Nina Amenta

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

To produce elongated ingots consisting of steel in an ingot mold, a plurality of electrodes are fused down in succession in a slag bath maintained in said ingot mold and having a depth of at least 0.9 and at most 1.8 times the square root of the diameter of said ingot mold in centimeters, provided that the depth is at least 4 centimeters. Each succeeding electrode is replaced for the preceding one within 150 seconds.

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.

Defining point-set surfaces

The MLS surface [Levin 2003], used for modeling and rendering with point clouds, was originally defined algorithmically as the output of a particular meshless construction. We give a new explicit definition in terms of the critical points of an energy function on lines determined by a vector field. This definition reveals connections to research in computer vision and computational topology.Variants of the MLS surface can be created by varying the vector field and the energy function. As an example, we define a similar surface determined by a cloud of surfels (points equipped with normals), rather than points.We also observe that some procedures described in the literature to take points in space onto the MLS surface fail to do so, and we describe a simple iterative procedure which does.

Surface reconstruction by Voronoi filtering

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.

A simple algorithm for homeomorphic surface reconstruction

Abstract 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

Optimal point placement for mesh smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.

Real-time parallel hashing on the GPU

We demonstrate an efficient data-parallel algorithm for building large hash tables of millions of elements in real-time. We consider two parallel algorithms for the construction: a classical sparse perfect hashing approach, and cuckoo hashing, which packs elements densely by allowing an element to be stored in one of multiple possible locations. Our construction is a hybrid approach that uses both algorithms. We measure the construction time, access time, and memory usage of our implementations and demonstrate real-time performance on large datasets: for 5 million key-value pairs, we construct a hash table in 35.7 ms using 1.42 times as much memory as the input data itself, and we can access all the elements in that hash table in 15.3 ms. For comparison, sorting the same data requires 36.6 ms, but accessing all the elements via binary search requires 79.5 ms. Furthermore, we show how our hashing methods can be applied to two graphics applications: 3D surface intersection for moving data and geometric hashing for image matching.

Incremental constructions con BRIO

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.

Case study: visualizing sets of evolutionary trees

We describe a visualization tool which allows a biologist to explore a large set of hypothetical evolutionary trees. Interacting with such a dataset allows the biologist to identify distinct hypotheses about how different species or organisms evolved, which would not have been clear from traditional analyses. Our system integrates a point-set visualization of the distribution of hypothetical trees with detail views of an individual tree, or of a consensus tree summarizing a subset of trees. Efficient algorithms were required for the key tasks of computing distances between trees, finding consensus trees, and laying out the point-set visualization.

Surface reconstruction from noisy point clouds

We show that a simple modification of the power crust algorithm for surface reconstruction produces correct outputs in presence of noise. This is proved using a fairly realistic noise model. Our theoretical results are related to the problem of computing a stable subset of the medial axis. We demostrate the effectiveness of our algorithm with a number of experimental results.