The problem of computing a piecewise linear approximation to a surface from a set of sample points is important in solid modeling, computer graphics and computer vision. A recent algorithm [1] using the Voronoi diagram of the sample points gave a guarantee on the distance of the output surface from the original sampled surface assuming the sample was `suciently dense'. We give a similar algorithm, simplifying the computation and the proof of the geometric guarantee. In addition, we guarantee that our output surface is homeomorphic to the original surface; to our knowledge this is the rst such topological guarantee for this problem. 1 Introduction A number of applications in CAD, computer graphics, computer vision and mathematical modeling involve the computation of a piecewise lin- Dept. of Computer Science, U. of Texas, Austin TX 78712. e-mail: amenta@cs.utexas.edu, supported by NSF grant CCR-9731977 y Dept. of Computer Science, U. of Texas, Austin, TX 78712. e-mail: sunghe...

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

A new active contour model for finding and mapping the outer cortex in brain images is developed. A cross-section of the brain cortex is modeled as a ribbon, and a constant speed mapping of its spine is sought. A variational formulation, an associated force balance condition, and a numerical approach are proposed to achieve this goal. The primary difference between this formulation and that of snakes is in the specification of the external force acting on the active contour. A study of the uniqueness and fidelity of solutions is made through convexity and frequency domain analyses, and a criterion for selection of the regularization coefficient is developed. Examples demonstrating the performance of this method on simulated and real data are provided.

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...

: We introduce a method for the construction of skeletal curves from an unorganized collection of scattered data points lying on a surface. These curves may have a tree like structure to capture branching shapes such as blood vessels. The skeletal curves can be used for different applications ranging from surface reconstruction to object recognition. As an input, the algorithm takes a set of 3D points. It returns a set of curves arranged in a tree structure. The only interaction needed is the selection of a data point which represent the root of the tree. A neighborhood graph is constructed over the set of points to compute geodesic distances between the root point and the other points. Connected level sets of the distance map are then extracted and organized in a tree structure. The centers of these levels sets constitute the skeletal curves. Key-words: visualization, skeletal curve, cylindrical decomposition, generalized cylinders, reconstruction (R'esum'e : tsvp) Anne.Verroust@in...

A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197--213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR-8921421 and under the Alan T. Waterman award, grant CCR-9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...

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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...

In this paper we summarize our experiences with 3D constrained Delaunay triangulation algorithms for industrial applications. In addition, we report a robust implementation process for constructing 3D constrained triangulations from initial unconstrained triangulations, based on a minimalist approach, in which we minimize the use of geometrical operations such as intersections. This is achieved by inserting Steiner points on missing constraining edges and faces in the initial unconstrained triangulations. This approach allowed the generation of tetrahedral meshes for arbitrarily complex 3D domains.

This paper describes efforts aimed at more objectively and accurately quantifying the local, regional and global function of the left ventricle (LV) of the heart from four--dimensional (4D) image data. Using our shape--based image analysis methods, point--wise myocardial motion vector fields between successive image frames through the entire cardiac cycle will be computed. Quantitative LV motion, thickening, and strain measurements will then be established from the point correspondence maps. In the paper, we will also briefly describe an in vivo experimental model which uses implanted imaging--opaque markers to validate the results of our image analysis methods. Finally, initial experimental results using image sequences from two different modalities will be presented. 1 Introduction Ischemic heart disease is a major clinical issue. Myocardial injury from ischemic heart disease is often regional, and it is the fundamental goal of many forms of cardiac imaging and image analysis method...