Showing papers on "Affine transformation published in 1992"

Hierarchical Model-Based Motion Estimation

Abstract: This paper describes a hierarchical estimation framework for the computation of diverse representations of motion information. The key features of the resulting framework (or family of algorithms) are a global model that constrains the overall structure of the motion estimated, a local model that is used in the estimation process, and a coarse-fine refinement strategy. Four specific motion models: affine flow, planar surface flow, rigid body motion, and general optical flow, are described along with their application to specific examples.

Some efficient solutions to the affine scheduling problem: I. One-dimensional time

Abstract: Programs and systems of recurrence equations may be represented as sets of actions which are to be executed subject to precedence constraints. In may cases, actions may be labelled by integral vectors in some iterations domains, and precedence constraints may be described by affine relations. A schedule for such a program is a function which assigns an execution data to each action. Knowledge of such a schedule allows one to estimate the intrinsic degree of parallelism of the program and to compile a parallel version for multiprocessor architectures or systolic arrays. This paper deals with the problem of finding closed form schedules as affine or piecewise affine functions of the iteration vector. An algorithm is presented which reduces the scheduling problem to a parametric linear program of small size, which can be readily solved by an efficient algorithm.

Some efficient solutions to the affine scheduling problem. Part II. Multidimensional time

Abstract: This paper extends the algorithms which were developed in Part I to cases in which there is no affine schedule, i.e. to problems whose parallel complexity is polynomial but not linear. The natural generalization is to multidimensional schedules with lexicographic ordering as temporal succession. Multidimensional affine schedules, are, in a sense, equivalent to polynomial schedules, and are much easier to handle automatically. Furthermore, there is a strong connection between multidimensional schedules and loop nests, which allows one to prove that a static control program always has a multidimensional schedule. Roughly, a larger dimension indicates less parallelism. In the algorithm which is presented here, this dimension is computed dynamically, and is just sufficient for scheduling the source program. The algorithm lends itself to a “divide and conquer” strategy. The paper gives some experimental evidence for the applicability, performances and limitations of the algorithm.

A three-frame algorithm for estimating two-component image motion

Abstract: A fundamental assumption made in formulating optical-flow algorithms, that motion at any point in an image can be represented as a single pattern component undergoing a simple translation, fails for a number of situations that commonly occur in real-world images. An alternative formulation of the local motion assumption in which there may be two distinct patterns undergoing coherent (e.g. affine) motion within a given local analysis region is proposed. An algorithm for the analysis of two-component motion in which tracking and nulling mechanisms applied to three consecutive image frames separate and estimate the individual components is given. Precise results are obtained, even for components that differ only slightly in velocity as well as for a faint component in the presence of a dominant, masking component. The algorithm provides precise motion estimates for a set of elementary two-motion configurations and is robust in the presence of noise. >

Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem

Abstract: Consider the affine variational inequality problem. It is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point. This bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem. This latter result improves upon a recent result of Luo and Tseng that further assumes the problem to be monotone.

Online minimization of transition systems (extended abstract)

Abstract: We are given a transition system implicitly through a compact representation and wish to perform simultaneously reachability analysis and minimization without constructing first the whole system graph. We present an algorithm for this problem that applies to general systems, provided we have appropriate primitive operations for manipulating blocks of states and we can determine termination; the number of operations needed to construct the minimal reachable graph is quadratic in the size of this graph. We specialize the method to obtain efficient algorithms for extended finite state machines that apply separable affine transformations on the variables.

A class of affine Wigner functions with extended covariance properties

Abstract: Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three‐parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.

Object recognition based on moment (or algebraic) invariants

Abstract: Toward the development of an object recognition and positioning system, able to deal with arbitrary shaped objects in cluttered environments, we introduce methods for checking the match of two arbitrary curves in 2D or surfaces in 3D, when each of these subobjects (i.e., regions) is in arbitrary position, and we also show how to efficiently compute explicit expressions for the coordinate transformation which makes two matching subobjects (i.e., regions) coincide. This is to be used for comparing an arbitrarily positioned subobject of sensed data with objects in a data base, where each stored object is described in some “standard” position. In both cases, matching and positioning, results are invariant with respect to viewer coordinate system, i.e., invariant to the arbitrary location and orientation of the object in the data set, or, more generally, to affine transformations of the objects in the data set, which means translation, rotation, and different stretchings in two (or three) directions, and these techniques apply to both 2D and 3D problems. The 3D Euclidean case is useful for the recognition and positioning of solid objects from range data, and the 2D affine case for the recognition and positioning of solid objects from projections, e.g., from curves in a single image, and in motion estimation. The matching of arbitrarily shaped regions is done by computing for each region a vector of centered moments. These vectors are viewpointdependent, but the dependence on the viewpoint is algebraic and well known. We then compute moment invariants, i.e., algebraic functions of the moments that are invariant to Euclidean or affine transformations of the data set. We present a new family of computationally efficient algorithms, based on matrix computations, for the evaluation of both Euclidean and affine algebraic moment invariants of data sets. The use of moment invariants greatly reduces the computation required for the matching, and hence initial object recognition. The approach to determining and computing these moment invariants is different than those used by the vision community previously. The method for computing the coordinate transformation which makes the two matching regions coincide provides an estimate of object position. The estimation of the matching transformation is based on the same matrix computation techniques introduced for the computation of invariants, it involves simple manipulations of the moment vectors, it neither requires costly iterative methods, nor going back to the data set. The use of geometric invariants in this application is equivalent to specifying a center and an orientation for an arbitrary data constellation in a region. These geometric invariant methods appear to be very important for dealing with the situation of a large number of different possible objects in the presence of occlusion and clutter. As we point out in this paper, each moment invariant also defines an algebraic invariant, i.e., an invariant algebraic function of the coefficients of the best fitting polynomial to the data. Hence, this paper also introduces a new design and computation approach to algebraic invariants.

