59 research outputs found
Flow Computations on Imprecise Terrains
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
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
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
A Sweep-Plane Algorithm for Calculating the Isolation of Mountains
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
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A framework for local terrain deformation based on diffusion theory
Terrains have a key role in making outdoor virtual scenes believable and immersive as they form the support for every other natural element in the scene. Although important, terrains are often given limited interactivity in real-time applications. However, in nature, terrains are dynamic and interact with the rest of the environment changing shape on different levels, from tracks left by a person running on a gravel soil (micro-scale), to avalanches on the side of a mountain (macro-scale).
The challenge in representing dynamic terrains correctly is that the soil that forms them is vastly heterogeneous and behaves differently depending on its composition. This heterogeneity introduces difficulties at different levels in dynamic terrains simulations, from modelling the large amount of different elements that compose the oil to simulating their dynamic behaviour.
This work presents a novel framework to simulate multi-material dynamic terrains by taking into account the soil composition and its heterogeneity. In the proposed framework soil information is obtained from a material description map applied to the terrain mesh. This information is used to compute deformations in the area of interaction using a novel mathematical model based on diffusion theory. The deformations are applied to the terrain mesh in different ways depending on the distance of the area of interaction from the camera and the soil material. Deformations away from the camera are simulated by dynamically displacing normals. While deformations in a neighbourhood of the camera are represented by displacing the terrain mesh, which is locally tessellated to better fit the displacement. For gravel based soils the terrain details are added near the camera by reconstructing the meshes of the small rocks from the texture image, thus simulating both micro and macro-structure of the terrain.
The outcome of the framework is a realistic interactive dynamic terrain animation in real-time
Algorithms for Triangles, Cones & Peaks
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 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 -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
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
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
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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