Elementos euclides1

Radon Transform in Tomographic Image Reconstruction

Ditte Iben Marcussen Camilla H. Trinderup

Kongens Lyngby 2009 IMM-B.Sc.-2009-1

Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 reception@imm.dtu.dk www.imm.dtu.dk

IMM-B.Sc.: ISSN 0909-3192

Abstract

There are differentmethods for approaching the inverse problem of reconstructing a tomographic image from its sinogram. The filtered backprojection, FBP, is one of them and is a widely used method in both engineering and especially medical science. But there are also other ways to reconstruct these images. Two other methods for reconstructing are the SVE/SVD and the ART methods. The first one is based on the singularvalue expansion of an operator, and the second one is based on constructing a linear system of equations, with a sparse matrix and then solving this with the iterative CGLS method. Both the SVE/SVD method and the ART method gives good results, and it is easy to compute these solutions when the preliminary work has been done. For the first method it is easy to determine a truncation parameter anddecrease the effect the noise has on the image. For the second method the quality of the solution can be increased by increasing the number of iterations in the CGLS solver. Comparing simulated reconstructions from all three methods gives a clear picture of which to prefer.

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Resum´ e

Der findes flere forskellige metoder til løse det inverse problem, der best˚ af at ar reconstruere ettomografisk billede fra dets sinogram. Filtered backprojection er den af de metoder, der er mest udbredt, især inden for billedbehandlingen i den medicinske verden. Men der findes ogs˚ andre metoder til at rekonstruere. a To andre er SVE/SVD og ART metoderne. Den første er baseret p˚ en singulær a værdi ekspansion af en operator, og den anden er baseret p˚ at konstruere et a lineært ligningssystem med en’sparse’ matrix, som s˚ kan løses med den iterative a CGLS metode. B˚ SVE/SVD’en og ART metoden giver gode resultater, og ade det er let at beregne disse løsninger, n˚ forarbejdet er gjort godt. For den ar første metode er det let at finde en god trunkeringsparameter, som nedsætter støjens effekt p˚ billedet. For den anden metode f˚ de gode løsninger da det er a as muligt at vælge antallet af iterationerfor CGLS metoden, s˚ løsningen kommer a numerisk tæt p˚ den eksakte løsning. a En sammenligning af simulerede rekonstruktioner fra hver af de tre metoder, vil hurtigt give et billede af hvilken der er at foretrække.

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Contents

Abstract Resum´ e 1 Introduction 2 Tomography 2.1 Basic Idea of Computed Tomography . . . . . . . . . . . . . . . . 3 The Radon Transform 3.1 The Radon Transform. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Sinogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Filtered Backprojection 13 4.1 The Fourier Slice Therorem . . . . . . . . . . . . . . . . . . . . . 13 4.2 Real Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Deriving the FBP Method . . . . . . . . . . . . . . . . . . . . .. 15 5 Singular Value Expansion 5.1 One-dimensional SVE . . . . . . . . . . . . . . . . . . . . . 5.2 The Picard Condition . . . . . . . . . . . . . . . . . . . . . 5.3 Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Two-dimensional SVE . . . . . . . . . . . . . . . . . . . . . 5.5 SVE of the Radon Transform . . . . . . . . . . . . . . . . . 5.6 Analytical results . . . . . .. . . . . . . . . . . . . . . . . . 5.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The Picard condition and the SVD of the Radon Transform 5.9 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Truncated SVD of the Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 25 26 28 29 30 33 35 35 37...