Showing papers on "Affine transformation published in 1990"
TL;DR: In this paper, a new method is presented that generalizes Siegel and Benson's (1982) resistant-fit theta-rho analysis so that more than two objects can be compared at the same time.
Abstract: Superimposition methods for comparing configurations of landmarks in two or more specimens are reviewed. These methods show differences in shape among specimens as residuals after rotation, translation, and scaling them so that they align as well as possible. A new method is presented that generalizes Siegel and Benson's (1982) resistant-fit theta-rho analysis so that more than two objects can be compared at the same time. Both least-squares and resistant-fit approaches are generalized to allow for affine transformations (uniform shape change). The methods are compared, using artificial data and data on 18 landmarks on the wings of 127 species of North American mosquitoes. Graphical techniques .are also presented to help sum- marize the patterns of differences in shape among the objects being compared. (Morphometrics; resistant-fit; least-squares; theta-rho analysis; rotational fit; affine transformations.) An important problem in morphomet- is now easy to display a transformation grid rics is that of comparing configurations of that maps the configuration of landmarks landmarks in two or more specimens. of one organism exactly into those of Thompson (1917) suggested an elegant ap- another. proach, using "transformation grids," that An alternative approach to fitting a mod- depicts the overall form of one organism el that completely describes the differences as a distortion in the shape of a reference between two organisms is to fit a very sim- organism. The basic idea was to place a ple model that only takes into consider- Cartesian coordinate grid over the refer- ation global parameters such as differences ence organism and then distort the image in rotation, translation, and scale. Geo- of the organism (including the grid) in var- metrically, this corresponds to superim- ious ways until the form of the second or- posing one organism on top of another so ganism was achieved. The differences in that its landmarks align as well as possible shapes of the two organisms are shown by (in some sense) with the positions of the the deviations of the fitted grid (usually corresponding landmarks on the second. bent and stretched in various ways) from Differences in shape are then shown by the original simple square grid. Thompson differences in positions of corresponding (1917) sketched the grids subjectively landmarks. Shape differences between two without an explicit specification of which organisms are found by studying these re- landmarks were used. Not all landmarks siduals. These methods are the subject of shown in pairs of drawings are located ex- the present paper. actly where the superimposed grids would Sneath (1967) investigated the problem imply they should be. This means that the of finding the optimal translation, rota- grids should be more complex than those tion, and size change of one object in order shown in Thompson (1917) in order to ac- for it to be superimposed on another. The curately show the differences between two two objects were represented as sets of x,y- organisms. Bookstein (1978) developed the coordinates of landmarks. A least-squares method of biorthogonal grid analysis criterion was used to measure the goodness which quantifies Thompson's approach and of fit of one object to another. Gower (1971) makes it objective. But it is complex and further developed Sneath's (1967) method has not been applied very often. A recent and expressed the operations in terms of b~ak-through (Bookstein, 1989) is the use matrix algebra. Siegel and Benson (1982) of methods based on thin-plate splines. It made the important observation that a
3,621 citations
TL;DR: The method of Fourier descriptors is extended to produce a set of normalized coefficients which are invariant under any affine transformation (translation, rotation, scaling, and shearing) and allows considerable robustness when applied to images of objects which rotate in all three dimensions.
Abstract: The method of Fourier descriptors is extended to produce a set of normalized coefficients which are invariant under any affine transformation (translation, rotation, scaling, and shearing). The method is based on a parameterized boundary description which is transformed to the Fourier domain and normalized there to eliminate dependencies on the affine transformation and on the starting point. Invariance to affine transforms allows considerable robustness when applied to images of objects which rotate in all three dimensions, as is demonstrated by processing silhouettes of aircraft maneuvering in three-space. >
449 citations
01 Nov 1990
TL;DR: Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design and two criteria turn out to be of special interest, the distance to linear structures and the Distance to affine functions, which are shown to be invariant under all affine transformations.
