This dissertation considers a method for processing two-dimensional (2-D) signals (e.g. imagery) by transformation to a coordinate space where the 2-D operation separates into orthogonal 1-D operations. After processing, the 2-D output is reconstructed by a second coordinate transformation. This approach is based on the Radon transform, which maps a two-dimensional Cartesian representation of a signal into a series of one-dimensional signals by line-integral projection. The mathematical principles of this transformation are well-known as the basis for medical computed tomography. This approach can process signals more rapidly than conventional digital processing and more flexibly and precisely than optical techniques. A new formulation of the Radon transform is introduced that employs a new transformation--the central-slice transform--to symmetrize the operations between the Cartesian and Radon representations of the signal and to aid in analyzing operations that may be susceptible to solution in this manner. It is well-known that 2-D Fourier transforms and convolutions can be performed by 1-D operations after Radon transformation, as proven by the central-slice and filter theorems. Demonstrations of these operations via Radon transforms are described. An optical system has been constructed to derive the line-integral projections of 2-D transmissive or reflective input data. Fourier transforms of the projections are derived by a surface-acoustic-wave chirp Fourier transformer, and filtering is performed in a surface-acoustic-wave convolver. Reconstruction of the processed 2-D signal is performed optically. The system can process 2-D imagery at approximately 5 frames/second, though rates to 30 frames/second are achievable if a faster image rotator is added. Other signal processing operations in Radon space are demonstrated, including Labeyrie stellar speckle interferometry, the Hartley transform, and the joint coordinate-frequency representations such as the Wigner distribution function. Other operations worthy of further study include derivation of the 2-D cepstrum, and several spectrum estimation algorithms.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 47-51).

This thesis presents an algorithm for image reconstruction from projections intended for use in a new class of volume imaging PET scanners. The algorithm is based on the inversion of the three dimensional Radon Transform as it applies to the truncated cylindrical detector geometry and is derived from the X-ray Transform inversion given by the Orlov recovery operator. The algorithm is tested using Monte Carlo simulations of several phantom geometries and employs a single iterative step to include all detected events in the reconstruction. The reconstructed images are good representations of the original objects, however the iterative step is a source of some significant artefacts in the images. Also discussed is the extension of the Radon Transform technique to a non-iterative method for three dimensional image reconstruction using all detected events. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

No description available.

A review of the applications of the Radon transform is presented, with emphasis on emission computed tomography and transmission computed tomography. The theory of the 2D and 3D Radon transforms, and the effects of attenuation for emission computed tomography are presented. The algebraic iterative methods, their importance and limitations are reviewed. Analytic solutions of the 2D problem the convolution and frequency filtering methods based on linear shift invariant theory, and the solution of the circular harmonic decomposition by integral transform theory--are reviewed. The relation between the invisible kernels, the inverse circular harmonic transform, and the consistency conditions are demonstrated. The discussion and review are extended to the 3D problem-convolution, frequency filtering, spherical harmonic transform solutions, and consistency conditions. The Cormack algorithm based on reconstruction with Zernike polynomials is reviewed. An analogous algorithm and set of reconstruction polynomials is developed for the spherical harmonic transform. The relations between the consistency conditions, boundary conditions and orthogonal basis functions for the 2D projection harmonics are delineated and extended to the 3D case. The equivalence of the inverse circular harmonic transform, the inverse Radon transform, and the inverse Cormack transform is presented. The use of the number of nodes of a projection harmonic as a filter is discussed. Numerical methods for the efficient implementation of angular harmonic algorithms based on orthogonal functions and stable recursion are presented. The derivation of a lower bound for the signal-to-noise ratio of the Cormack algorithm is derived.

Thesis (doctoral)--Université de Rennes, 2004. / Includes bibliographical references (p. [131]-[136]) . Also available on the World Wide Web.

