm for homeomorphic surface reconstruction

Incremental constructions con BRIO