Abstract: Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design. The classification is set up in terms of the largest transformation group leaving a criterion invariant. In this respect two criteria turn out to be of special interest, the distance to linear structures and the distance to affine functions, which are shown to be invariant under all affine transformations. With regard to these criteria an optimum class of functions is considered. These functions simultaneously have maximum distance to affine functions and maximum distance to linear structures, as well as minimum correlation to affine functions. The functions with these properties are proved to coincide with certain functions known in combinatorial theory, where they are called bent functions. They are shown to have practical applications for block ciphers as well as stream ciphers. In particular they give rise to a new solution of the correlation problem.
424 citations
TL;DR: In this paper, conditions under which DEA models are translation invariant are established under which affine displacement does not alter the efficient frontier for models incorporating the convexity constraint, which affords a ready solution to the problems of scaling and the presence of zero values which arise in Data Envelopment Analysis.
379 citations
TL;DR: In this paper, the masses and three-point couplings for all affine Toda theories are calculated and the exact factorisable S-matrices are conjectured on the basis of the classical masses and couplings.
349 citations
TL;DR: The research demonstrated that the determination of structure from motion in actual human observers may be restricted to the use of first order temporal relations, which are available within 2-frame apparent motion sequences.
Abstract: The research described in the present article was designed to identify the minimal conditions for the visual perception of 3-dimensional structure from motion by comparing the theoretical limitations of ideal observers with the perceptual performance of actual human subjects on a variety of psychophysical tasks. The research began with a mathematical analysis, which showed that 2-frame apparent motion sequences are theoretically sufficient to distinguish between rigid and nonrigid motion and to identify structural properties of an object that remain invariant under affine transformations, but that 3 or more distinct frames are theoretically necessary to adequately specify properties of euclidean structure such as the relative 3-dimensional lengths or angles between nonparallel line segments. A series of four experiments was then performed to verify the psychological validity of this analysis. The results demonstrated that the determination of structure from motion in actual human observers may be restricted to the use of first order temporal relations, which are available within 2-frame apparent motion sequences. That is to say, the accuracy of observers’ judgments did not improve in any of these experiments as the number of distinct frames in an apparent motion sequence was increased from 2 to 8, and performance on tasks involving affine structure was of an order of magnitude greater than performance on similar tasks involving euclidean structure.
193 citations
05 Dec 1990
TL;DR: An observer for state-affine systems is constructed and it is shown that it depends on the inputs of the system, and some topological properties are given.
Abstract: The problem of synthesis of observers for nonlinear systems is considered. An observer for state-affine systems is constructed and it is shown that it depends on the inputs of the system. The inputs for which the observer converges are classified, and some topological properties are given. >
141 citations
TL;DR: The familiar Fenchel-Moreau-Rockafellar duality scheme deduced from a conjugation is shown to be applicable to quasi-convex problems and quasi-affine functions take the place of affine functions.
Abstract: The familiar Fenchel-Moreau-Rockafellar duality scheme deduced from a conjugation is shown to be applicable to quasi-convex problems. Here quasi-affine functions take the place of affine functions. The links with other quasi-convex dualities are examined.
132 citations
01 May 1990
TL;DR: The problem of computing a translation that minimizes the Hausdorff distance between two sets of points is considered and the results for the one-dimensional case are applied to the problem of comparing polygons under general affine transformations, where they extend the recent results of Arkin et al on polygon resemblance under rigid body motion.
Abstract: @2. The results for the one-dimensional case are applied to the problem of comparing polygons under general affine transformations, where we extend the recent results of Arkin et al on polygon resemblance under rigid body motion. The two-dimensional case is closely related to the problem of finding an approximate congruence between two points sets under translation in the plane, as considered by Alt et al.
129 citations
TL;DR: In this article, it was shown that the local controllability along a reference trajectory of an affine control system implies the local control along the corresponding reference trajectory for the original system.