Method for 2-D affine transformation of images

Abstract: Affine image transformations are performed in an interleaved manner, whereby coordinate transformations and intensity calculations are alternately performed incrementally on small portions of an image. The pixels are processed in rows such that after coordinates of a first pixel are determined for reference, each pixel in a row, and then pixels in vertically adjacent rows, are processed relative to the coordinates of the previously processed adjacent pixels. After coordinate transformation to produce affine translation, rotation, skew, and/or scaling, intermediate metapixels are vertically split and shifted to eliminate holes and overlaps. Intensity values of output metapixels are calculated as being proportional to the sum of scaled portions of the intermediate metapixels which cover the output pixels respectively.

The dimension of self-affine fractals II

Abstract: A family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

Feature predictive vector quantization of multispectral images

Abstract: A compression method for multispectral data sets is proposed where a small subset of image bands is initially vector quantized. The remaining bands are predicted from the quantized images. Two different types of predictors are examined, an affine predictor and a new nonlinear predictor. The residual (error) images are encoded at a second stage based on the magnitude of the errors. This scheme simultaneously exploits both spatial and spectral correlation inherent in multispectral images. Simulation results on an image set from the Thematic Mapper with seven spectral bands provide a comparison of the affine predictor with the nonlinear predictor. It is shown that the nonlinear predictor provides significantly improved performance compared to the affine predictor. Image compression ratios between 15 and 25 are achieved with remarkably good image quality. >

Exact S-Matrices for Nonsimply-Laced Affine Toda Theories

Abstract: We derive exact, factorized, purely elastic scattering matrices for affine Toda theories based on the nonsimply-laced Lie algebras and superalgebras.

Partitioning tree image representation and generation from 3D geometric models

Abstract: While almost all research on image representation has assumed an underlying discrete space, the most common sources of images have the structure of the continuum. Although employing discrete space representations leads to simple algorithms, among its costs are quantization errors, significant verbosity and lack of structural information. A neglected alternative is the use of continuous space representations. In this paper we discuss one such representation and algorithms for its generation from views of 3D continuous space geometric models. For this we use binary, space partitioning trees for representing both the model and the image. Our approach falls under the general rubric of visible surface algorithms, providing an objectspace algorithm which under certain conditions requires only sub-linear time for a partitioning tree represented model, and in general exploits occlusion so that the computational cost converges toward the complexity of the image as the depth complexity increases. Visible edges can also be generated as a step following visible surface determination. However, an important contextual difference is that the resulting image trees are used in subsequent continuous space operations. These include affine transformations, set operations, and metric calculations, which can be used to provide image compositing, incremental image modification in a sequence of frames, and facilitating matching for computer vision/robotics. Image trees can also be used with the hemicube and light buffer illumination methods as a replacement for regular grids, thereby providing exact rather than approximate visibility. Discrete vs. Continuous Space We have come to think of images as synonymous with a 2D array of pixels. However, this is an artifact of the transducers we use to convert between the physical domain and the informational domain. Physical space at the resolution with which we are concerned is most effectively modeled mathematically as being continuous; that is, as having the structure of the Reals (or at least the Rationals) as opposed to the structure of the Integers. Modeling space as being defined on a regular lattice, while simple, is verbose and induces quantization which reduces accuracy and can introduce visible artifacts. Using nothing other than a lattice for the representation provides no image dependent structure such as edges. Consider applying to a discrete image an affine transformation, an elementary spatial operation. The solution for this is developed by reasoning not merely in discrete space but in the continuous domain as well: samples are used to reconstruct a "virtual" continuous function which is then resampled. However, the quantization effects can become rather apparent should the transform entail a significant increase in size and a rotation by some small angle, despite the use of high quality filters. This is due to such factors as ringing, blurring, aliasing, and anisotropic effects which cannot all be simultaneously minimized (see, for example, [Mitchell and Netravali 88]). More importantly, discontinuities become increasingly smeared as one increases the size, since the convolution assumes a band-limited signal, i.e. an image with no edges. This has practical implications when texture mapping is used to define the color of surfaces in 3D: since a texture map can be enlarged arbitrarily, a brick texture, for example, will become diffuse instead of exhibiting distinctly separate bricks. Now consider applying affine transformations to images represented by quadtrees, a spatial structure, developed within the context of a finite discrete space, for reducing verbosity and inducing structure on an image. The algorithm for constructing the new quadtree of the transformed image seems relatively complicated when compared to the corresponding algorithms for continuous space representations: it must resample each transformed leaf node and construct an entirely new tree. In contrast, boundary representations, simplical decompositions, or binary space partitioning trees only require transforming points and/or

