Yuanxin Liu

Streaming computation of Delaunay triangulations

We show how to greatly accelerate algorithms that compute Delaunay triangulations of huge, well-distributed point sets in 2D and 3D by exploiting the natural spatial coherence in a stream of points. We achieve large performance gains by introducing spatial finalization into point streams: we partition space into regions, and augment a stream of input points with finalization tags that indicate when a point is the last in its region. By extending an incremental algorithm for Delaunay triangulation to use finalization tags and produce streaming mesh output, we compute a billion-triangle terrain representation for the Neuse River system from 11.2 GB of LIDAR data in 48 minutes using only 70 MB of memory on a laptop with two hard drives. This is a factor of twelve faster than the previous fastest out-of-core Delaunay triangulation software.

Testing Homotopy for Paths in the Plane

Abstract In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For paths specified by n line segments with obstacles described by n points, several standard ways achieve quadratic running time. For simple paths, our algorithm runs in O(n log n) time, which we show is tight. For self-intersecting paths the problem is related to Hopcroft’s problem; our algorithm runs in O(n3/2log n) time.

Generating raster DEM from mass points via TIN streaming

It is difficult to generate raster Digital Elevation Models (DEMs) from terrain mass point data sets too large to fit into memory, such as those obtained by LIDAR. We describe prototype tools for streaming DEM generation that use memory and disk I/O very efficiently. From 500 million bare-earth LIDAR double precision points (11.2 GB) our tool can, in just over an hour on a standard laptop with two hard drives, produce a 50,394 × 30,500 raster DEM with 20 foot post spacing in 16 bit binary BIL format (3 GB), using less than 100 MB of main memory and less than 300 MB of temporary disk space.

Flooding Triangulated Terrain

We extend pit filling and basin hierarchy computation to TIN terrain models. These operations are relatively easy to implement in drainage computations based on networks (e.g., raster D8 or Voronoi dual) but robustness issues make them difficult to implement in an otherwise appealing model of water flow on a continuous surface such as a TIN. We suggest a consistent solution of the robustness issues, then augment the basin hierarchy graph with different functions for how basins fill and spill to simplify the watershed graph to the essentials. Our solutions can be tuned by choosing a small number of intuitive parameters to suit applications that require a data-dependent selection of basin hierarchies.

Testing Homotopy for paths in the plane

In this paper we present an efficient algorithm to test if two given paths are homotopic; that is, whether they wind around obstacles in the plane in the same way. For simple paths specified by n line segments with obstacles described by n points, our algorithm runs in O(n log n) time, which we show is tight. For self-intersecting paths the problem is related to Hopcrofts problem.

Quadratic and cubic b-splines by generalizing higher-order voronoi diagrams

A long-standing problem in spline theory has been to generalize classic B-splines to the multivariate setting, and its full solution will have broad impact. We initiate a study of triangulations that generalize the duals of higher order Voronoi diagrams, and show that these can serve as a foundation for a family of multivariate splines that generalize the classic univariate B-splines. This paper focuseson Voronoi diagrams of orders two and three, which produce families of quadratic and cubic bivariate B-splines. We believe that these families are the most general bivariate B-splines to date and supportour belief by demonstrating that a classic quadratic box spline, the Zwart-Powell (ZP) element, is contained in our family. Our work is directly based on that of Neamtu, who established the fascinating connection between splines and higher order Voronoi diagrams.

Sphere-based Computation of Delaunay Diagrams on Points from 4d Grids

The Delaunay diagram in d dimensions is the dual of the Voronoi diagram of a set of input sites. If we assume no degeneracies in the input, i.e. no d + 2 sites are co-spherical, then the diagram is a triangulation. Because this assumption is common, and can be enforced by symbolic perturbation, we often forget that Delaunay diagrams need not be triangulations. Input sets chosen from integer grids are common in scientific visualization applications, however, and these often have many degeneracies. Perturbation signifcantly increases the size of the Delaunay and dual Voronoi diagrams - a single 4D cube becomes 16 to 24 simplices, so one dual vertex becomes many. Our result is a sphere-based algorithm for direct, incremental computation of the Delaunay diagram in 4D. For input with many degeneracies, its speed is comparable to our fastest Delaunay triangulation program, yet it computes the exact Delaunay diagram.