Abstract: Graded approximations of an affine control system are defined and their properties are investigated. In particular, it is proved that the local controllability along a reference trajectory of an approximating system implies the local controllability along the corresponding reference trajectory of the original system.Using graded approximations, sufficient conditions of local controllability along a reference trajectory that generalize some known results are given.
122 citations
TL;DR: A projection operator which is not identically zero, but which is guaranteed to vanish on all the proper components of the system f"i=0, which fills the role of a general affine projection operator or variable elimination ''black box'' which can be used for arbitrary polynomial systems.
TL;DR: In this article, the authors study boson representations of affine Kac-Moody algebras and give an explicit description of primary fields and intertwining operators, using vertex operators.
Abstract: We study boson representations of the affine Kac-Moody algebras and give an explicit description of primary fields and intertwining operators, using vertex operators. We establish the resolution of the irreducible module, consisting of boson representations, and point out the connection with Virasoro algebra. All these give new bosonization procedures for Wess-Zumino-Witten (WZW) models and mathematical backgrounds for the integral representation of correlation functions in WZW models on the plane and on the torus.
[...]
TL;DR: In this article, a characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine and all hyperplanes of the fully classified polar spaces are determined.
Abstract: Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of ‘locally polar spaces’. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.
TL;DR: The camera calibration process relates camera system measurements (pixels) to known reference points in a three-dimensional world coordinate system using a linear affine transformation as a map between the rectified camera coordinates and the geometrically projected coordinates on the image plane ofknown reference points.
Abstract: The camera calibration process relates camera system measurements (pixels) to known reference points in a three-dimensional world coordinate system The calibration process is viewed as consisting of two independent phases: the first is removing geometrical camera distortion so that rectangular calibration grids are straightened in the image plane, and the second is using a linear affine transformation as a map between the rectified camera coordinates and the geometrically projected coordinates on the image plane of known reference points Phase one is camera-dependent, and in some systems may be unnecessary Phase two is concerned with a generic model that includes 12 extrinsic variables and up to five intrinsic parameters General methods handling additional constraints on the intrinsic variables in a manner consistent with explicit satisfaction of all six constraints on the orthogonal rotation matrix are presented The use of coplanar and noncoplanar calibration points is described >
Book•
01 Jan 1990
TL;DR: In this article, the Cavalieri principle and other prerequisites for invariance with respect to translations and Euclidean motions of point processes have been discussed, and the Haar measures on groups of affine transformations have been shown to be invariant to translations.
Abstract: Preface 1. Cavalieri principle and other prerequisites 2. Measures invariant with respect to translations 3. Measures invariant with respect to Euclidean motions 4. Haar measures on groups of affine transformations 5. Combinatorial integral geometry 6. Basic integrals 7. Stochastic point processes 8. Palm distributions of point processes 9. Poisson-generated geometrical processes 10. Section through planar geometrical processes References Index.
01 Nov 1990
TL;DR: The two-pass algorithm for affine image transformations to a three-pass affine volume transformation algorithm is extended and the mathematics of the decomposition is derived and the issues involved in resampling the transformed volume are discussed.
Abstract: Affine transformations of volume arrays are important for implementing volume modeling and orthographic viewing transformations and for registering multiple data sets. In this paper we extend the two-pass algorithm for affine image transformations to a three-pass affine volume transformation algorithm. The mathematics of the decomposition is derived and the issues involved in resampling the transformed volume are discussed.
01 Jul 1990
TL;DR: In this paper, the affine arclength measure has been shown to be a better choice for non-degenerate curves in ℝn than for degenerate ones.
Abstract: This article deals with several related questions in harmonic analysis which are well understood for non-degenerate curves in ℝn, but poorly understood in the degenerate case. These questions invariably involve a positive ‘reference’ measure on the curve. In the non-degenerate case the choice of measure is not particularly critical and it is usually taken to be the Euclidean arclength measure. Since the questions considered here are invariant under the group of affine motions (of determinant 1), the correct choice of reference measure is the affine arclength measure. We refer the reader to Guggenheimer [8] for information on affine geometry. When the curve has degeneracies, the choice of measure becomes critical and it is the affine arclength measure which yields the most powerful results. From the Euclidean point of view the affine arclength measure has correspondingly little mass near the degeneracies and thus compensates automatically for the poor behaviour there. This principle should also be valid for general submanifolds of ℝn but alas the affine geometry of submanifolds is itself not well understood in general.
