We introduce the contour spectrum, a user interface component that improves qualitative user interaction and provides real-time exact quantification in the visualization of isocontours. The contour spectrum is a signature consisting of a variety of scalar data and contour attributes, computed over the range of scalar values w 2!.We explore the use of surface area, volume, and gradientintegral of the contour that are shown to be univariate B-spline functions of the scalar value w for multi-dimensional unstructured triangular grids. These quantitative properties are calculated in real-time and presented to the user as a collection of signature graphs (plots of functions of w) to assist in selecting relevant isovalues w 0 for informative visualization. For timevarying data, these quantitative properties can also be computed over time, and displayed using a 2D interface, giving the user an overview of the time-varying function, and allowing interaction in both isovalue and timestep. The effectiveness of the current system and potential extensions are discussed.

We consider online routing algorithms for finding paths between the vertices of plane graphs.

In this paper we show how to traverse a subdivision and to report all cells, edges and vertices, without making use of mark bits in the structure or a stack. We do this by performing a depth-first search on the subdivision, using local criteria for deciding what is the next cell to visit. Our method is extremely simple and provably correct. The algorithm has applications in the field of Geographic Information Systems (GIS), where traversing subdivisions is a common operation, but modifying the database is unwanted or impossible. We show how to adapt our algorithm to answer related queries, such as windowing queries and reporting connected subsets of cells that have a common attribute. Finally, we show how to extend our algorithm such that it can handle convex 3-dimensional subdivisions. Keywords: Subdivisions, traversal algorithms, topological data structure, windowing, three dimensions. 1 Introduction The basic spatial vector data structure in any geographic information system is the...

We wish to extract the topology from scanned maps. In previous work [GNY96] this was done by extracting a skeleton from the Voronoi diagram, but this required vertex labelling and was only useable for polygon maps. We wished to take the crust algorithm of Amenta, Bern and Eppstein [ABE98] and modify it to extract the skeleton from unlabelled vertices. We find that by reducing the algorithm to a local test on the original Voronoi diagram we may extract both a crust and a skeleton simultaneously, using a variant of the Quad-Edge structure of [GS85]. We show that this crust has the properties of the original, and that the resulting skeleton has many practical uses. We illustrate the usefulness of the combined diagram with various applications.

A data structure is presented to store a triangulated irregular network digital elevation model, from which isolines (contour lines) can be extracted very efficiently. If the network is based on n points, then for any elevation, the isolines can be obtained in O(log n + k) query time, where k is the number of line segments that form the isolines. This compares favorably with O(n) time by straightforward computation. When a structured representation of the isolines is needed, the same query time applies. For a fully topological representation (with adjacency), the query requires additional O(c log c) or O(c log log n) time, where c is the number of connected components of isolines. In all three cases, the required data structure has only linear size.

We describe an algorithm for enumerating all vertices, edges and faces of a planar subdivision stored in any of the usual pointer-based representations, while using only a constant amount of memory beyond that required to store the subdivision. The algorithm is a refinement of a method introduced by de Berg et al (1997), that reduces the worst case running time from O(n²) to O(n log n). We also give experimental results that show that our modified algorithm runs faster not only in the worst case, but also in many realistic cases.

For 2D or 3D meshes that represent the domain of continuous function to the reals, the contours---or isosurfaces---of a specified value are an important way to visualize the function. To find such contours, a seed set can be used for the starting points from which the traversal of the contours can begin. This paper gives the first methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(n log n) time in 2D for meshes with n elements, and in O(n 2 ) time in higher dimensions. The additional storage overhead is proportial to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in roughly quadratic time. Since in practice this can be excessive, we develop a simple approximation algorithm giving a seed set of size a...

Abstract. In many applications of terrain analysis, pits or local minima are considered artifacts that must be removed before the terrain can be used. Most of the existing methods for local minima removal work only for raster terrains. In this paper we consider algorithms to remove local minima from polyhedral terrains, by modifying the heights of the vertices. To limit the changes introduced to the terrain, we try to minimize the total displacement of the vertices. Two approaches to remove local minima are analyzed: lifting vertices and lowering vertices. For the former we show that all local minima in a terrain with n vertices can be removedintheoptimalwayinO(n log n) time. For the latter we prove that the problem is NP-hard, and present an approximation algorithm with factor 2 ln k, wherek is the number of local minima in the terrain. 1

The concept of a “Marine GIS ” is in some ways an oxymoron: a GIS (Geographical Information System) almost presupposes that “Geography ” equals “Land”. The classical GIS structure is based on manual cartography, with registered transparent overlays of different themes, and the manual extraction of

Visual representation techniques enable perception and exploration of scientific data. Following the topological landscapes metaphor of Weber et al., we provide a new algorithm for visualizing scalar functions defined on simply connected domains of arbitrary dimension. For a potentially high dimensional scalar field, our algorithm produces a collection of, in some sense complete, two-dimensional terrain models whose contour trees and corresponding topological persistences are identical to those of the input scalar field. The algorithm exactly preserves the volume of each region corresponding to an arc in the contour tree. We also introduce an efficiently computable metric on terrain models we generate. Based on this metric, we develop a tool that can help the users to explore the space of possible terrain models. Categories and Subject Descriptors (according to ACM CCS): Generation—View algorithms