Chandrajit L. Bajaj

Automatic reconstruction of surfaces and scalar fields from 3D scans

We present an efficient and uniform approach for the automatic reconstruction of surfaces of CAD (computer aided design) models and scalar fields defined on them, from an unorganized collection of scanned point data. A possible application is the rapid computer model reconstruction of an existing part or prototype from a three dimensional (3D) points scan of its surface. Color, texture or some scalar material property of the physical part, define natural scalar fields over the surface of the CAD model. Our reconstruction algorithm does not impose any convexity or differentiability restrictions on the surface of the original physical part or the scalar field function, except that it assumes that there is a sufficient sampling of the input point data to unambiguously reconstruct the CAD model. Compared to earlier methods our algorithm has the advantages of simplicity, efficiency and uniformity (both CAD model and scalar field reconstruction). The simplicity and efficiency of our approach is based on several novel uses of appropriate sub-structures (alpha shapes) of a three-dimensional Delaunay Triangulation, its dual the three-dimensional Voronoi diagram, and dual uses of trivariate Bernstein-Bezier forms. The boundary of the CAD model is modeled using implicit cubic Bernstein-Bezier patches, while the scalar field is reconstructed with functional cubic Bernstein-Bezier patches. CR

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.

The transfer function bake-off

A liquid fuel pumping apparatus for supplying fuel to an internal combustion engine includes a member which is movable against the action of a spring system by means of centrifugally operable weights. The member is coupled to a fuel control rod of the pumping apparatus, and the spring system includes first and second springs. The first spring is preloaded so as to be deflected by the weights only when the speed of the engine attains a predetermined value. The other spring is deflected by the weights at lower engine speeds and also acts to transmit the force exerted by the weights to the first spring.

Tracing surface intersections

Abstract We consider the problem of tracing the intersection of surfaces given either implicitly or parametrically. We give a numerical tracing procedure in which a third-order Taylor approximant is constructed for taking steps of variable length, and the points so found are improved by Newton iteration. We show how this construction relates to local parametrizations of the curve at singularities, and discuss our experience with the method. For plane curves, given implicitly, we show how desingularization techniques can be incorporated to trace correctly through all types of singularities. An implementation of this method is also discussed.

Anisotropic diffusion of surfaces and functions on surfaces

We present a unified anisotropic geometric diffusion PDE model for smoothing (fairing) out noise both in triangulated two-manifold surface meshes in IR3 and functions defined on these surface meshes, while enhancing curve features on both by careful choice of an anisotropic diffusion tensor. We combine the C1 limit representation of Loops subdivision for triangular surface meshes and vector functions on the surface mesh with the established diffusion model to arrive at a discretized version of the diffusion problem in the spatial direction. The time direction discretization then leads to a sparse linear system of equations. Iteratively solving the sparse linear system yields a sequence of faired (smoothed) meshes as well as faired functions.

Arbitrary topology shape reconstruction from planar cross sections

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 difficult 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 specific 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.

The algebraic degree of geometric optimization problems

In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.

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.

Progressive compression and transmission of arbitrary triangular meshes

The recent growth in the size and availability of large triangular surface models has generated interest in compact multi-resolution progressive representation and data transmission. An ongoing challenge is to design an efficient data structure that encompasses both compactness of geometric representations and visual quality of progressive representations. We introduce a topological layering based data structure and an encoding scheme to build a compact progressive representation of an arbitrary triangular mesh (a 2D simplicial complex in 3D) with attached attribute data. This compact representation is composed of multiple levels of detail that can be progressively transmitted and displayed. The global topology, which is the number of holes and connected components, can be flexibly changed among successive levels while still achieving guaranteed size of the coarsest level mesh for very complex models. The flexibility in our encoding scheme also allows topology preserving progressivity.