Computing Realistic Terrains from Imprecise Elevations

It is ideal for triangulated terrains to have characteristics or properties that are realistic. In the imprecise terrain model, each vertex of a triangulated terrain has an imprecise eleva- tion value only known to lie within some interval. Under some objective function, the goal is to compute a precise terrain by assigning a single elevation value to each point, so that the objective function is optimized. This thesis examines various objectives, such as minimizing the number of local extrema and minimizing the terrain’s surface area. We give algorithms in some cases, hardness results in other cases. Specifically, we consider four objectives: (1) minimizing the number of local extrema; (2) optimizing coplanar features; (3) minimizing the surface area; (4) minimizing the maximum steepness. Problem (1) is known to be NP-hard, but we give an algorithm for a special case. For problem (2) we give an NP-hardness proof for the general case and a positive result for a special case. Meanwhile, problems (3) and (4) can be approximated using Second Order Cone Programming. We also consider versions of these problems for terrains one dimension down, where the output is a polyline. Here we give very efficient algorithms for all objective functions considered. Finally, we go beyond terrains and briefly consider the distant representatives problem, where the goal is to choose precise points from segments to be as far from each other as possible. For this problem, we give a parameterized algorithm for vertical segments, prove NP-hardness for unit horizontal segments, and show hardness of approximation for vertical and horizontal segments

Spatially Coherent RANSAC for Multi-Model Fitting

RANSAC [15, 38, 1] is a reliable method for fitting parametric models to sparse data with many outliers. Originally designed for extracting a single model, RANSAC also has variants for fitting multiple models when supported by data. Our main insight is that, in practice, inliers for each model are often spatially coherent — all previous RANSAC-based methods ignore this. Our new method fits an unspecified number of models to data by combining ideas of random sampling and spatial regularization. As in basic RANSAC, we randomly sample data points to generate a set of proposed models (labels). We formulate model selection and inlier classification as a single problem — labeling of triangulated data points. Geometric fit errors and spatial coherence are combined in one MRF-based energy. In contrast to basic RANSAC, inlier classification does not depend on a fixed threshold. Moreover, our optimization framework allows iterative re-estimation of models/inliers with a clear stopping criteria and convergence guarantees. We show that our new method, SCO- RANSAC, can significantly improve results on synthetic and real data supporting multiple linear, affine, and homographic models

Hydrologic modeling using triangulated irregular networks : terrain representation, flood forecasting and catchment response

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2003.Includes bibliographical references.Numerical models are modern tools for capturing the spatial and temporal variability in the land-surface hydrologic response to rainfall and understanding the physical relations between internal watershed processes and observed streamflow. This thesis presents the development and application of a distributed hydrologic model distinguished by its representation of topography through a triangulated irregular network (TIN) and its coupling of the surface and subsurface processes leading to the catchment response. As a research tool for hydrologic forecasting and experimentation, the TIN-based Real-time Integrated Basin Simulator (tRIBS) fully incorporates spatial heterogeneities in basin topography, surface descriptors and hydrometeorological forcing to produce dynamic maps of hydrologic states and fluxes. These capabilities allow investigation of theoretical questions and practical problems in hydrologic science and water resources engineering. Three related themes are developed in this thesis. First, a set of methods are developed for constructing TIN topographic models from raster digital elevation models (DEM) for hydrologic and geomorphic applications. A new approach for representing a steady-state estimate of a particular watershed process within the physical mesh is introduced. Hydrologic comparisons utilizing different terrain models are made to investigate the suitable level of detail required for capturing process dynamics accurately. Second, the TIN-based model is utilized in conjunction with a rainfall forecasting algorithm to assess the space-time flood predictability. For two hydrometeorological case studies, the forecast skill is assessed as a function of rainfall forecast lead time, catchment scale and the spatial variability in the quantitative precipitation forecasts (QPF). Third, the surface and subsurface runoff response in a complex basin is investigated with respect to changes in storm properties and the initial water table position.The partitioning of rainfall into runoff production mechanisms is found to be a causative factor in the nonlinearity and scale-dependence observed in the basin hydrograph response. The model applications presented in this thesis highlight the advantages of TIN- based modeling for hydrologic forecasting and process-oriented studies over complex terrain. In particular, the multi-resolution and multi-scale capabilities are encouraging for a range of applied and scientific problems in catchment hydrology.by Enrique R. Vivoni.Ph.D

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

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version