59 research outputs found

    Flow Computations on Imprecise Terrains

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    We study the computation of the flow of water on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time

    Flow computations on imprecise terrains

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    We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x, y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.Peer ReviewedPostprint (published version

    Flow computations on imprecise terrains

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    Abstract. We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x, y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time. Rose knew almost everything that water can do, there are an awful lot when you think what. Gertrude Stein, The World is Round

    I/O-efficient removal of noise from terrain data

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    A Sweep-Plane Algorithm for Calculating the Isolation of Mountains

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    One established metric to classify the significance of a mountain peak is its isolation. It specifies the distance between a peak and the closest point of higher elevation. Peaks with high isolation dominate their surroundings and provide a nice view from the top. With the availability of worldwide Digital Elevation Models (DEMs), the isolation of all mountain peaks can be computed automatically. Previous algorithms run in worst case time that is quadratic in the input size. We present a novel sweep-plane algorithm that runs in time ?(nlog n+pT_NN) where n is the input size, p the number of considered peaks and T_NN the time for a 2D nearest-neighbor query in an appropriate geometric search tree. We refine this to a two-level approach that has high locality and good parallel scalability. Our implementation reduces the time for calculating the isolation of every peak on Earth from hours to minutes while improving precision

    Algorithms for Triangles, Cones & Peaks

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    Three different geometric objects are at the center of this dissertation: triangles, cones and peaks. In computational geometry, triangles are the most basic shape for planar subdivisions. Particularly, Delaunay triangulations are a widely used for manifold applications in engineering, geographic information systems, telecommunication networks, etc. We present two novel parallel algorithms to construct the Delaunay triangulation of a given point set. Yao graphs are geometric spanners that connect each point of a given set to its nearest neighbor in each of kk cones drawn around it. They are used to aid the construction of Euclidean minimum spanning trees or in wireless networks for topology control and routing. We present the first implementation of an optimal O(nlogn)\mathcal{O}(n \log n)-time sweepline algorithm to construct Yao graphs. One metric to quantify the importance of a mountain peak is its isolation. Isolation measures the distance between a peak and the closest point of higher elevation. Computing this metric from high-resolution digital elevation models (DEMs) requires efficient algorithms. We present a novel sweep-plane algorithm that can calculate the isolation of all peaks on Earth in mere minutes

    2D and 3D surface image processing algorithms and their applications

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    This doctoral dissertation work aims to develop algorithms for 2D image segmentation application of solar filament disappearance detection, 3D mesh simplification, and 3D image warping in pre-surgery simulation. Filament area detection in solar images is an image segmentation problem. A thresholding and region growing combined method is proposed and applied in this application. Based on the filament area detection results, filament disappearances are reported in real time. The solar images in 1999 are processed with this proposed system and three statistical results of filaments are presented. 3D images can be obtained by passive and active range sensing. An image registration process finds the transformation between each pair of range views. To model an object, a common reference frame in which all views can be transformed must be defined. After the registration, the range views should be integrated into a non-redundant model. Optimization is necessary to obtain a complete 3D model. One single surface representation can better fit to the data. It may be further simplified for rendering, storing and transmitting efficiently, or the representation can be converted to some other formats. This work proposes an efficient algorithm for solving the mesh simplification problem, approximating an arbitrary mesh by a simplified mesh. The algorithm uses Root Mean Square distance error metric to decide the facet curvature. Two vertices of one edge and the surrounding vertices decide the average plane. The simplification results are excellent and the computation speed is fast. The algorithm is compared with six other major simplification algorithms. Image morphing is used for all methods that gradually and continuously deform a source image into a target image, while producing the in-between models. Image warping is a continuous deformation of a: graphical object. A morphing process is usually composed of warping and interpolation. This work develops a direct-manipulation-of-free-form-deformation-based method and application for pre-surgical planning. The developed user interface provides a friendly interactive tool in the plastic surgery. Nose augmentation surgery is presented as an example. Displacement vector and lattices resulting in different resolution are used to obtain various deformation results. During the deformation, the volume change of the model is also considered based on a simplified skin-muscle model

    Computational and Theoretical Issues of Multiparameter Persistent Homology for Data Analysis

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    The basic goal of topological data analysis is to apply topology-based descriptors to understand and describe the shape of data. In this context, homology is one of the most relevant topological descriptors, well-appreciated for its discrete nature, computability and dimension independence. A further development is provided by persistent homology, which allows to track homological features along a oneparameter increasing sequence of spaces. Multiparameter persistent homology, also called multipersistent homology, is an extension of the theory of persistent homology motivated by the need of analyzing data naturally described by several parameters, such as vector-valued functions. Multipersistent homology presents several issues in terms of feasibility of computations over real-sized data and theoretical challenges in the evaluation of possible descriptors. The focus of this thesis is in the interplay between persistent homology theory and discrete Morse Theory. Discrete Morse theory provides methods for reducing the computational cost of homology and persistent homology by considering the discrete Morse complex generated by the discrete Morse gradient in place of the original complex. The work of this thesis addresses the problem of computing multipersistent homology, to make such tool usable in real application domains. This requires both computational optimizations towards the applications to real-world data, and theoretical insights for finding and interpreting suitable descriptors. Our computational contribution consists in proposing a new Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility of our preprocessing over real datasets, and evaluate the impact of the proposed algorithm as a preprocessing for computing multipersistent homology. A theoretical contribution of this thesis consists in proposing a new notion of optimality for such a preprocessing in the multiparameter context. We show that the proposed notion generalizes an already known optimality notion from the one-parameter case. Under this definition, we show that the algorithm we propose as a preprocessing is optimal in low dimensional domains. In the last part of the thesis, we consider preliminary applications of the proposed algorithm in the context of topology-based multivariate visualization by tracking critical features generated by a discrete gradient field compatible with the multiple scalar fields under study. We discuss (dis)similarities of such critical features with the state-of-the-art techniques in topology-based multivariate data visualization

    Collection of abstracts of the 24th European Workshop on Computational Geometry

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    International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
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