A standard isosurface extraction algorithm for a 3D image is the Marching Cubes algorithm [4]. The image is assumed to be defined on a regular lattice of size N0×N1×N2 with integer points (x, y, z) where 0 ≤ x < N0, 0 ≤ y < N1, and 0 ≤ z < N2. The image values themselves are F (x, y, z). An isosurface is of the form F (x, y, z) = c for some specified level value c where x, y, and z are treated as continuous variables. A voxel in the image is a rectangular solid whose corners are eight neighboring lattice points (x0, y0, z0), (x0 +1, y0, z0), (x0, y0+1, z0), (x0+1, y0+1, z0), (x0, y0, z0+1), (x0+1, y0, z0+1), (x0, y0+1, z0+1), and (x0+1, y0+1, z0+1). The Marching Cubes algorithm analyzes each voxel in the image and determines if the isosurface intersects it. If so, the algorithm produces a triangle mesh for the voxel that is intended to approximate that portion of the isosurface inside the voxel. By selecting a level value that cannot be an image value, for example by selecting a non–integer value when the image has only integer values, the voxel analysis requires determining the signs of G(x, y, z) = F (x, y, z) − c at the eight corners, each sign positive or negative. If two adjacent corners have opposite signs, and if the image values are assumed to be linear along the edge connecting the corners, the isosurface G(x, y, z) = 0 must intersect the edge in a single point somewhere along the edge. The complexity of the surface of intersection is related to the sign changes that occur on all the edges of the voxel.

The SLERP of quaternions is a common operation in keyframe animation. The operation can be a significant bottleneck in an animation-heavy application. The standard implementation of SLERP for an FPU typically involves trigonometric function evaluations, divisions, and branching. Commonly available SIMD implementations will modify the standard implementation and obtain a moderate speed-up. This paper provides a novel approach to computing SLERP, using only multiplications and additions. The algorithm is based on ideas from Chebyshev polynomials, power series solutions for differential equations, and error balancing using the Chebyshev Equioscillation Theorem and the associated Remez Algorithm. Implementations are provided for the FPU and SIMD. When SLERPing a pair of quaternions in parallel using four time samples, performance measurements show a 10-fold acceleration per SLERP over the standard SLERP implementation on an FPU. Source code is available online.

A novel interactive segmentation method has been developed which uses estimated boundaries, generated from cores, to initialize a scale-space boundary evolution process in greyscale medical images. Presented is an important addition to core extraction methodology that improves core generation for objects that are in the presence of interfering objects. The boundary at the scale of the core (BASOC) and its associated width information, both derived from the core, are used to initialize the second stage of the segmentation process. In this automatic refinement stage, the BASOC is allowed to evolve in a spline-snake-like manner that makes use of object-relevant width information to make robust measurements of local edge positions.

A classic problem in computer graphics is to decompose a simple polygon into a collection of triangles whose vertices are only those of the simple polygon. By definition, a simple polygon is an ordered sequence of n points, ~ V0 through ~ Vn−1. Consecutive vertices are connected by an edge 〈~ Vi, ~ Vi+1〉, 0 ≤ i ≤ n − 2, and an edge 〈~ Vn−1, ~ V0〉 connects the first and last points. Each vertex shares exactly two edges. The only place where edges are allowed to intersect are at the vertices. A typical simple polygon is shown in Figure 1

This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1].

From the Publisher:Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more.If you have a mathematics degree, this book will save you time and trouble. If you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed, and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices.KEY FEATURES: * Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors.* Covers problems relevant for both 2D and 3D graphics programming.* Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you.* Provides the math and geometry background you need to understand the solutions and put them to work.* Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode.* Resources associated with the book are available at the companion Web site.Author Biography: Philip Schneider leads a modeling and dynamic simulation software group at Walt Disney Feature Animation. Prior to that, his work at Apple and Digital Equipment Corporation in 3D graphics ranged from low-level interfaces to graphics libraries and interactive applications. He holds an M.S. in Computer Science from the University of Washington.Dave Eberly is the president of Magic Software, Inc., a company known for its free source code and documentation for computer graphics, image analysis, and numerical methods. Previously, he was the director of engineering at Numerical Design Limited, the company responsible for the real-time 3D game engine, NetImmerse. His background includes a B.A. in mathematics from Bloomsburg University, M.S. and Ph.D. degrees in mathematics from the University of Colorado at Boulder, and M.S. and Ph.D. degrees in computer science from the University of North Carolina at Chapel Hill. Dave is author of 3D Game Engine Design, co-author with Philip Schneider of Geometric Tools for Computer Graphics, and author of the forthcoming Game Physics (Spring 2003).

