A technique for rendering images Of volumes containing mixtures of materials is presented. The shading model allows both the interior of a material and the boundary between materials to be colored. Image projection is performed by simulating the absorption of light along the ray path to the eye. The algorithms used are designed to avoid artifacts caused by aliasing and quantization and can be efficiently implemented on an image computer. Images from a variety of applications are shown.

This paper presents a new afgorithm for smoothly blending between two 2-D polygonal shapes. The algorithm is based on a physical model wherein one of the shapes is considered to be constructed of wire, and a solution is found whereby the first shape can be bent and/or stretched into the second shape with a minimum amount of work. The resulting solution tends to associate regions on the two shapes which look alike. If the two polYgons have m and n vertices respectively, the afgorithm is O(mn). The algorithm avoids local shape inversions in whkh intermediate

In computed tomography, magnetic resonance imaging and ultrasound imaging, reconstruction of the 3D object from the 2D scalar-valued slices obtained by the imaging system is di cult because of the large spacings between the 2D slices. The aliasing that results from this undersampling in the direction orthogonal to the slices leads to two problems known as the correspondence problem and the tiling problem. A third problem, known as the branching problem, arises because of the structure of the objects being imaged in these applications. Existing reconstruction algorithms typically address only one or two of these problems. In this paper, we approach all three of these problems simultaneously. This is accomplished by imposing a set of three constraints on the reconstructed surface and then deriving precise correspondence and tiling rules from these constraints. The constraints ensure that the regions tiled by these rules obey physical constructs and have a natural appearance. Regions which cannot be tiled by these rules without breaking one or more constraints are tiled with their medial axis (edge Voronoi diagram). Our implementation of the above approach generates triangles of 3D isosurfaces from input which is either a set of contour data or a volume of image slices. Results obtained with synthetic and actual medical data are presented. There are still speci c cases in which our new approach can generate distorted results, but these cases are much less likely to occur than those which cause distortions in other tiling approaches. 2 1

In this paper we present a new technique for piecewiselinear surface reconstruction from a series of parallel polygonal cross-sections. This is an important problem in medical imaging, surface reconstruction from topographic data, and other applications. We reduce the problem, as in most previous works, to a series of problems of piecewise-linear interpolation between each pair of successive slices. Our algorithm uses a partial curve matching technique for matching parts of the contours, an optimal triangulation of 3-D polygons for resolving the unmatched parts, and a minimum spanning tree heuristic for interpolating between non simply connected regions. Unlike previous attempts at solving this problem, our algorithm seems to handle successfully any kind of data. It allows multiple contours in each slice, with any hierarchy of contour nesting, and avoids the introduction of counter-intuitive bridges between contours, proposed in some earlier papers to handle interpolation between multi...

In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching technique for matching parts of the defects, and an optimal triangulation of 3-D polygons for resolving the unmatched parts. It is also shown that finding a consistent set of partial curve matches with maximum score, a subproblem which is related to our repairing process, is NP-Hard. Experimental results on several polyhedra are presented. Keywords: CAD, polyhedra, gap filling, curve matching, geometric hashing, triangulation. 1 Introduction The problem studied in this paper is the detection and repair of "gaps" in the boundary of a polyhedron. This problem usually appears in polyhedral approximations of CAD objects, whose boundaries are described using curved entities of higher leve...

Contents 1 Introduction 2 2 Map Data Modeling 4 2.1 Two-Dimensional Spatial Entities and Relationships . . . . . . . . . . . . . . . . . . . . . 4 2.2 Raster and Vector Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Subdivisions as Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Topological Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Multiresolution Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Map data processing 8 3.1 Spatial Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Map Overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Geometric Problems in Map Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Map Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . .

Surface reconstruction from parallel slices is a well researched problem in solid modeling and computer graphics. The importance of the problem stems from its wide applicability such as in medical imaging for organ modeling, and in topography for terrain modeling. As pointed out in earlier literature, the three major issues for such surface reconstruction are tiling problem, correspondence problem and branching problem. Many of the earlier approaches concentrated primarily on the tiling problem, where the main concern is to generate a non self-intersecting surface connecting two contours with certain optimization. Lately, a few approaches take a more wholistic view to address all of them. In this paper we revisit one such method based on Delaunay triangulations. This method seems more promising and appropriate in handling correspondence and branching problem due to the inherent ability of Delaunay triangulations to capture proximity. Further, a non self-intersecting tiling is automatic...

This paper describes an e cient method for constructing a tiling between a pair of planar contours. The problem is of interest in a number of domains, including medical imaging, biological research and geological reconstructions. Our method, based on ideas from multiresolution analysis and wavelets, requires O(n) space and appears to require O(n log n) time for average inputs, compared to the O(n 2) space and O(n 2 log n) time required by the optimizing algorithm of Fuchs, Kedem and Uselton 1. The results computed by our algorithm are in many cases nearly the same as those of the optimizing algorithm, but at a small fraction of the computational cost. The performance improvement makes the algorithm usable for large contours in an interactive system. The use of multiresolution analysis provides an e cient mechanism for data compression by discarding wavelet coe cients smaller than a threshold value during reconstruction. The amount of detail lost can be controlled by appropriate choice of the threshold value. The use of lower resolution approximations to the original contours yields signi cant savings in the time required to display a reconstructed object, and in the space required to store it.

We present an algorithm for reconstructing a solid model from a series of planar cross-sections. In most previous works the layers are assumed to be independent: each layer is interpolated separately and the concatenation of the interpolated layers is considered the solution to the whole problem. The resulting surface can therefore exhibit abrupt changes. The main contribution of this work is avoiding this assumption. We use the slopes of triangles created in the interpolation of neighboring layers to guide the interpolation of the current layer. As a result, consecutive layers are connected smoothly. We also discuss in this paper various objective functions which aim to optimize the reconstruction and evaluate these functions using various criteria. Keywords: surface reconstruction, interpolation, triangulation. 1 Introduction The reconstruction of a polyhedral object from a series of cross-sections has been an intriguing problem during the last couple of decades. The main...

ix Acknowledgements x Acknowledgements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x Publication History : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x 1. Introduction 1 1.1 Introduction to Direct Volume Rendering : : : : : : : : : : : : : : : : : : : 2 1.1.1 Volumetric Grids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.1.2 Image-space Rendering Algorithms : : : : : : : : : : : : : : : : : : : 4 1.1.3 Object-space Rendering Algorithms : : : : : : : : : : : : : : : : : : 5 1.1.4 Shear Transformations : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.1.5 Complexity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.2 Motivation for Parallel Direct Volume Rendering : : : : : : : : : : : : : : : 8 1.2.1 Scalability Is Important : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.3 Context for Use : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 1.3.1 Distributed Graphical Us...