A Convolutional Forward and Back-Projection Model for Fan-Beam Geometry Kai Zhang, Student Member, IEEE, Alireza Entezari, Senior Member, IEEE
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A Box Spline Calculus for Computed Tomography
A BOX SPLINE CALCULUS FOR COMPUTED TOMOGRAPHY Alireza Entezari1, Michael Unser2 1: CISE Department, University of Florida, Gainesville, FL 32611-6120, USA 2: Biomedical Imaging Group, EPFL, CH-1015, Lausanne, Switzerland ABSTRACT cess on them. For a parallel projection geometry, the suitable model is the Radon transform (or the X-ray transform in 3- B-splines are attractive basis functions for the continuous- D). Current reconstruction algorithms typically use the natu- domain representation of biomedical images and volumes. In ral (square) pixel basis or some blob (or Kaiser-Bessel) func- this paper, we prove that the extended family of box splines tions which have the advantage of being circularly symmetric. are closed under the Radon transform and derive explicit for- While these particular choices simplify the derivation of the mulae for their transforms. Our results are general; they cover forward model, they are not as favorable from an approxima- all known brands of compactly-supported box splines (tensor- tion theoretic point of view. Both bases have—at best—a first product B-splines, separable or not) in any dimensions. In order of approximation which implies that the rate of decay particular, we prove that the 2-D Radon transform of an N- of the discretization error as the reconstruction grid gets finer direction box spline is generally a (non-uniform) polynomial is relatively slow [2, 3]. In principle, choosing higher-order spline of degree N − 1. The proposed framework allows for a B-spline basis functions would be more advantageous since proper discretization of a variety of 2-D and 3-D tomographic these are optimal in the sense of their support being minimal reconstruction problems in a box spline basis. -
Efficient Sampling Geometries and Reconstruction Algorithms for Estimation of Diffusion Propagators
EFFICIENT SAMPLING GEOMETRIES AND RECONSTRUCTION ALGORITHMS FOR ESTIMATION OF DIFFUSION PROPAGATORS By WENXING YE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 c 2012 Wenxing Ye 2 To my parents 3 ACKNOWLEDGMENTS I owe a great thank you to many people who helped me finish my dissertation. First and foremost, I would like to thank Dr. Baba Vemuri, my advisor, who introduced me into the field of medical image processing and guided me through my Ph.D. study using his tireless inspiration and rich academic experience. I am also thankful that Dr. Vemuri offered me full research assistantship with which I could put all my focus on the research and not worry about financial problems. Dr. Vemuri built the example of a passionate and serious scholar for me. I would also like to appreciate Dr. Alireza Entezari, my co-advisor, who is always available for discussion when I need his technical advise. He selflessly shared his experience and knowledge so that I avoided making many mistakes. From him, I learned the importance of mathematical thinking and carefulness in solving a research problem. The memories are still clear that he worked together with me from morning to the night before the conference deadlines. It is my luck to have him as a patient friend and knowledgeable mentor. Furthermore, I would like to express my sincere appreciation to Dr. Anand Rangarajan, Dr. Jay Gopalakrishnan and Dr. Stephen Blackband for serving as my Ph.D. -
A Box Spline Calculus for Computed Tomography
A BOX SPLINE CALCULUS FOR COMPUTED TOMOGRAPHY Alireza Entezari1, Michael Unser2 1: CISE Department, University of Florida, Gainesville, FL 32611-6120, USA 2: Biomedical Imaging Group, EPFL, CH-1015, Lausanne, Switzerland ABSTRACT cess on them. For a parallel projection geometry, the suitable model is the Radon transform (or the X-ray transform in 3- B-splines are attractive basis functions for the continuous- D). Current reconstruction algorithms typically use the natu- domain representation of biomedical images and volumes. In ral (square) pixel basis or some blob (or Kaiser-Bessel) func- this paper, we prove that the extended family of box splines tions which have the advantage of being circularly symmetric. are closed under the Radon transform and derive explicit for- While these particular choices simplify the derivation of the mulae for their transforms. Our results are general; they cover forward model, they are not as favorable from an approxima- all known brands of compactly-supported box splines (tensor- tion theoretic point of view. Both bases have—at best—a first product B-splines, separable or not) in any dimensions. In order of approximation which implies that the rate of decay particular, we prove that the 2-D Radon transform of an N- of the discretization error as the reconstruction grid gets finer direction box spline is generally a (non-uniform) polynomial is relatively slow [2, 3]. In principle, choosing higher-order spline of degree N − 1. The proposed framework allows for a B-spline basis functions would be more advantageous since proper discretization of a variety of 2-D and 3-D tomographic these are optimal in the sense of their support being minimal reconstruction problems in a box spline basis. -
Data Processing on the Body-Centered Cubic Lattice
DATA PROCESSING ON THE BODY-CENTERED CUBIC LATTICE by Usman Raza Alim M.Sc., Rochester Institute of Technology, 2007 B.Sc., University of Rochester, 2001 a Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computing Science Faculty of Applied Sciences c Usman Raza Alim 2012 SIMON FRASER UNIVERSITY Summer 2012 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced without authorization under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Usman Raza Alim Degree: Doctor of Philosophy Title of Thesis: Data Processing on the Body-Centered Cubic Lattice Examining Committee: Dr. Hao (Richard) Zhang Chair Dr. Torsten M¨oller, Senior Supervisor Professor of Computing Science Dr. Steve Ruuth, Supervisor Professor (Applied and Computational Mathematics) Dr. Ghassan Hamarneh, Internal Examiner Associate Professor of Computing Science Dr. Michael Unser, External Examiner Professor of Image Processing Ecole´ Polytechnique F´ed´erale de Lausanne (EPFL) Switzerland Date Approved: ii Partial Copyright Licence iii Abstract The body-centered cubic (BCC) lattice is the optimal three-dimensional sampling lattice. Its optimality stems from the fact that its dual, the face-centered cubic (FCC) lattice, achieves the highest sphere-packing efficiency. In order to approximate a scalar-valued function from samples that reside on a BCC lattice, spline-like compact kernels have been recently proposed. The lattice translates of an admissible BCC kernel form a shift-invariant approximation space that yields higher quality approximations as compared to similar spline- like spaces associated with the ubiquitous Cartesian cubic (CC) lattice. -
A Box Spline Calculus for the Discretization of Computed
1532 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 8, AUGUST 2012 A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems Alireza Entezari*, Member, IEEE, Masih Nilchian, Student Member, IEEE, and Michael Unser, Fellow, IEEE Abstract—B-splines are attractive basis functions for the contin- gebraic reconstruction (ART) or the statistical OSEM method, uous-domain representation of biomedical images and volumes. which offer greater flexibility. In this paper, we prove that the extended family of box splines are A crucial step in the design of iterative reconstruction algo- closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known rithms is the discretization of the forward model. This is usu- brands of compactly-supported box splines (tensor-product ally achieved by selecting image-domain basis functions and B-splines, separable or not) in any number of dimensions. The by mathematically simulating the acquisition process on them. proposed box spline approach extends to non-Cartesian lattices For a parallel projection geometry, the suitable model is the used for discretizing the image space. In particular, we prove Radon transform (or the X-ray transform in 3-D). Current re- that the 2-D Radon transform of an -direction box spline is generally a (nonuniform) polynomial spline of degree . construction algorithms typically use the natural (square) pixel The proposed framework allows for a proper discretization of a basis or some blob (or Kaiser–Bessel) functions which have the variety of tomographic reconstruction problems in a box spline advantage of being circularly symmetric.