Showing papers on "Affine transformation published in 1987"

Robust Model Predictive Control

Abstract: Concepts of model predictive control are extended to uncertain linear systems. An on-line optimizing control scheme is developed which has as its objective the minimization of the worst-case tracking error for a family of linear plants. For uncertainty descriptions which provide impulse response models as affine functions of uncertain parameters, it is shown that the required minimax optimization can be recast as a linear program. Situations which lead to such an uncertainty description are discussed. An example is presented to demonstrate the properties of the proposed control scheme.

Three-dimensional model matching from an unconstrained viewpoint

Abstract: It is demonstrated that the affine viewing transformation is a reasonable approximation to perspective. A group of image vertices and edges, called the vertex-pair, which fully determines the affine transformation between a three-dimensional model and a two-dimensional image is defined. A clustering approach, which produces a set of consistent assignments between vertex-pairs in the model and in the image is described. A number of experimental results on outdoor images are presented.

The Affine Permutation Groups of Rank Three

Abstract: (iii) G is an ajfine group, that is, the socle of G is a vector space V, where V s= {ZpY for some prime p and n=p ; moreover, if Go is the stabilizer of the zero vector in V then G = VG0, Go is an irreducible subgroup of GLd(p), and GQ has exactly two orbits on the non-zero vectors of V. The rank 3 groups under (i) are given by the classification of the 2-transitive groups with simple socle (see § 5 of [5]). Those under (ii) are determined in [22] when the socle socG is a classical group, in [3] when socG is an alternating group, and in [27] when soc G is an exceptional group of Lie type or a sporadic group. The object of this paper is to determine completely the groups satisfying (iii), thus completing the classification of the finite primitive groups of rank 3. Our proof uses the classification of finite simple groups. The 2-transitive affine groups have been determined by Hering in [15]. Our methods can easily be used to give a proof of Hering's result, and we present this proof in Appendix 1, at the end of the paper. The finite soluble primitive groups of rank 3 are automatically affine groups, and these are determined by Foulser in [11]. We shall make use of his results and of some of his methods in one special case in our proof, which we call the 'extraspecial case'. Our main result is

Methods and apparatus for image compression by iterated function system

Abstract: A method and apparatus for obtaining highly compressed images employing an iterated function system (IFS). An original input or target image is subdivided into regions having similar characteristics. Contractive copies or maps of a particular region, which are the results of affine transformations to the region, are generated and tiled with respect to the input image until the entire region is covered and a collage is formed. Each region is processed in like manner. The affine transformation coefficients or IFS codes completely represent the input image, and are stored or transmitted. To generate an image from the IFS codes, a decoding system is disclosed. One disclosed method involves a chaotic dynamical system. A random iteration of the IFS codes is performed until an attractor, which is the target image, emerges and stabilizes. Another disclosed deterministic method repeatedly and successively applies the IFS codes to an arbitrary starting image until the attractor emerges. Also disclosed are various methods for representing and compressing the color information of an image, including a method for employing an additional spatial dimension in the mappings and a method for employing an arbitrary probabilistic measure for the color rendering.

Skitters and jacks: interactive 3D positioning tools

Abstract: Let scene composition be the precise placement of shapes relative to each other, using affine transformations. By this definition, the steps of scene composition are the selection of objects to be moved, the choice of transformation, and the specification of the parameters of the transformation. These parameters can be divided into two classes: anchors (such as an axis of rotation) and end conditions (such as a number of degrees to rotate). I discuss the advantages of using Cartesian coordinate frames to describe both kinds of parameters. Coordinate frames used in this way are called jacks. I also describe an interactive technique for placing jacks, using a three-dimensional cursor, called a skitter.

Complete affine locally flat manifolds with a free fundamental group

Abstract: Examples of free noncommutative subgroups of the affine group A(3), which act properly discontinuously on ℝ3, are constructed in the paper These examples refute a conjecture of Milnor to the effect that the fundamental group of any complete affine locally flat manifold contains a solvable subgroup of finite index

Expanders obtained from affine transformations

Abstract: A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.

Wiener measures for path integrals with affine kinematic variables

Abstract: The results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp(−iTH) can be written as a well‐defined Wiener integral, involving Wiener measure on the Lobachevsky half‐plane, in the limit that the diffusion constant diverges. This approach works for a wide class of Hamiltonians, including, e.g., −d2/dx2+V(x) on L2(R+), with V sufficiently singular at x=0.

