Showing papers on "Affine transformation published in 1989"

The mapping of linear recurrence equations on regular arrays

Abstract: The parallelization of many algorithms can be obtained using space-time transformations which are applied on nested do-loops or on recurrence equations. In this paper, we analyze systems of linear recurrence equations, a generalization of uniform recurrence equations. The first part of the paper describes a method for finding automatically whether such a system can be scheduled by an affine timing function, independent of the size parameter of the algorithm. In the second part, we describe a powerful method that makes it possible to transform linear recurrences into uniform recurrence equations. Both parts rely on results on integral convex polyhedra. Our results are illustrated on the Gauss elimination algorithm and on the Gauss-Jordan diagonalization algorithm.

Method and apparatus for non-affine image warping

Abstract: A method and apparatus for two-pass image transformation, providing a general solution to execute arbitrary warping of an image. A bicubic mesh is created, by splines or other suitable means, and is used to create displacement tables for X and Y displacement. Alternatively, the displacement tables can be generated directly. The displacement tables represent the movement of each pixel from an original location in the source image to a new location in the destination image. One of the displacement maps is applied to the source image to create an intermediate image and to the other displacement map to create a resampled displacement map. The resampled map is then applied to the intermediate image to create the destination image. By resampling, compensation for altered location points is done automatically. In this manner, no inversion of the underlying equations and functions is required.

The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories

Abstract: This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called A-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of P-trajectories and A-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and A-trajectories. It shows that projective transformations map P-trajectories into P-trajectories. It presents Karmarkar's interpretation of A-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, /x2 restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of A-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that P-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The A-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

An Implementation of a Primal-Dual Interior Point Method for Linear Programming

Abstract: The purpose of this paper is to describe in detail an implementation of a primal-dual interior point method for solving linear programming problems. Preliminary computational results indicate that this implementation compares favorably with a comparable implementation of a dual affine interior point method, and with MINOS 5.0, a state-of-the-art implementation of the simplex method. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A geometric characterization of parametric cubic curves

Abstract: In this paper, we analyze planar parametric cubic curves to determine conditions for loops, cusps, or inflection points. By expressing the curve to be analyzed as a linear combination of control points, it can be transformed such that three of the control points are mapped to specific locations on the plane. We call this image curve the canonical curve. Affine maps do not affect inflection points, cusps, or loops, so the analysis can be applied to the canonical curve instead of the original one. Since the first three points are fixed, the canonical curve is completely characterized by the position of its fourth point. The analysis therefore reduces to observing which region of the canonical plane the fourth point occupies. We demonstrate that for all parametric cubes expressed in this form, the boundaries of these regions are tonics and straight lines. Special cases include Bezier curves, B-splines, and Beta-splines. Such a characterization forms the basis for an easy and efficient solution to this problem.

Incentive strategies and equilibria for dynamic games with delayed information

Abstract: The construction of time-lag incentive strategies for continuous time convex problems is considered. The strategies are affine in the data available and they are represented by means of Stieltjes measures. It is shown how incentive strategies can be used as equilibrium strategies in symmetric games where the decision makers are cooperative.

A survey of applications of an affine invariant norm

Abstract: A survey of applications of an affine invariant norm to several areas of Computer Aided Geometric Design is presented. The main purpose for using an affine invariant norm is to obtain methods and techniques which are not affected by affine transformations of the input data. This means, for example, that the artificial choices of the origin or the units of measurement should have no effect on the final results of the method. Applications in the areas of scattered data interpolation, knot spacing for parametric curve interpolation, and triangulations and tessellations are covered.

From equilibrium spin models to probabilistic cellular automata

Abstract: The general equivalence betweenD-dimensional probabilistic cellular automata (PCA) and (D+1)-dimensional equilibrium spin models satisfying a “disorder condition” is first described in a pedagogical way and then used to analyze the phase diagrams, the critical behavior, and the universality classes of some automata. Diagrammatic representations of time-dependent correlation functions of PCA are introduced. Two important classes of PCA are singled out for which these correlation functions simplify: (1) “Quasi-Hamiltonian” automata, which have a current-carrying steady state, and for which some correlation functions are those of aD-dimensional static model. PCA satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes. (2) “Linear” (and more generally “affine”) PCA for which the diagrammatics reduces to a random walk problem closely related to (D+1)-dimensional directed SAWs: both problems display a critical behavior with mean-field exponents in any dimension. The correlation length and effective velocity of propagation of excitations can be calculated for affine PCA, as is shown on an explicitD=1 example. We conclude with some remarks on nonlinear PCA, for which the diagrammatics is related to reaction-diffusion processes, and which belong in some cases to the universality class of Reggeon field theory.

