Exact Geosedics and Shortest Paths on Polyhedral Surface

We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.National Institute for Biomedical Imaging and Bioengineering (R01 EB001550

A framework for working with digitized cultural heritage artefacts

In this paper, we present our work in designing, implementing, and evaluating a set of 3D interactive spatial measurement tools in the context of Cultural Heritage Toolbox (CH Toolbox), a framework for computer-aided cultural heritage research. Our application utilizes a bi-manual, spaceball and mouse driven user interface to help the user manage visualized 3D models digitized from real artifacts. We have developed a virtual radius estimator, useful for analyzing incomplete pieces of radial artifacts, and a virtual tape measure, useful in measurement of geodesic distances between two points on the surface of an artifact. We tested the tools on the special case of pottery analysis

A minimalistic approach for fast computation of geodesic distances on triangular meshes

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes.Comment: Preprint submitted to Computers & Graphic

Iso-level tool path planning for free-form surfaces

The aim of tool path planning is to maximize the efficiency against some given precision criteria. In practice, scallop height should be kept constant to avoid unnecessary cutting, while the tool path should be smooth enough to maintain a high feed rate. However, iso-scallop and smoothness often conflict with each other. Existing methods smooth iso-scallop paths one-by-one, which make the final tool path far from being globally optimal. This paper proposes a new framework for tool path optimization. It views a family of iso-level curves of a scalar function defined over the surface as tool path so that desired tool path can be generated by finding the function that minimizes certain energy functional and different objectives can be considered simultaneously. We use the framework to plan globally optimal tool path with respect to iso-scallop and smoothness. The energy functionals for planning iso-scallop, smoothness, and optimal tool path are respectively derived, and the path topology is studied too. Experimental results are given to show effectiveness of the proposed methods

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure