Accurate terrain representation with appropriate preservation of important terrain characteristics, especially slope steepness is becoming more crucial and fundamental as the geographical models are getting more complex and commonly used. Based on our earlier success in Overdetermined Laplacian Partial Differential Equation (ODETLAP), which allows for compact yet accurate compression of the Digital Elevation Model (DEM), we propose a new terrain compression technique which focuses on improving slope accuracy in compression of high resolution terrain data. With high slope accuracy and a high compression ratio, this technique will help all sorts of geographical applications that require a high precision in slope yet also have strict constraints on data size. Our proposed technique has the following contribution: we modify the ODETLAP system by adding slope equations for some key points picked automatically so that we can compress the elevation without explicitly storing slope values. By adding these slope equations, we can perturb the elevation in such a way that when slope is computed from the reconstructed surface, they are accurate. Note we are not storing the slope explicitly, instead we only store the elevation difference at a few locations. Since the ultimate goal is to have a compact terrain representation, encoding is also an integral part of this research. We have used Run Length Encoding (RLE) and linear prediction in the past, which gave us substantial file size reduction. In addition to that, we also propose a Minimum Spanning Tree based encoding scheme that takes advantage of the spatial correlation between selected points. Our technique is able to achieve a 1:10 compression at the cost of 4.23 degree of RMS slope error and 3.30 meters of RMS elevation error.

Accurate terrain representation with appropriate preservation of important terrain characteristics, especially slope steepness, is becoming more crucial and fundamental as the geographical models are becoming more complex. Based on our earlier success with Overdetermined Laplacian Partial Differential Equations (ODETLAP), which allows for compact yet accurate compression of the Digital Elevation Model (DEM), we propose a new terrain compression technique that focuses on improving slope accuracy in compression of high resolution terrain data. With high slope accuracy and a high compression ratio, this technique will help geographical applications that require a high precision in slope yet also have strict constraints on data size. Our proposed technique has the following contribution: we modify the ODETLAP system by adding slope equations for some key points picked automatically so that we can compress the elevation without explicitly storing slope values. By adding these slope equations, we can perturb the elevation in such a way that when slope is computed from the reconstructed surface, they are accurate. Note we are not storing the slope explicitly, instead we only store the elevation difference at a few locations. Since the ultimate goal is to have a compact terrain representation, encoding is also an integral part of this research. We have used Run Length Encoding (RLE) and linear prediction in the past, which gave us substantial file size reduction. In addition to that, we also propose a Minimum Spanning Tree based encoding scheme that takes advantage of the spatial correlation between selected points. On a typical test, our technique is able to achieve a 1:10 compression at the cost of 4.23 degree of RMS slope error and 3.30 meters of RMS elevation error.

The distribution and management of spatial data require strategies for handling large amount of terrain data that are now available. Especially, data like LIDAR and Digital Elevation Model (DEM) which have been used in a large group of diversified users. In this paper, we propose a progressive terrain data transmission scheme based on the Over-determined Laplacian PDE (ODETLAP) which can achieve a compromise of high compression ratio and accuracy. The ODETLAP can be thought of as a compressor of original terrain data and using run length encoding as well as linear prediction we can reach a higher compression ratio. In general, this technique is capable of reducing a hilly DEM dataset to 1% of its original binary size and a mountainous, to 3%. The accuracy loss in elevation and slope are also discussed.

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.

We report on variants of the ODETLAP lossy terrain compression method where the reconstructed terrain has accurate slope as well as elevation. Slope is important for applications such as mobility, visibility and hydrology. One variant involves selecting a regular grid of points instead of selecting the most important points, requiring more points but which take less space. Another variant adds a new type of equation to the overdetermined system to force the slope of the reconstructed surface to be close to the original surface’s slope. Tests on six datasets with elevation ranges from 505m to 1040m, compressed at ratios from 146:1 to 1046:1 relative to the original binary file size, showed RMS elevation errors of 10m and slope errors of 3 to 10 degrees. The reconstructed terrain also supports planning optimal paths that avoid observers’ viewsheds. Paths planned on the reconstructed terrain were only 5% to 20% more expensive than paths planned on the original terrain. Tradeoffs between compressed data size and output accuracy are possible. Therefore storing terrain data on portable devices or transmitting over slow links and then using it in applications is more feasible.

We present an algorithm for path planning on complex terrain in the presence of observers, and define several metrics related to path planning to evaluate the quality of various terrain compression strategies. The path-planning algorithm simulates a smugglers and border guards scenario. First, we place observers on a terrain so as to optimize their visible coverage area. Next, we compute a path that a smuggler would take to minimize detection by an observer, path length, and uphill movement. The smuggler is allowed the full range of Euclidean motion on the 2-dimensional plane, unlike alternate path planning schemes that strictly avoid obstacles. We use two runs of the A* algorithm to efficiently compute this path. Our path-planning routine is used to evaluate the quality of terrain compression on the smugglers and border guards scenario. We also present an extension of the ODETLAP compression method to include slope equations, rather than using elevation data only, to specifically target the compression of terrain slopes. Effective terrain compression strategies have become even more crucial with limited bandwidth capacity and the increasing size of elevation datasets. We introduce new application-specific error metrics for evaluating lossy terrain compression. The target terrain applications are the optimal placement of observers on a landscape and the navigation through the terrain by smugglers. The error metrics compare the observer visibility and the cost of the optimal smuggler’s route on the reconstructed terrain to the original terrain.

We extend Laplacian PDE by adding a new equation to form an over-determined system so that we can control the relative importance of smoothness and accuracy in the reconstructed surface. Benefits of the method include the ability to process isolated, scattered elevation points and the fact that reconstructed surface could generate local maxima, which is not possible in the original Laplacian PDE by the maximum principle. We use certain geometric algorithm including Triangulate Irregular Network, Visibility test, Level Set Component that discovers important points which reflect the terrain structure and use our extended Laplacian PDE to approximate the terrain from these points. We present experiments and measurements using different metrics and our method gives convincing results.

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.

We examine a smugglers and border guards scenario. We place observers on a terrain so as to optimize their visible coverage area. Then we compute a path that a smuggler would take so as to avoid detection, while also minimizing the path length. We also examine how our results are affected by using a lossy representation of the terrain instead. We propose three new application-specific error metrics for evaluating terrain compression. Our target terrain applications are the optimal placement of observers on a landscape and the navigation through the terrain by smugglers. Instead of using standard metrics such as average or maximum elevation error, we seek to optimize our compression on the specific real-world application of smugglers and border guards.

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.

We present a new form of terrain compression to preserve the hydrological information that is lost using standard terrain simplication techniques. First, we compute the drainage by using a system of linear equations to determine the amount of water owing into each cell. The ow is then computed on the inverted terrain which provides an approximation of the ridge network. The drainage and ridge networks are simplied using the Douglas-Peucker algorithm, selecting the most signicant points. These points represent our compressed version of the hydrology. To uncompress, we use Over-determined Laplacian Partial Dierential Equations (ODETLAP) to \ll in" the missing data points. Our results show that the ow and watersheds on the reconstructed terrain are typically better than original ODETLAP point selection and lossy JPEG2000 compression.