Almost periodic solutions of affine stochastic evolution equations

Abstract: In this paper the problem of the existence of almost periodic in distribution solutions of almost periodic affine Ito equations of evolution type is studied. Under the hypotheses that the linear part of the equation is exponentially stable in mean square and the relative compactness of the one-dimensional distributions of the unique bounded solution of the equation, it is proved the almost periodicity of the one-dimensional distributions of the solution

Global Convergence Property of the Affine Scaling Methods for Primal Degenerate Linear Programming Problems

Abstract: In this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy. The behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous LP problems and Karmarkar's method. It is shown that the step-size 1/8, where the displacement vector is normalized with respect to the distance in the scaled space, is sufficient to guarantee the global convergence for dual nondegenerate LP problems. The result can be regarded as a counterpart to Dikin's global convergence result on the affine scaling method assuming primal nondegeneracy.

New variants of the POCS method using affine subspaces of finite codimension with applications to irregular sampling

Abstract: The POCS-method (projection onto convex subsets) has been proposed as an efficient way of recovering a band-limited signal from irregular sampling values. However, both the ordinary POCS-method (which uses one sampling point at a given time, i.e. consists of a succession of projections onto affine hyperplanes) and the one-step method (which uses all sampling values at the same time) become extremely slow if the number of sampling points gets large. Already for midsize 2D-problems (e.g. 128 X 128 images) one may easy run into memory problems. Based on the theory of pseudo-inverse matrices new efficient variants of the POCS- method (so to say intermediate versions) are described, which make use of a finite number of sampling points at each step. Depending on the computational environment appropriate strategies of designing those families of sampling points (either many families with few points, or few families with many points, overlapping families or disjoint ones...) have to be found. We also report on numerical results for these algorithms.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Robust face identification scheme: KL expansion of an invariant feature space

Abstract: This paper proposes a new approach for extracting features from face images that offer robust face identification against image variations. We combine the K-L expansion technique with two new operations that transform the face pattern into an invariant feature space. The two operations are the affine transformation which yields a standard face view from the input face image, and its transformation into the Fourier spectrum domain, which develops the property of shift-invariance. Although the basic idea of applying the K-L expansion to extract features for face recognition originates from the eigenface approach proposed by Turk and Pentland our scheme offers superior performance due to the transformation into the invariant feature space. The performance of the two schemes for face identification against various imaging conditions is compared.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Second-order optic flow

Abstract: We investigate the structure of the second-order spatial differential structure of optic flow for the case of a rigid surface in parallel projection. We disregard such problems as that of noise and that of how to obtain optic flow from image flow in the first place. The focus of the investigation is on the essential geometric structures. It is shown that the second-order structure is conveniently represented as a tensor that allows one to obtain the second-order directional derivative of the flow for arbitrary directions in the field of view. This second-order directional derivative in an arbitrary direction turns out to be a vector in the epipolar direction with a magnitude that is proportional to the normal curvature of the surface in the direction of derivation. Our analysis involves only affine concepts and constructions; metric concepts do not enter into the analysis. The second-order structure immediately reveals the projection of Dupin’s indicatrix, which captures important affine aspects of shape. When the second-order structure is combined with the first-order differential structure and after the introduction of a metric, only one degree of ambiguity is left: The rate of turn about an axis in the frontoparallel plane can be traded against the height of relief. Finally we construct hypothetical neural implementations for the second-order flow analysis. A simple, center–surround organization suffices, although one may not pool over orientations.

Shape from periodic texture using the spectrogram

Abstract: It is shown how local spatial image frequency is related to the surface normal of a textured surface. It is found that the Fourier power spectra of any two similarly textured patches on a plane are approximately related to each other by an affine transformation. The transformation parameters are a function of the plane's surface normal. This relationship is used as the basis of an algorithm for finding surface normals of textured shapes using the spectrogram, which is one type of local spatial frequency representation. The relationship is validated by testing the algorithm on real textures. By analyzing shape and texture in terms of the local spatial frequency representation, the advantages of the representation for the shape-from-texture problem can be exploited. Specifically, the algorithm requires no feature detection and can give correct results even when the texture is aliased. >

Inhomogeneous quantum groups

Abstract: Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsU q (N), SO q (N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.

Quantum affine symmetry in vertex models

Abstract: We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.

Decoding geometric Goppa codes using an extra place

Abstract: Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1/2(d* − 1) − s errors, where d* is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa codes, namely C Ω (D, mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n 3 ), where n is the length of codewords