Chun, Chun Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 73-75). / Abstracts in English and Chinese. / ABSTRACT --- p.i / 摘要 --- p.ii / ACKNOWLEDGEMENTS --- p.iii / TABLE OF CONTENTS --- p.iv / LIST OF FIGURES --- p.v / LIST OF TABLES --- p.vii / Chapter 1 --- INTRODUCTION --- p.01 / Chapter 1.1 --- Iris Texture as A Biometric Password --- p.01 / Chapter 1.1.1 --- Advantages --- p.02 / Chapter 1.1.2 --- Previous Work --- p.04 / Chapter 1.1.3 --- Operation Procedures --- p.06 / Chapter 1.1.3.1 --- Image Acquisition --- p.07 / Chapter 1.1.3.2 --- Iris Localization --- p.07 / Chapter 1.1.3.3 --- Image Encoding and Database Matching --- p.08 / Chapter 1.2 --- Motivation and Research Objective --- p.09 / Chapter 1.3 --- Thesis Outline --- p.11 / Chapter 2 --- IMAGE ACQUISITION --- p.13 / Chapter 2.1 --- Difficulties on Image Acquisition --- p.13 / Chapter 2.2 --- Our Iris Image Acquisition Setting --- p.13 / Chapter 3 --- PRE-PROCESSING --- p.15 / Chapter 3.1 --- Isolating the Region of Interest --- p.15 / Chapter 3.1.1 --- Conscious and Unconscious Recognition --- p.15 / Chapter 3.1.2 --- Iris Boundary Detection --- p.16 / Chapter 3.2 --- Iris-ring Unfolding and Normalization --- p.19 / Chapter 3.2.1 --- Eccentric-polar Coordinate System --- p.20 / Chapter 3.2.2 --- Iris-ring Unfolding --- p.22 / Chapter 3.2.3 --- Normalization --- p.22 / Chapter 3.3 --- Data Binarization --- p.24 / Chapter 4 --- RADON TRANSFORM BASED ENCODING AND MATCHING --- p.27 / Chapter 4.1 --- Radon Transform based Encoding --- p.28 / Chapter 4.2 --- Iris Code Matching --- p.32 / Chapter 4.2.1 --- Regional Correlation --- p.32 / Chapter 5 --- PALM-TOP IMPLEMENTATION ON COMPUTING PLATFORM --- p.36 / Chapter 5.1 --- Image Acquisition --- p.37 / Chapter 5.1.1 --- Desktop Version --- p.37 / Chapter 5.1.2 --- Palm-top Version --- p.37 / Chapter 5.2 --- Iris Localization --- p.39 / Chapter 5.2.1 --- Desktop Version --- p.39 / Chapter 5.2.2 --- Palm-top Version --- p.39 / Chapter 5.3 --- Image Encoding --- p.41 / Chapter 5.3.1 --- Desktop Version --- p.41 / Chapter 5.3.2 --- Palm-top Version --- p.41 / Chapter 5.4 --- Palm-Top Computer Application --- p.42 / Chapter 5.4.1 --- Palm-top Computer Setting --- p.42 / Chapter 5.4.2 --- Software Selection --- p.42 / Chapter 5.4.3 --- Technical Problems --- p.43 / Chapter 5.4.3.1 --- Problem 1: Memory Limitation --- p.43 / Chapter 5.4.3.2 --- Problem 2: Image Format --- p.44 / Chapter 5.4.3.3 --- Problem 3: Origin of Image --- p.44 / Chapter 5.5 --- Our Iris Recognition Platform --- p.44 / Chapter 6 --- EXPERIMENTAL RESULTS --- p.47 / Chapter 6.1 --- The Test Data --- p.47 / Chapter 6.2 --- Experiment One: Eccentric Polar Coordinates System Recognition Performance --- p.48 / Chapter 6.2.1 --- Performance Measure of Recognition --- p.48 / Chapter 6.2.2 --- Experimental Result --- p.49 / Chapter 6.3 --- Experiment Two: Radon Transform-based Recognition System Performance --- p.53 / Chapter 6.3.1 --- Intra-group Similarity vs. Inter-group Similarity --- p.54 / Chapter 6.3.2 --- Performance Comparison with an Existing System --- p.57 / Chapter 6.4 --- Experiment Three: The Resolution of Image in the Eccentric-polar Coordinates System --- p.58 / Chapter 7 --- CONCLUSION AND FUTURE WORK --- p.62 / Chapter 7.1 --- Conclusion --- p.62 / Chapter 7.2 --- Future Work --- p.63 / APPENDIX A --- p.66 / APPENDIX B --- p.67 / BIBLIOGRAPHY --- p.73

Families of linear functionals on a vector space that are mapped to each other by a group of symmetries of the space have a significant amount of structure. This results in computational redundancies which can be used to make computing the entire family of functionals at once more efficient than applying each in turn. This thesis explores asymptotic complexity results for a few such families: contingency tables and unranked choice data. These are used to explore the framework of Radon transform diagrams, which promise to allow general theorems about linear summary statistics to be stated and proved.

Thesis (M.S.)--University of Texas at El Paso, 2009. / Title from title screen. Vita. CD-ROM. Includes bibliographical references. Also available online.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 98-99). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.