We study the problem of maintaining the contour tree T of a terrain Sigma, represented as a triangulated xy-monotone surface, as the heights of its vertices vary continuously with time. We characterize the combinatorial changes in T and how they relate to topological changes in Sigma. We present a kinetic data structure (KDS) for maintaining T efficiently. It maintains certificates that fail, i.e., an event occurs, only when the heights of two adjacent vertices become equal or two saddle vertices appear on the same contour. Assuming that the heights of two vertices of Sigma become equal only O(1) times and these instances can be computed in O(1) time, the KDS processes O(kappa + n) events, where n is the number of vertices in Sigma and kappa is the number of events at which the combinatorial structure of T changes, and processes each event in O(log n) time. The KDS can be extended to maintain an augmented contour tree and a join/split tree.

Computing paths over a terrain that are highly occluded with respect to observers is an important problem in GIS. Given a fast algorithm for computing the visibility map, the path-planning step becomes the bottleneck. In this paper, we present an approach for quickly computing occluded paths over a terrain using a sparse network, a sparse 1-dimensional network over the terrain. We present different strategies for constructing the sparse network. Experimental results show that our approach results in significantly improved time for computing highly occluded paths between two query points, and that the different strategies offer a tradeoff between higher-quality paths and lower preprocessing times. Furthermore, there are strategies that achieve near-optimal paths with small preprocessing cost.

Understanding the locations of highly occluded paths on a terrain is an important GIS problem. In this paper we present a model and a fast algorithm for computing highly occluded paths on a terrain. It does not assume the observer locations to be known and yields a path likely to be occluded under a rational observer strategy. We present experimental results that examine several different observer strategies. The repeated visibility map computations necessary for our model is expedited using a fast algorithm for calculating approximate visibility maps that models the decrease in observational fidelity as distance increases. The algorithm computes a multiresolution approximate visibility map and makes use of a graphics processing unit (GPU) to speed up computation. We present experimental results on terrrain data sets with up to 144 million points.

A fundamental problem in analyzing trajectory data is to identify common patterns between pairs or among groups of trajectories. In this paper, we consider the problem of matching similar portions between a pair of trajectories, each observed as a sequence of points sampled from it. We present new measures of trajectory similarity --- both local and global --- between a pair of trajectories to distinguish between similar and dissimilar portions. We then use this model to perform segmentation of a set of trajectories into fragments, contiguous portions of trajectories shared by many of them.

Understanding the impact of climate and land-use on forest ecosystems involves modeling and simulating complex spatial interactions at many different scales. With this goal in mind, we have developed an individual-based, spatially explicit forest simulator, which incorporates fine-scale processes that influence forest dynamics. In this paper we present new, faster algorithms for computing understory light and for dispersal of seeds --- the two most computationally intensive submodules in our simulator. By exploiting temporal coherence, we circumvent the problem of doing the entire simulation at each step. We provide experimental results that support the efficiency and efficacy of our approach.

With modern focus on LiDAR technology the amount of topographic data, in the form of massive point clouds, has increased dramatically. Furthermore, due to the popularity of LiDAR, repeated surveys of the same areas are becoming more common. This trend will only increase as topographic changes prompt surveys over already scanned terrain, in which case we obtain large spatio-temporal data sets.

A terrain M can be represented as a triangulation of the plane along with a height function associated with the vertices (and linearly interpolated within the edges and triangles) of M. We investigate the problem of answering contour queries on M: Given a height ℓ and a triangle f of M that intersects the level set of M at height ℓ, report the list of the edges of the connected component of this level set that intersect f, sorted in clockwise or counter-clockwise order. Contour queries are different from level-set queries in that only one contour (connected component of the level set) out of all those that may exist is expected to be reported. We present an I/O-efficient data structure of linear size that answers a contour query in O(logB N + T/B) I/Os, where N is the number of triangles in the terrain and T is the number of edges in the output contour. The data structure can be constructed using O(Sort(N)) I/Os.

We consider the problem of automatically cleaning massive sonar data point clouds, that is, the problem of automat-ically removing noisy points that for example appear as a result of scans of (shoals of) fish, multiple reflections, scan-ner self-reflections, refraction in gas bubbles, and so on. We describe a new algorithm that avoids the problems of previous local-neighbourhood based algorithms. Our algo-rithm is theoretically I/O-efficient, that is, it is capable of efficiently processing massive sonar point clouds that do not fit in internal memory but must reside on disk. The algo-rithm is also relatively simple and thus practically efficient, partly due to the development of a new simple algorithm for computing the connected components of a graph embedded in the plane. A version of our cleaning algorithm has already been incorporated in a commercial product. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical algo-rithms and problems—Geometrical problems and computa-tions

With modern LiDAR technology the amount of topographic data, in the form of massive point clouds, has increased dramatically. One of the most fundamental GIS tasks is to construct a grid digital elevation model (DEM) from these 3D point clouds. In this paper we present a simple yet very fast algorithm for constructing a grid DEM from massive point clouds using natural neighbor interpolation (NNI). We use a graphics processing unit (GPU) to significantly speed up the computation. To handle the large data sets and to deal with graphics hardware limitations clever blocking schemes are used to partition the point cloud. For example, using standard desktop computers and graphics hardware, we construct a high-resolution grid with 150 million cells from two billion points in less than thirty-seven minutes. This is about one-tenth of the time required for the same computer to perform a standard linear interpolation, which produces a much less smooth surface.

A terrain M is the graph of a bivariate function. We assume that M is represented as a triangulated surface with N vertices. A contour (or isoline) of M is a connected component of a level set of M. Generically, each contour is a closed polygonal curve; at "critical" levels these curves may touch each other or collapse to a point. We present I/O efficient algorithms for the following two problems related to computing contours of M:

We consider the problem of extracting a river network and a watershed hierarchy from a terrain given as a set of irregularly spaced points. We describe TERRASTREAM, a "pipelined" solution that consists of four main stages: construction of a digital elevation model (DEM), hydrological conditioning, extraction of river networks, and construction of a watershed hierarchy. Our approach has several advantages over existing methods. First, we design and implement the pipeline so that each stage is scalable to massive data sets; a single non-scalable stage would create a bottleneck and limit overall scalability. Second, we develop the algorithms in a general framework so that they work for both TIN and grid DEMs. Furthermore, TERRASTREAM is flexible and allows users to choose from various models and parameters, yet our pipeline is designed to reduce (or eliminate) the need for manual intervention between stages.