Efficient neighbourhood computing for discrete rigid transformation graph search

International audienceRigid transformations are involved in a wide variety of image processing applications, including image registration. In this context, we recently proposed to deal with the associated optimization problem from a purely discrete point of view, using the notion of discrete rigid transformation (DRT) graph. In particular, a local search scheme within the DRT graph to compute a locally optimal solution without any numerical approximation was formerly proposed. In this article, we extend this study, with the purpose to reduce the algorithmic complexity of the proposed optimization scheme. To this end, we propose a novel algorithmic framework for just-in-time computation of sub-graphs of interest within the DRT graph. Experimental results illustrate the potential usefulness of our approach for image registration

Continuum quantum systems as limits of discrete quantum systems. IV. Affine canonical transforms

Cataloged from PDF version of article.Affine canonical transforms, complex-order Fourier transforms, and their associated coherent states appear in two scenarios: finite-discrete and continuum. We examine the relationship between the two scenarios, making systematic use of inductive limits, which were developed in the preceding articles in this series. © 2003 American Institute of Physics

Harmonic analysis in finite phase space

Cataloged from PDF version of article.The Wigner distribution and linear canonical transforms are important tools for optics, signal processing, quantum mechanics, and mathematics. In this thesis, we study the discrete versions of Wigner distributions and linear canonical transforms. In the definition of a discrete entity we focus on two aspects: structural analogy and continuum approximation and/or limits. Based on this framework, the tradeoffs are analyzed and a compromise for a discrete Wigner distribution that meets both objectives to a high degree is presented by consolidating sampling theory and the algebraic approach. Such a compromise is necessary since it is impossible to meet the conditions to the highest possible degree. The differences between discrete and continuous time-frequency analysis are also discussed in a group theoretical perspective. In the second part of the thesis, the discrete versions of linear canonical transforms are reviewed and their connections to the continuous theory is established. As a special case the discrete fractional Fourier transform is defined and its properties are derived.Korkmaz, SayitM.S

The discrete fractional Fourier transform

Ankara : Department of Electrical and Electronic Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1998.Thesis (Master's) -- Bilkent University, 1998.Includes bibliographical references leaves 92-96.In this work, the discrete counterpart of the continuous Fractional Fourier Transform (FrFT) is proposed, discussed and consolidated. The discrete transform generalizes the Discrete Fourier Transform (DFT) to arbitrary orders, in the same sense that the continuous FrFT generalizes the continuous time Fourier Transform. The definition proposed satisfies the requirements of unitarity, additivity of the orders and reduction to DFT. The definition proposed tends to the continuous transform as the dimension of the discrete transform matrix increases and provides a good approximation to the continuous FrFT for the finite dimensional matrices. Simulation results and some properties of the discrete FrFT are also discussed.Candan, ÇağatayM.S

Rigid transformations on 2D digital images : combinatorial and topological analysis

In this thesis, we study rigid transformations in the context of computer imagery. In particular, we develop a fully discrete framework for handling such transformations. Rigid transformations, initially defined in the continuous domain, are involved in a wide range of digital image processing applications. In this context, the induced digital rigid transformations present different geometrical and topological properties with respect to their continuous analogues. In order to overcome the issues raised by these differences, we propose to formulate rigid transformations on digital images in a fully discrete framework. In this framework, Euclidean rigid transformations producing the same digital rigid transformation are put in the same equivalence class. Moreover, the relationship between these classes can be modeled as a graph structure. We prove that this graph has a polynomial space complexity with respect to the size of the considered image, and presents useful structural properties. In particular, it allows us to generate incrementally all digital rigid transformations without numerical approximation. This structure constitutes a theoretical tool to investigate the relationships between geometry and topology in the context of digital images. It is also interesting from the methodological point of view, as we illustrate by its use for assessing the topological behavior of images under rigid transformationsDans cette thèse, nous étudions les transformations rigides dans le contexte de l'imagerie numérique. En particulier, nous développons un cadre purement discret pour traiter ces transformations. Les transformations rigides, initialement définies dans le domaine continu, sont impliquées dans de nombreuses applications de traitement d'images numériques. Dans ce contexte, les transformations rigides digitales induites présentent des propriétés géométriques et topologiques différentes par rapport à leurs analogues continues. Afin de s'affranchir des problèmes inhérents à ces différences, nous proposons de formuler ces transformations rigides dans un cadre purement discret. Dans ce cadre, les transformations rigides sont regroupées en classes correspondant chacune à une transformation digitale donnée. De plus, les relations entre ces classes de transformations peuvent être modélisées par une structure de graphe. Nous prouvons que ce graphe présente une complexité spatiale polynômiale par rapport à la taille de l'image. Il présente également des propriétés structurelles intéressantes. En particulier, il permet de générer de manière progressive toute transformation rigide digitale, et ce sans approximation numérique. Cette structure constitue un outil théorique pour l'étude des relations entre la géométrie et la topologie dans le contexte de l'imagerie numérique. Elle présente aussi un intérêt méthodologique, comme l'illustre son utilisation pour l'évaluation du comportement topologique des images sous des transformations rigide

Understanding Discrete Rotations

The concept of rotations in continuous-time, continuousfrequency is extended to discrete-time, discrete-frequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a prime-length signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory. 1. INTRODUCTION The Wigner distribution satisfies many desirable properties, among them bei..