Joint triangulations and triangulation maps

Abstract: In rubber-sheeting applications in cartography, it is useful to seek piecewise-linear homeomorphisms (PLH maps) between rectangular regions which map an arbitrary sequence of n points {p1, p2, …,pn} from the interior of one rectangle to a corresponding sequence {q1, q2, …, qn} of n points in the interior of the second region. This paper proves that it is always possible to find such PLH maps and describes then in terms of a joint triangulation of the domain and the range rectangular regions.One naive approach to finding a PLH map is to triangulate (in any fashion) the domain rectangle on its n points and four corners and to define a piecewise affine map on each triangle up11p12p13 to be the unique affine map that sends the three vertices p11, p12, p13 of the triangle to the three corresponding vertices q11, q12, q13 of the image triangle uq11q12q13. Such piecewise affine maps send triangles to triangles, agree on shared edges, and thus extend globally, and will be called triangulation maps. The shortcoming of building transformations in this fashion is that the resulting triangulation map need not be one-to-one, although there is a simple test to determine if such a map is one-to-one (see Theorem 2 below). If the map is one-to-one, then the image triangles will form a triangulation of the range space; and we will have a joint triangulation. If the map is not one-to-one, then there will be folding over of triangles. It may be possible to alleviate this folding by choosing a different triangulation of the n domain points, or it may be the case that no triangulation of the n domain points will work. (See figures 5 and 6 below). We show that it will be possible, in all cases, to rectify the folding by adding appropriate additional triangulation vertex pairs {pn+1, pn+2, …, pn+m} and {qn+1, qn+2, …, qn+m} and retriangulating (see Theorem 1 below). This paper examines conditions for triangulation maps to be homeomorphisms and explores different ways of modifying triangulations and triangulation maps to make them joint triangulations and homeomorphisms.The paper concludes with a section on alternative constructive approaches to the open problem of finding joint triangulations on the original sequences of vertex pairs without augmenting those sequences of pairs.The existence proofs in this paper do not solve computational geometry problems per se; instead they permit us to formulate new computational geometry problems. The problems we pose are of interest to us because of a particular application in automated cartography.

Orthogonality and spacetime geometry

Abstract: This book examines the geometrical notion of orthogonality, and shows how to use it as the primitive concept on which to base a metric structure in affine geometry. The subject has a long history, and an extensive literature, but whatever novelty there may be in the study presented here comes from its focus on geometries hav- ing lines that are self-orthogonal, or even singular (orthogonal to all lines). The most significant examples concern four-dimensional special-relativistic spacetime (Minkowskian geometry), and its var- ious sub-geometries, and these will be prominent throughout. But the project is intended as an exercise in the foundations of geome- try that does not presume a knowledge of physics, and so, in order to provide the appropriate intuitive background, an initial chapter has been included that gives a description of the different types of line (timelike, spacelike, lightlike) that occur in spacetime, and the physical meaning of the orthogonality relations that hold between them.The coordinatisation of affine spaces makes use of constructions from projective geometry, including standard results about the ma- trix represent ability of certain projective transformations (involu- tions, polarities). I have tried to make the work sufficiently self- contained that it may be used as the basis for a course at the ad- vanced undergraduate level, assuming only an elementary knowledge of linear and abstract algebra.

Method for producing a geometrical transformation on a video image and devices for carrying out said method

Abstract: Special effects are produced on video images by means of a method of geometrical transformation which can involve a translation, a rotation, an affine transformation or an effect of perspective. The method consists: in locating each point of the initial image by means of coordinates (X, Y, O) in a cartesian reference frame designated as a movable reference frame and related to the initial image; in representing each point of the initial image by a binary word; in storing the initial image in a storage device having two read address inputs and one data output, the function of the device being to deliver a binary word corresponding to the point with coordinates (X, Y, O) in the initial image when read address values X and Y are applied respectively to the two inputs: in causing each point of the transformed image resulting from an initial-image transformation to be located by means of coordinates (X3, Y3, O) in a cartesian reference frame which is related to the transformed image and designated as a fixed reference frame; in representing each point of the transformed image by a binary word M supplied by the storage device by applying values of X and Y to the read address inputs of the device, these values being computed as a function of the coordinates (X3, Y3, O) of said point and as a function of the geometrical transformation parameters.

Piecewise affine functions as a difference of two convex functions

Abstract: We give different representations of piecewise affine linear functions as a difference of convex piecewise affine linear functions.

