W. Randolph Franklin

Parallel ODETLAP for terrain compression and reconstruction

We introduce a parallel approximation of an Over-determined Laplacian Partial Differential Equation solver (ODETLAP) applied to the compression and restoration of terrain data used for Geographical Information Systems (GIS). ODETLAP can be used to reconstruct a compressed elevation map, or to generate a dense regular grid from airborne Light Detection and Ranging (LIDAR) point cloud data. With previous methods, the time to execute ODETLAP does not scale well with the size of the input elevation map, resulting in running times that are prohibitively long for large data sets. Our algorithm divides the data set into patches, runs ODETLAP on each patch, and then merges the patches together. This method gives two distinct speed improvements. First, we provide scalability by reducing the complexity such that the execution time grows almost linearly with the size of the input, even when run on a single processor. Second, we are able to calculate ODETLAP on the patches concurrently in a parallel or distributed environment. Our new patch-based implementation takes 2 seconds to run ODETLAP on an 800 x 800 elevation map using 128 processors, while the original version of ODETLAP takes nearly 10 minutes on a single processor (271 times longer). We demonstrate the effectiveness of the new algorithm by running it on data sets as large as 16000 x 16000 on a cluster of computers. We also discuss our preliminary results from running on an IBM Blue Gene/L system with 32,768 processors.

Tradeoffs when Multiple Observer Siting on Large Terrain Cells

This paper demonstrates a toolkit for multiple observer siting to maximize their joint viewshed, on high-resolution gridded terrains, up to 2402 × 2402, with the viewsheds’ radii of up to 1000. It shows that approximate (rather than exact) visibility indexes of observers are sufficient for siting multiple observers. It also shows that, when selecting potential observers, geographic dispersion is more important than maximum estimated visibility, and it quantifies this. Applications of optimal multiple observer siting include radio towers, terrain observation, and mitigation of environmental visual nuisances.

Surface compression using over-determined Laplacian approximation

We describe a surface compression technique to lossily compress elevation datasets. Our approach first approximates the uncompressed terrain using an over-determined system of linear equations based on the Laplacian partial differential equation. Then the approximation is refined with respect to the uncompressed terrain using an error metric. These two steps work alternately until we find an approximation that is good enough. We then further compress the result to achieve a better overall compression ratio. We present experiments and measurements using different metrics and our method gives convincing results.

Efficient viewshed computation on terrain in external memory

The recent availability of detailed geographic data permits terrain applications to process large areas at high resolution. However the required massive data processing presents significant challenges, demanding algorithms optimized for both data movement and computation. One such application is viewshed computation, that is, to determine all the points visible from a given point p. In this paper, we present an efficient algorithm to compute viewsheds on terrain stored in external memory. In the usual case where the observer’s radius of interest is smaller than the terrain size, the algorithm complexity is θ(scan(n2)) where n2 is the number of points in an n × n DEM and scan(n2) is the minimum number of I/O operations required to read n2 contiguous items from external memory. This is much faster than existing published algorithms.

Smugglers and border guards: the GeoStar project at RPI

We present the GeoStar project at RPI, which researches various terrain (i.e., elevation) representations and operations thereon. This work is motivated by the large amounts of hi-res data now available. The purpose of each representation is to lossily compress terrain while maintaining important properties. Our ODETLAP representation generalizes a Laplacian partial differential equation by using two inconsistent equations for each known point in the grid, as well as one equation for each unknown point. The surface is reconstructed from a carefully-chosen small set of known points. Our second representation segments the terrain into a set of regions, each of which is simply described. Our third representation has the most long term potential: scooping, which forms the terrain by emulating surface water erosion. Siting hundreds of observers, such as border guards, so that their viewsheds jointly cover the maximum terrain is our first operation. This process allows both observer and target to be above the local terrain, and the observer to have a finite radius of interest. Planning a path so that a smuggler may get from point A to point B while maximally avoiding the border guards is our second operation. The path metric includes path length, distance traveled uphill, and amount of time visible to a guard. The quality of our representations is determined, not only by their RMS elevation error, but by how accurately they support these operations.

