Robert H. Lewis

Solving the recognition problem for six lines using the Dixon resultant

The “Six-line Problem” arises in computer vision and in the automated analysis of images. Given a three-dimensional (3D) object, one extracts geometric features (for example six lines) and then, via techniques from algebraic geometry and geometric invariant theory, produces a set of 3D invariants that represents that feature set. Suppose that later an object is encountered in an image (for example, a photograph taken by a camera modeled by standard perspective projection, i.e. a “pinhole” camera), and suppose further that six lines are extracted from the object appearing in the image. The problem is to decide if the object in the image is the original 3D object. To answer this question two-dimensional (2D) invariants are computed from the lines in the image. One can show that conditions for geometric consistency between the 3D object features and the 2D image features can be expressed as a set of polynomial equations in the combined set of two- and three-dimensional invariants. The object in the image is geometrically consistent with the original object if the set of equations has a solution. One well known method to attack such sets of equations is with resultants. Unfortunately, the size and complexity of this problem made it appear overwhelming until recently. This paper will describe a solution obtained using our own variant of the Cayley–Dixon–Kapur–Saxena–Yang resultant. There is reason to believe that the resultant technique we employ here may solve other complex polynomial systems.

Heuristics to accelerate the Dixon resultant

The Dixon resultant method solves a system of polynomial equations by computing its resultant. It constructs a square matrix whose determinant (det) is a multiple of the resultant (res). The naive way to proceed is to compute det, factor it, and identify res. But often det is too large to compute or factor, even though res is relatively small. In this paper we describe three heuristic methods that often overcome these problems. The first, although sometimes useful by itself, is often a subprocedure of the second two. The second may be used on any polynomial system to discover factors of det without producing the complete determinant. The third applies when res appears as a factor of det in a certain exponential pattern. This occurs in some symmetrical systems of equations. We show examples from computational chemistry, signal processing, dynamical systems, quantifier elimination, and pure mathematics.

Conic tangency equations and Apollonius problems in biochemistry and pharmacology

The Apollonius Circle Problem dates to Greek antiquity, circa 250 B.C. Given three circles in the plane, find or construct a circle tangent to all three. This was generalized by replacing some circles with straight lines. Viete [Canon mathematicus seu Ad triangula cum adpendicibus, Lutetiae: Apud Ioannem Mettayer, Mathematicis typographum regium, sub signo D. Ioannis, regione Collegij Laodicensis, p. 1579] solved the problem using circle inversions before 1580. Two generations later, Descartes considered a special case in which all four circles are mutually tangent to each other (i.e. pairwise). In this paper, we consider the general case in two and three dimensions, and further generalizations with ellipsoids and lines. We believe, we are the first to explicitly find the polynomial equations for the parameters of the solution sphere in these generalized cases. Doing so is quite a challenge for the best computer algebra systems. We report later some comparative times using various computer algebra systems on some of these problems. We also consider conic tangency equations for general conics in two and three dimensions.Apollonius problems are of interest in their own right. However, the motivation for this work came originally from medical research, specifically the problem of computing the medial axis of the space around a molecule: obtaining the position and radius of a sphere which touches four known spheres or ellipsoids.

Algorithmic search for flexibility using resultants of polynomial systems

This paper describes the recent convergence of four topics: polynomial systems, flexibility of three dimensional objects, computational chemistry, and computer algebra. We discuss a way to solve systems of polynomial equations with resultants. Using ideas of Bricard, we find a system of polynomial equations that models a configuration of quadrilaterals that is equivalent to some three dimensional structures. These structures are of interest in computational chemistry, as they represent molecules. We then describe an algorithm that examines the resultant and determines ways that the structure can be flexible.

Isomorphism classes and derived series of certain almost-free groups

Baumslagdefined a family of groups that are of interest because they closely resemble free groups, yet are not free. It was known that each group in this family hasthe samelower central series of quotients and the same first two terms in the derived seriesof quotients as does the free group F on two generators. We have verified that there are different isomorphism types among the groups in the family, and that the third terms in the derived seriesof quotients are often distinct from that of F. Our basic technique isto count the number of homomorphisms from the groups of interest to a target group.

Support Vector Machines (SVM)

In statistical learning theory (regression, classification, etc.) there are many regression models, such as algebraic polynomials,

Homology and cell structure of nilpotent spaces

Let A and X denote finitely dominated nilpotent CW complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex. We investigate the set of possible cofibers of maps A -> X, and find a severe restriction. When it is met and the fundamental group is finite, X can be constructed from A by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes. We show that the Euler characteristic of X must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes. Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups. 0. Introduction. The topological concept of nilpotency is a generalization of that of simple connectivity. Since nilpotent complexes satisfy the Whitehead Theorem, one would like to derive their homotopy type by starting with their homology groups. We are therefore interested in the following sort of questions, all of which have affirmative or easy answers for simply connected complexes. Given nilpotent complexes A and X with the same finitely generated fundamental

Algebraic method to speed up robust algorithms: example of laser-scanned point clouds

Surface reconstruction from point clouds generated by laser scanning technology has become a fundamental task in many fields of geosciences, such as robotics, computer vision, digital photogrammetry, computational geometry, digital building modelling, forest planning and operational activities. Point clouds produced by laser scanning, however, are limited due to the occurrence of occlusions, multiple reflectance and noise, and off-surface points (outliers), thus necessitating the need for robust fitting techniques. In this contribution, a fast, non-iterative and data invariant algebraic algorithm with constant O(1) complexity that fits planes to point clouds in the total least squares sense using Gaussian-type error distribution is proposed. The maximum likelihood estimator method is used, resulting in a multivariate polynomial system that is solved in an algebraic way. It is shown that for plane fitting when datasets are affected heavily by outliers, the proposed algebraic method can be embedded into the framework of robust methods like the Danish or the RANdom SAmple Consensus methods and computed in parallel to provide rigorous algebraic fitting with significantly reduced running times. Compared to the embedded traditional singular value decomposition and principal component analysis approaches, the performance of the proposed algebraic algorithm demonstrated its efficiency on both synthetic data and real laser-scanned measurements. The evaluation of a symbolic algebraic formula is practically independent of the values of its coefficients; however, the computation of the coefficients depends on the complexity of the data. Since the main advantage of the symbolic solution is its non-requirement of numerical iteration, the data complexity will have weak influence on the speed-up. The novelty of the proposed method is the use of algebraic technique in a robust plane fitting algorithm that could be applied to remote sensing data analysis/delineation/classification. In general, the method could be applied to most plane fitting problems in the geoscience field.

Comparing acceleration techniques for the Dixon and Macaulay resultants

The Bezout-Dixon resultant method for solving systems of polynomial equations lends itself to various heuristic acceleration techniques, previously reported by the present author, which can be extraordinarily effective. In this paper we will discuss how well these techniques apply to the Macaulay resultant. In brief, we find that they do work there with some difficulties, but the Dixon method is greatly superior. That they work at all is surprising and begs theoretical explanation.

Flexibility of Bricard's linkages and other structures via resultants and computer algebra

Flexibility of structures is extremely important for chemistry and robotics. Following our earlier work, we study flexibility using polynomial equations, resultants, and a symbolic algorithm of our creation that analyzes the resultant. We show that the software solves a classic arrangement of quadrilaterals in the plane due to Bricard. We fill in several gaps in Bricards work and discover new flexible arrangements that he was apparently unaware of. This provides strong evidence for the maturity of the software, and is a wonderful example of mathematical discovery via computer assisted experiment.