A two-parameter family of pension contribution functions and stochastic optimization☆

Abstract: In a previous article the author has suggested a linear function of A(t) (present value of future benefits) and F(t) (fund) as pension contribution function in place of the form given in Trowbridge (1963) which is a one-parameter family of funding methods. Here we provide some theoretical justification for such a method by showing that, in the simplified model of this paper, the optimal solution of a stochastic control problem yields, as contribution function, an affine function of A(t) and F(t).

Affine collineations in Robertson–Walker space‐time

Abstract: In a recent paper [M. L. Bedran and B. Lesche, J. Math. Phys. 27, 2360 (1986)], an attempt was made to find an affine collineation in Robertson–Walker space‐time. However, only a homothetic affine collineation was found, in the case of a linear scale factor, R(t)∼t. It is pointed out that another homothetic affine collineation exists when R(t)∼tb, and a proper (nonhomothetic) affine collineation in the Einstein static space‐times has been found.

On discrete chamber-transitive automorphism groups of affine buildings

Abstract: 1. Introduction. Let A be the affine building of a simple adjoint algebraic group Q of relative rank > 2 over a locally compact local field K. Let Aut A (resp. E Aut A) denote the group of type-preserving (resp. of all) auto-morphisms of A. Note that E Aut A contains the group $(K) of ÜT-rational points of §. We will be interested in discrete subgroups of Aut A which are chamber-transitive on A. It is extremely rare that such groups exist and, as can therefore be expected, exceptions are interesting phenomena; our purpose is to list them all (see the theorem below). In order to describe them we must first introduce some notation. Let ƒ be a quadratic form in n variables over Q p with coefficients in Z. We let Pfi(/, Z[l/p]) denote the intersection PSO(/, Q p)'nPGL(n, Z[l/p]) within PGL(n,Q p), and similarly PGO(/, Z[l/p]) = PGO(/,Q p) nPGL(n,Z[l/p]). In the following list, T will always be a chamber-transitive subgroup of Aut A. The fundamental quadratic form (over Z) of the root lattice of type A n , B n , E n , normalized so that the long roots have squared length 2, will be denoted by a n ,6 n ,e n , respectively; note that b n is Yl\" x l-(i) Let ƒ = eg, &7,ci6,66*^6* or as, and let A be the affine building of PSO(/,Q 2). Here T can be any group between r min = Pfi(/,Z[l/2]) and r ma x = PGO(/,Z[l/2]) fi Aut A. The quotient r max /r min is elementary abelian of order 1, 1, 1, 4, 2, or 2, respectively, and r max is generated by Train and reflections. (ii) Let ƒ = &5,e6, or b' e = Xa x ? + ^ x h an(l let A be the building of PSO(/,Q 3). The group r max (/) = PGO(/, Z[l/3]) n Aut A has 3, 5, or 9 conjugacy classes of chamber-transitive subgroups T. Passage mod 2 maps r m ax(b5) onto the symmetric group S5, and the preimages in r max (&5) of S5, A5, or a group of order 20 form the 3 desired conjugacy classes of groups T. The forms e& and b' e are rationally equivalent, and hence the buildings they define over Q3 are the \"same\" ; with suitable identifications of buildings and groups, T b = r max (ee) …

On the linear span of binary sequences obtained from finite geometries

Abstract: A class of periodic binary sequences that are obtained from the incidence vectors of hyperplanes in finite geometries is defined, and a general method to determine their linear spans (the length of the shortest linear recursion over GF(2) satisfied by the sequence) is described. In particular, we show that the projective and affine hyperplane sequences of odd order both have full linear span. Another application involves the parity sequence of order n, which has period pn - 1 and linear span vL(s) where v = (pn - 1)/(p - 1) and L(s) is the linear span of a parity sequence of order 1. The determination of the linear span of the parity sequence of order 1 leads to an interesting open problem involving primes.

Affine processes: a test of isotropy based on level sets

Abstract: The class of affine processes is defined as a reasonable alternative to test the null hypothesis that a given process is isotropic. Estimators for the affinity parameters based only on one level set of the observed process are provided, and they are shown to be consistent under partial independence assumptions. Then asymptotic procedures are proposed to test the isotropy of a stationary process with parameter in ${\bf R}^2 $.

On the controllability of piecewise-linear hypersurface systems

Abstract: This paper investigates the global controllability of piecewise-linear (hypersurface) systems, which are defined as control systems that are subject to affine dynamics on each of the components of a finite polyhedral partition. Various new tools are developed for the study of the problem including a classification of the facets of the polyhedra in the partition. Necessary and sufficient conditions for complete controllability are obtained via the study of a suitably defined controllability connection matrix of polyhedra.