Voronoi diagrams with barriers and on polyhedra for minimal path planning

Two generalizations of the Voronoi diagram in two dimensions (E2) are presented in this paper. The first allows impenetrable barriers that the shortest path must go around. The barriers are straight line segments that may be combined into polygons and even mazes. Each region of the diagram delimits a set of points that have not only the same closest existing point, but have the same topology of shortest path. The edges of this diagram, which has linear complexity in the number of input points and barrier lines, may be hyperbolic sections as well as straight lines. The second construction considers the Voronoi diagram on the surface of a convex polyhedron, given a set of fixed source points on it. Each face is partitioned into regions, such that the shortest path to any goal point in a given region from the closest fixed source point travels over the same sequence of faces to the same closest point.

3-D graphic display of discrete spatial data by prism maps

An efficient algorithm for displaying 3-D scenes showing a discrete spatially varying surface is described. Given a 2-D map or planar graph composed of polygons where each polygon has a positive real number attribute, a prism is erected on each polygon with height proportional to that attribute. The resulting 3-D scene is plotted with shading and hidden lines removed. Thus the spatial variation of the attribute may be quickly and intuitively grasped by the nontechnical observer. This has applications to areas such as geography if the map is a cartographic map, or to physics if the map diagrams the periodic table. The algorithm takes time O(N*log(N)) where N is the number of edges in the map. Most of the calculations can be done without knowing the prism heights so extra plots with different attributes for the prisms can be produced quickly. This algorithm has been implemented and tested on maps of up to 12000 edges.

A New Method for Computing the Drainage Network Based on Raising the Level of an Ocean Surrounding the Terrain

We present a new and faster internal memory method to compute the drainage network, that is, the flow direction and accumulation on terrains represented by raster elevation matrix. The main idea is to surround the terrain by water (as an island) and then to raise the outside water level step by step, with depressions filled when the water reaches their boundary. This process avoids the very time-consuming depression filling step used by most of the methods to compute flow routing, that is, the flow direction and accumulated flow. The execution time of our method is very fast, and linear in the terrain size. Tests have shown that our method can process large terrains more than 100 times faster than other recent methods.

More efficient terrain viewshed computation on massive datasets using external memory

We present a better algorithm and implementation for external memory viewshed computation. It is about four times faster than the most recent and most efficient published methods. Ours is also much simpler. Since processing large datasets can take hours, this improvement is significant. To reduce the total number of I/O operations, our method is based on subdividing the terrain into blocks which are stored in a special data structure managed as a cache memory. The viewshed is that region of the terrain that is visible by a fixed observer, who may be on or above the terrain. Its applications range from visual nuisance abatement to radio transmitter siting and surveillance.

Evaluation of algorithms to display vector plots on raster devices

Abstract Rasterization is the process of converting lines intended to be drawn on a vector plotter to a set of pixels that can be displayed on raster device. It is common operation that has been adequately solved by ad hoc routines. However, plotters now have accuracies of 2000 by 2000 or better and maps such as the World Data Bank II have over 6,000,000 points. Hence, more sophisticated algorithms are now needed. This paper analyzes the theoretical foundations of the problem, considering contraints such as: (1) the amount of main memory available, (2) the cost of secondary I/O, (3) the number of line vectors to be converted in the plot, (4) their total length, and (5) whether or not they are in main memory, virtual memory, or secondary memory. The 11 algorithms considered include: (1) sorting the lines completely, (2) bucket sorting them to within strips of the raster screen, (3) splitting long lines to make all the lines about the same length, (4) making several passes through the whole set (or specified subsets) of the lines to pick out the ones needed for any raster, (5) linking the lines in a list so they can be efficiently sorted and deleted, and (6) immediately splitting the lines into pixels and then sorting the pixels. Then some useful generalizations of the rasterizations of black and white straight line drawing are considered. They include rasterizing: (1) curved lines defined by equations such as circles, (2) straight lines derived from cross-hatching areas, (3) halftone shaded areas, and (4) with color displays. For usual constraints, the optimal method is: Divide the screen into strips, with each strip using all available memory, split any lines that cross a boundary, bucket sort the lines by strip, and read them and set the pixels, one strip at a time.