David Avis

Automated 3-D Extraction of Inner and Outer Surfaces of Cerebral Cortex from MRI

Automatic computer processing of large multidimensional images such as those produced by magnetic resonance imaging (MRI) is greatly aided by deformable models, which are used to extract, identify, and quantify specific neuroanatomic structures. A general method of deforming polyhedra is presented here, with two novel features. First, explicit prevention of self-intersecting surface geometries is provided, unlike conventional deformable models, which use regularization constraints to discourage but not necessarily prevent such behavior. Second, deformation of multiple surfaces with intersurface proximity constraints allows each surface to help guide other surfaces into place using model-based constraints such as expected thickness of an anatomic surface. These two features are used advantageously to identify automatically the total surface of the outer and inner boundaries of cerebral cortical gray matter from normal human MR images, accurately locating the depths of the sulci, even where noise and partial volume artifacts in the image obscure the visibility of sulci. The extracted surfaces are enforced to be simple two-dimensional manifolds (having the topology of a sphere), even though the data may have topological holes. This automatic 3-D cortex segmentation technique has been applied to 150 normal subjects, simultaneously extracting both the gray/white and gray/cerebrospinal fluid interface from each individual. The collection of surfaces has been used to create a spatial map of the mean and standard deviation for the location and the thickness of cortical gray matter. Three alternative criteria for defining cortical thickness at each cortical location were developed and compared. These results are shown to corroborate published postmortem and in vivo measurements of cortical thickness.

Reverse search for enumeration

Abstract The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing 1. (i) all triangulations of a set of n points in the plane. 2. (ii) all cells in a hyperplane arrangement in Rd, 3. (iii) all spanning trees of a graph, 4. (iv) all Euclidean (noncrossing) trees spanning a set of n points in the plane. 5. (v) all connected induced subgraphs of a graph, and 6. (vi) all topological orderings of an acyclic graph. Finally, we propose a new algorithm for the 0 1 integer programming problem which can be considered as an alternative to the branch-and-bound algorithm.

A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

We present a new pivot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a)Virtually no additional storage is required beyond the input data.(b)The output list produced is free of duplicates.(c)The algorithm is extremely simple, requires no data structures, and handles all degenerate cases.(d)The running time is output sensitive for nondegenerate inputs.(e)The algorithm is easy to parallelize efficiently. For example, the algorithm finds thev vertices of a polyhedron inRd defined by a nondegenerate system ofn inequalities (or, dually, thev facets of the convex hull ofn points inRd, where each facet contains exactlyd given points) in timeO(ndv) andO(nd) space. Thev vertices in a simple arrangement ofn hyperplanes inRd can be found inO(n2dv) time andO(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.

Multiple surface identification and matching in magnetic resonance images

An iterative algorithm is presented for simultaneous deformation of multiple curves and surfaces to an MRI, with inter-surface constraints and self-intersection avoidance. The resulting robust segmentation, combined with local curvature matching, automatically creates surfaces of MRI datasets with a common mapping to surface parametric space.

A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm

This paper describes an improved implementation of the reverse search vertex enumeration/convex hull algorithm for d-dimensional convex polyhedra. The implementation uses a lexicographic ratio test to resolve degeneracy, works on bounded or unbounded polyhedra and uses exact arithmetic with all integer pivoting. It can also be used to compute the volume of the convex hull of a set of points. For a polyhedron with m inequalities indvariables and known extreme point, it finds all bases in time O(md) 2 per basis. This implementation can handle problems of quite large size, especially for simple polyhedra (where each basis corresponds to a vertex and the complexity reduces to O (md) per vertex). Computational experience is included in the paper, including a comparison with an earlier implementation.

A Survey of Heuristics for the Weighted Matching Problem

This survey paper reviews results on heuristics for two weighted matching problems: matchings where the vertices are points in the plane and weights are Euclidean distances, and the assignment problem. Several heuristics are described in detail and results are given for worst-case ratio bounds, absolute bounds, and expected bounds. Applications to practical problems and some mathematical complements are also included.

A linear algorithm for finding the convex hull of a simple polygon

The problem of determining the convex hull of a set of n points in the plane has recently received a good deal of attention. Several algorithms for the general problem with worst case complexity D(n log n) have appeared (e.g., [3,4,6]). The special case where the points form the vertices of a simple polygon has long been considered easier. Indeed, Sklansky [S] has proposed an O(n) algorithm, but a. recently published counter example of Bykat [2] shows that the algorithm can sometimes fail. A slightly different counterexample can be constructed for a similar algorithm of Shamos [4]. In this note we present and prove the validity of a new linear time algorithm for this problem.

On a convex hull algorithm for polygons and its application to triangulation problems

Abstract A frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases. Due to its simplicity the algorithm is, nevertheless, attractive. In this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons. Some application areas where such polygons occur are briefly discussed. In addition, it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0( n ) time.

An efficient algorithm for decomposing a polygon into star-shaped polygons

Abstract In this paper we show how a theorem in plane geometry can be converted into a O(n log n) algorithm for decomposing a polygon into star-shaped subsets. The computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods.

ON THE EXTREME RAYS OF THE METRIC CONE

Introduction. A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equations