Graded approximations and controllability along a trajectory

Abstract: A class of approximating nilpotent systems of an affine control system is defined. The systems of this class are used to study the local controllability along a trajectory. >

Implementation of a Dual Affine Interior Point Algorithm for Linear Programming

Abstract: The dual affine interior point method is extended to handle variables with simple upper bounds as well as free variables. During execution, variables which appear to be going to zero are fixed at zero, and rows with slack variables bounded away from zero are removed. A variant of the big-M artificial variable method to attain feasibility is derived. The simplex method is used to recover an optimal basis upon completion of the algorithm, and the effects of scaling are discussed. Computational experience on a variety of problems is presented. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A comparative study of three reconstruction methods for a limited-view computer tomography problem

Abstract: Three novel methods are used to reconstruct a simulated image from a set of incomplete data spanning a 160 degrees angular range. These methods are the squashing affine transformation of J.A. Reeds and L.A. Shepp (1987), the circular interpolation method derived from the theory of J.J. Clark, M.R. Plamer, and P.D. Lawrence (1985), and the geometry-free reconstruction using the theory of convex projections. These methods are briefly explained, and their reconstructions are compared for the case of limited angular views. >

A fast algorithm for computing parametric rational functions

Abstract: An algorithm for fast computation of a class of parametric rational functions whose coefficients are assumed to be affine functions of some underlying parameters is described. The algorithm has been implemented in Microsoft Quick BASIC 4.0. Execution times are minimal. >

Essential orders and the non-linear decoupling problem

Abstract: The concept of essential orders has recently been introduced into linear system theory and has been shown to be useful in the solution of static and dynamic decoupling problems (Commauit et al. 1986, De Luca et al. 1985). This paper presents a non-trivial extension to the class of non-linear affine systems of some of these definitions and results. It is shown that these invariant integers give the minimal order for a regular dynamic compensator that decouples a right-invertible system.

A coordinate-free approach to geometric programming

Abstract: In this paper it is shown that traditional (coordinate-based) approaches to geometric programming lead to programs that are geometrically ambiguous, and potentially geometrically invalid. To combat these deficiencies, a geometric algebra and an associated coordinate-free abstract data type are outlined. The algebra and the abstract data type are founded on two basic principles: affine/Euclidean geometry and coordinate-freedom.

An exact formula for the measure dimensions associated with a class of piecewise linear maps

Abstract: An exact formula for the various measure dimensions of attractors associated with contracting similitudes is given. An example is constructed showing that for more general affine maps the various measure dimensions are not always equal.

Hölder exponents and box dimension for self-affine fractal functions

Abstract: We consider some self-affine fractal functions previously studied by Barnsley et al. The graphs of these functions are invariant under certain affine scalings, and we extend their definition to allow the use of nonlinear scalings. The Holder exponent, h, for these fractal functions is calculated and we show that there is a larger Holder exponent, h λ, defined at almost every point (with respect to Lebesgue measure). For a class of such functions defined using linear affinities these exponents are related to the box dimension D B of the graph by h≤2-D B≤h λ.

On the finite element approximation of functions with noninteger derivatives

Abstract: The paper is devoted to the study of error estimates in the polynomial approxiamations of functions interpolation by affine families of finite elements treated in [4] is extended to functions from fractional Sobolev spaces .

Almost periodic solutions of affine ito equations

Abstract: We discuss the problem of the existence of almost periodic in distribution solutions of affine stochastic differential equations with almost periodic coefficients. We prove that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous L2 -bounded solution of the affine system has the onedimensional distributions almost periodic. An analogous result is shown for the asymptotic almost periodic case

Approximation of Measures by Markov Processes and Homogeneous Affine Iterated Function Systems

Abstract: It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system.

Existence of homoenergetic affine flows for the Boltzmann equation

Abstract: An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.

Affine structure of facially symmetric spaces

Abstract: In [ 7 ], the authors proposed the problem of giving a geometric characterization of those Banach spaces which admit an algebraic structure. Motivated by the geometry imposed by measuring processes on the set of observables of a quantum mechanical system, they introduced the category of facially symmetric spaces . A discrete spectral theorem for an arbitrary element in the dual of a reflexive facially symmetric space was obtained by using the basic notions of orthogonality, protective unit, norm exposed face, symmetric face, generalized tripotent and generalized Peirce projection , which were introduced and developed in this purely geometric setting.

Scheduling a System of Nonsingular Affine Recurrence Equations onto a Processor Array

Abstract: Most work on the problem of scheduling computations onto a systolic array is restricted to systems of uniform recurrence equations. In this paper, this restriction is relaxed to include systems of affine recurrence equations. In this broader class, a sufficient condition is given for the system to be computable. Necessary and sufficient conditions are given for the existence of an affine schedule, along with a procedure that constructs the schedule vector, when one exists.