Believing that figural zoom invariance and the cross-figural boundary linking implied by medial loci are important aspects of object shape, we present the mathematics of and algorithms for the extraction of medial loci directly from image intensities. The medial loci called cores are defined as generalized maxima in scale space of a form of medial information that is invariant to translation, rotation, and, in particular, zoom. These loci are very insensitive to image disturbances, in strong contrast to previously available medial loci, as demonstrated in a companion paper. Core-related geometric properties and image object representations are laid out which, together with the aforementioned insensitivities, allow the core to be used effectively for a variety of image analysis objectives.

The authors present an algorithm, called marching cores, that generates cores of 3D medical images and also generalizes to finding implicitly defined manifolds of codimension greater than one. As one marches along the core, one use medialness kernels to generate new medialness values and then find ridges in the extended medial space using the geometric definition of height ridges and mathematical models of manifold intersections. Results from both a test image and a CT image illustrate the algorithm.

The linear and cyclic convolutions are computed as polynomial products. Group algebras are generalizations of polynomials and polynomial arithmetic based on algebraic groups. The multiplication in group algebra is called “group convolution.” The selection of coefficients from the cyclic convolution to generate output for the linear convolution can be extended to other functions called “value functions,” to allow for a wider variety of outputs. Group convolution of a signal requires selecting a single base group to work with. For images, group convolution is naturally based on the direct product of groups. Matrix representation of group algebra elements is discussed. The chapter discusses direct product group algebras. The ideas extend to convolution of data sets whose dimension is larger than two. Semidirect product group algebras that are used mainly in studying group algebras based on dihedral groups are discussed. The chapter also discusses inverting group algebra elements. Two methods for inverting elements are explained. In applications where nonzero and noninvertible group algebra elements may be a problem, a way to eliminate these elements via ideals and quotient rings is presented. Matrices whose elements are from group algebra are discussed. The chapter focuses on those matrices that have the convolution property—that is, those matrices for which the transform of a convolution is the product of the individual transforms. The chapter presents examples of the application of group algebras to edge detection in signals and images. The examples illustrate the use of direct product cyclic groups (the standard convolution case), direct product dihedral groups, and direct product quaternions. The chapter describes the process to implement group algebras in an object-oriented programming language.

The multiscale medial axis (MMA) is a principled means of describing both the spatial and width properties of objects in grey-scale images. We describe its computation and provide an example of its use in an image registration task.

There are two basic principles associated with multiscale geometry: 1) geometry involves analysis that is invariant to certain spatial transformations, including translation, rotation, and zoom, and 2) the dimension of scale is as critical as the dimensions giving spatial position, corresponding intuitively to level of detail in image space. Considered together, image space and scale is called scale space. Three families of methods based on these principles are achieving impressive results in image analysis —particularlyin their insensitivity to irrelevant detail (including image noise) and intensity blurring, and in their ability to produce stable object descriptions and pixel classifications into objects. The three families are multiscale medial axis or core-based analysis (CBA), variable conductance difftision (VCD), and multiscale geometric statistical pattern recognition (MGSPR). This pair of tutorials (morning and afternoon) covered the basic mathematics of multiscale geometry as well as all three of these families of methods. It included algorithms for computation and also illustrative results of the applications of the methods to both 2D and 3D medical images of various modalities. The morning tutorial covered the mathematics of diffusion and scale space and the definition, effect on scale space geometry, and application of cores. The afternoon tutorial covered the mathematics, algorithms, and applications of variable conductance diffusion, including approaches involving MGSPR, and it covered algorithms for segmenting objects both via VCD and via CBA.

