Daniel R. Schikore

Contour trees and small seed sets for isosurface traversal

For 2D or 3D meshes that represent the domain of continuous function to the reals, the contours|or isosurfaces|of a speci ed value are an important way to visualize the function. To nd 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 rst 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 logn) time in 2D for meshes with n elements, and in O(n) 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 at most twice the size of the minimum. It requires O(n log n) time and linear storage once the contour tree is known. We also give experimental results, showing the size of the seed sets for several data sets.

The contour spectrum

The authors 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 /spl omega//spl isin/R. They explore the use of surface, area, volume, and gradient integral of the contour that are shown to be univariate B-spline functions of the scalar value /spl omega/ 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 /spl omega/) to assist in selecting relevant isovalues /spl omega//sub 0/ for informative visualization. For time-varying 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 time step. The effectiveness of the current system and potential extensions are discussed.

Automatic Reconstruction of 3D CAD Models from Digital Scans

We present an approach for the reconstruction and approximation of 3D CAD models from an unorganized collection of points. Applications include rapid reverse engineering of existing objects for use in a virtual prototyping environment, including computer aided design and manufacturing. Our reconstruction approach is flexible enough to permit interpolation of both smooth surfaces and sharp features, while placing few restrictions on the geometry or topology of the object. Our algorithm is based on alpha-shapes to compute an initial triangle mesh approximating the surface of the object. A mesh reduction technique is applied to the dense triangle mesh to build a simplified approximation, while retaining important topological and geometric characteristics of the model. The reduced mesh is interpolated with piecewise algebraic surface patches which approximate the original points. The process is fully automatic, and the reconstruction is guaranteed to be homeomorphic and error bounded with respect to the original model when certain sampling requirements are satisfied. The resulting model is suitable for typical CAD modeling and analysis applications.

Topology preserving data simplification with error bounds

Abstract Many approaches to simplification of triangulated terrains and surfaces have been proposed which permit bounds on the error introduced. A few algorithms additionally bound errors in auxiliary functions defined over the triangulation. We present an approach to simplification of scalar fields over unstructured grids which preserves the topology of functions defined over the triangulation, in addition to bounding of the errors. The topology of a 2D scalar field is defined by critical points (local maxima, local minima, saddle points), in addition to integral curves between them, which together segment the field into regions which vary monotonically. By preserving this shape description, we guarantee that isocontours of the scalar function maintain the correct topology in the simplified model. Methods for topology preserving simplification by both point-insertion (refinement) and point-deletion (coarsening) are presented and compared.

Visualization of scalar topology for structural enhancement

Scalar fields arise in every scientific application. Existing scalar visualization techniques require that the user infers the global scalar structure from what is frequently an insufficient display of information. We present a visualization technique which numerically detects the structure at all scales, removing from the user the responsibility of extracting information implicit in the data, and presenting the structure explicitly for analysis. We further demonstrate how scalar topology detection proves useful for correct visualization and image processing applications such as image co-registration, isocontouring, and mesh compression.

Error-bounded reduction of triangle meshes with multivariate data

Interactive visualization is complicated by the complexity of the objects being visualized. Sampled or computed scientific data is often dense, in order to capture high frequency components in measured data or to accurately model a physical process. Common visualization techniques such as isosurfacing on such large meshes generate more geometric primitives than can be rendered in an interactive environment. Geometric mesh reduction techniques have been developed in order to reduce the size of a mesh with little compromise in image quality. Similar techniques have been used for functional surfaces (terrain maps) which take advantage of the planar projection. We extend these methods to arbitrary surfaces in 3D and to any number of variables defined over the mesh by developing a algorithm for mapping from a surface mesh to a reduced representation and measuring the introduced error in both the geometry and the multivariate data. Furthermore, through error propagation, our algorithm ensures that the errors in both the geometric representation and multivariate data do not exceed a user-specified upper bound.

Distributed and collaborative visualization

Visualization typically involves large computational tasks, often performed on supercomputers. The results of these tasks are usually analyzed by a design team consisting of several members. Our goal is to depart from traditional single-user systems and build a low-cost scientific visualization environment that enables computer-supported cooperative work in the distributed setting. A synchronously conferenced collaborative visualization environment would let multiple users on a network of workstations and supercomputers share large data sets, simultaneously view visualizations of the data, and interact with multiple views while varying parameters. Such an environment would support collaboration in both the problem-solving phase and the review phase of design tasks. In this article we describe two distributed visualization algorithms and the facilities that enable collaborative visualization. These are all implemented on top of the distribution and collaboration mechanisms of an environment called Shastra, executing on a set of low-cost networked workstations.<<ETX>>

A Case Study of Isosurface Extraction Algorithm Performance

Isosurface extraction is an important and useful visualization method. Over the past ten years, the field has seen numerous isosurface techniques published, leaving the user in a quandary about which one should be used. Some papers have published complexity analysis of the techniques, yet empirical evidence comparing different methods is lacking. This case study presents a comparative study of several representative isosurface extraction algorithms. It reports and analyzes empirical measurements of execution times and memory behavior for each algorithm. The results show that asymptotically optimal techniques may not be the best choice when implemented on modern Computer architectures

Advances in three-dimensional reconstruction of the experimental spinal cord injury

Three-dimensional (3D) computer reconstruction is an ideal tool for evaluating the centralized pathology of mammalian spinal cord injury (SCI) where multiple anatomical features are embedded within each other. Here, we evaluate three different reconstruction algorithms to three-dimensionally visualize SCIs. We also show for the first time, that determination of the volume and surface area of pathological features is possible using the reconstructed 3D images themselves. We compare these measurements to those calculated by older morphometric approaches. Finally, we demonstrate dynamic navigation into a 3D spinal cord reconstruction.

A triangulation-based object reconstruction method

Reconstructing the shape of a 3D object from a digital scan of its surface has a range of applications, such as reverse engineering, authoring 3D synthetic worlds, shape analysis, 3D faxing and tailor-fit modeling. Input data might come in dfierent forms, depending on the scanning device used. It is usually comprised of the location (zI, yl, zi) of points on the surface of the object, and at times additional topological and geometric information, as well as measures of other physical properties. The sampling provided by recent scanning devices (such as the Lzser range scanner) is dense, in the sense that the resolution is much smaller than the sise of shape features of interest. Often multiple scans are required to capture the entire object’s surface. We make no assumptions on spatiaJ relations among sample points, and assume that the input is a large, but unorganized, collection of measurements. Our goal is to reconstruct a boundary representation of the object, based on implicit polynomial surface patches of low degree, that has the tiesired geometric continuity and approximates the data within a user-specified parameter E. For a discussion of related prior work the reader is referred to [6]. In [I], we presented a method based on alpha-shapes, to build an initial piecewise-linear reconstruction, followed by an incremental, adaptive piecewise polynomial fitting of the signed distance function defined by the alpha-shape. The method relied on the user to select a good a-value. The final reconstructed model was represented as a collection of C] -smooth implicit algebraic patches. A more up-to-date, detailed description of the algorithm can be found in [2]. In this paper, and the accompanying video presentation,