TL;DR: In this paper, the stability robustness of polynomials with coefficients which are affine functions of the parameter perturbations is investigated and a simple and numerically effective procedure, which is based on the Hahn-Banach theorem of convex analysis and which is applicable for any arbitrary norm, is obtained.
03 Apr 1990
TL;DR: It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution, which makes it a versatile tool for the analysis of nonstationary signals.
Abstract: A formalism of signal energy representations depending on time and scale is presented. Precise links between time-frequency and time-scale energy distributions are provided. It is known that a full description of the former is given by Cohen's class, which can be described as a generalization of the spectrogram appropriately parameterized by a smoothing function acting on the Wigner-Ville distribution. A full description of the latter is given, resulting in a class of representations in which the smoothing of the Wigner-Ville distribution is scale-dependent. Through proper choice of the smoothing function, interesting properties may be imposed on the representation, which makes it a versatile tool for the analysis of nonstationary signals. Also, specific choices allow known definitions to be recovered (including the Bertrands' and the energetic version of the wavelet transform, referred to as the scalogram). Another very flexible choice uses separable smoothing functions. It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution. >
TL;DR: The method of steepest descent with scaling with scaling (affine scaling) applied to the potential functionq logc′x — ∑i=1n logxi solves the linear programming problem in polynomial time forq ⩾ n.
Abstract: The method of steepest descent with scaling (affine scaling) applied to the potential functionq logcźx -- źi=1n logxi solves the linear programming problem in polynomial time forq ź n. Ifq = n, then the algorithm terminates in no more than O(n2L) iterations; if q ź n + $$\sqrt n $$ withq = O(n) then it takes no more than O(nL) iterations. A modified algorithm using rank-1 updates for matrix inversions achieves respectively O(n4L) and O(n3.5L) arithmetic computions.
TL;DR: The results show the effects of spatio-temporal discretisation and internal noise on the performance of this algorithm for situations in which the information about the affine structure is contained in only two successive images.
Abstract: We have developed an algorithm to extract local affine motion parallax structure of a varying image irradiance pattern in Part I. In Part II, we present computational results of this algorithm for situations in which the information about the affine structure is contained in only two successive images. This applies to a large class of problems (e.g. two-image motion sequences and stereoscopic vision). The results show the effects of spatio-temporal discretisation and internal noise on the performance of this algorithm.
TL;DR: In this paper, the Lyapunov spectrum of the homogeneous affine delay equation is used to decompose the state space into finite-dimensional and finite-codimensional subspaces.
Abstract: A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.
TL;DR: Error bounds and upper Lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution because of the polyhedral characterization of the solution set as the intersection of the feasible region with a single linear affine inequality constraint.
Abstract: Error bounds and upper Lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution. The existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate. Thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem, the residual that bounds the distance from a solution point consists of the complementarity condition alone, whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it. This and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region {z∣Mz + q ≥ 0, z ≥ 0} with a single linear affine inequality constraint.
TL;DR: In this article, the affine transformations of volume arrays are used for volume modeling and orthographic viewing transformations and for registering multiple data sets, and the authors extend the t-approximation algorithm.
Abstract: Affine transformations of volume arrays are important for implementing volume modeling and orthographic viewing transformations and for registering multiple data sets. In this paper we extend the t...
TL;DR: In this paper, the affine surfaces in R3 which are affine spheres and have flat affine metrics are classified and those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defining by (x2+y2)z=1.
Abstract: Nondegenerate affine surfaces in R3 which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x2+y2)z=1.