Image segmentation is a process which identifies and labels objects in an image. The goals of this dissertation are to produce an algorithm for segmenting an image in the way that a front-end vision system does, using the local geometry induced by the intensity values of the image, to create multiscale representations of the objects that allow exploration of the details of the image via an interactive computer system, and to provide a formal geometric foundation for multiscale image analysis. The geometric concept of ridges is discussed. These structures are used to describe the shape properties of objects in an image. Various definitions are given for d-dimensional ridge structures of n-dimensional images. Ridges are used in conjunction with multiscale methods to segment an image. The output of the segmentation is a single tree and the objects in the image are represented as unions and differences of subtrees. The tree and image are used as input to a visualization tool which allows the user to explore the image interactively. A formal foundation for multiscale analysis is given which involves non-Euclidean geometry. Metric selection for scale space is naturally related to invariances required for a particular application. The anisotropic diffusion process for generating multiscale data is automatically determined by the metric. Moreover, the metric is used as an aid in developing fast, stable, and adaptive numerical algorithms for solving the nonlinear diffusion equations. This geometric foundation for multiscale analysis provides a natural set of tools for extracting information about objects in an image.

The authors develop a class of filters called weighted majority of m values with minimum range (WMMR/sup m/) that have the same impulse rejection properties as the median. They demonstrate a subclass of these filters (WMMR) that may be optimized for edge enhancement in one dimension in that their output converges to the closest perfect edge. One of these filters is shown to restore a class of noisy edges to the closest perfect edge on one pass. Applications in one and two dimensions are discussed and a two-dimensional simulation is provided comparing the WMMR to other filters for smoothing and edge enhancement. >

Surface area and volume are essential measurements in the morphometric assessment of anatomical structures. New algorithms were developed to measure (1) distance along a curve, (2) surface area, and (3) volume using data extracted from tomographic images as a geometrical surface model. The model is a list of coordinates and normal vectors for each voxel or point gathered from the surface of a selected object. The resulting surface-based pointlist is also used for high-speed rendering of surfaces. Differential arclength and surface area are measured with high numeric precision by using the absolute value of the maximum component of the unit normal vector (MUNC) to approximate their values. These differential values are summed to measure distance along a curve and surface area. A discrete form of the Divergence theorem, also using the MUNC, is used to calculate volume. The intrinsic accuracy of the measurement algorithms was evaluated using computer generated pointlists of circles, ellipses, spheres, and ellipsoids. Compared to standard measurement techniques, the new algorithms provided the greatest accuracy and least shape-related bias for measurement of distance, surface area, and volume. Feasibility of using the new algorithms to measure physical objects was tested with CT images of spherical, egg-shaped, and irregular shaped objects. The Dividing Cubes algorithm was used to segment and create pointlists from the CT data. Volume and surface area measurements from CT data compared extremely well with reference values for most objects tested (errors less than 2%).

Local convergence results of the convex minorant (CM) algorithm to obtain the nonparametric maximum-likelihood estimator of a distribution under interval-censored observations are given. We also provide a variation of the CM algorithm, which yields global convergence. The algorithm is illustrated with data on AIDS survival time in 92 members of the U.S. Air Force.

The median filter is a nonlinear filter that preserves edges and eliminates impulses. The initial papers on the median filter, by Tyan and Gallagher and Wise, concentrated on the development of their impulse rejection properties, the set of fixed points, and convergence. It was shown that the fixed points of the median filter are the class of LOMO (locally monotone) signals, that they converge within a finite number of iterations, and that they would reject burst of up to n aberrant values in each nonoverlapping segment of length 2n + 1. Their initial work has led to many research papers in robust signal processing in the presence of edges. The order-statistic (OS) filter is very similar to the FIR filter with the exception that it orders the values in each window before weighing them. The WMMRc filters weight the c ordered values in the window with minimum range. If more than one set of values in the window have the minimum range, the average of the possible outputs is taken. If c is not specified it is assumed to be N + 1 for a window of length 2N + 1. For OS and WMMR filters with convex (sum to one and are nonnegative) weights, fixed point results are derived similar to those of Gallagher and Wise for the median filter, i.e., the fixed points are completely classified under the assumption of a finite length signal with constant boundaries.

Given a sample with mean x̄ and second moment s2, Anscombe in 1950 conjectured that the maximum likelihood equations for the two-parameter negative binomial distribution have a unique solution if and only if s2 > x̄. We give a proof of his conjecture.