Tomographic reconstruction of shock layer flows A thesis submitted for the degree of Doctor of Philosophy at The Australian National University May 2005 Rado Faletič Department of Physics Faculty of Science
c Rado Faletič Typeset in Palatino by TEX and LATEX 2ε.
Except where otherwise indicated, this thesis is my own original work. Rado Faletič 3 rd May 2005
Acknowledgements Thanks and appreciation goes to the following people who have been instrumental in helping me complete this PhD: Dr. Frank Houwing my supervisor who has been encouraging and positive throughout the whole project; Dr. Neil Mudford gave me many constructive ideas, and pearls of wisdom about all manner of things; Dr. Antony Searle helped me through many roadblocks in writing my computer code; Dr. Matt Gaston shared the beginning of my PhD journey with me; Dr. Russell Boyce provided the data for the hyperboloid phase maps; Dr. Malcolm Sambridge who taught me the principles of tomography; Mal Cheney & Richard Hermens helped me focus on the important things, and are first class people builders; and Tati & Mum for supporting me through my first degree, which enabled this second degree to be possible. Most importantly, my legendary wife HJ. Thanks all! iii
Abstract The tomographic reconstruction of hypersonic flows faces two challenges. Firstly, techniques used in the past, such as the Direct Fourier Method (DFM) or various backprojection techniques, have only been able to reconstruct areas of the flow which are upstream of any opaque objects, such as a model. Secondly, shock waves create sharp discontinuities in flow properties, which can be difficult to reconstruct both in position and in magnitude. This thesis will present a reconstruction method, utilising geometric ray-tracing and a sparse matrix iterative solver, which is capable of overcoming both of these challenges. It will be shown, through testing with phantom objects described in imaging and tomographic literature, that the results are comparable to those produced by the DFM technique. Finally, the method will be used to reconstruct three dimensional density fields from interferometric shock tunnel images, with good resolution. iv
Contents Acknowledgements Abstract Symbols & abbreviations iii iv viii 1 Introduction 1 2 Tomography 5 2.1 Interferometry............................... 6 2.2 The Radon transform........................... 10 2.3 Fourier slice method........................... 12 2.4 Backprojection methods......................... 15 2.5 Matrix inversion.............................. 17 2.5.1 Parameterisation......................... 17 2.5.2 Basis functions.......................... 19 2.5.2.1 Block parameterisation................ 19 2.5.2.2 Ray-tracing....................... 21 2.5.2.3 Polynomials...................... 22 2.6 Abel transform.............................. 23 2.7 Other inversion methods......................... 24 2.7.1 Fan projections.......................... 24 2.7.2 Additional physics........................ 25 2.7.3 Vector tomography........................ 25 2.7.4 Wavelets.............................. 26 3 Methods 27 3.1 Lenna.................................... 27 3.2 Projections................................. 28 3.3 DFM.................................... 29 3.3.1 One dimensional Fourier transform.............. 30 v
Contents vi 3.3.2 Placing slices in two dimensional Fourier space....... 32 3.3.3 Interpolation in two dimensional Fourier space....... 32 3.3.4 Reverse two dimensional Fourier transform......... 35 3.4 MI-RLS................................... 38 3.4.1 Grid................................ 39 3.4.1.1 cell............................ 40 3.4.1.2 neighbour....................... 42 3.4.1.3 surface cell and surface neighbours........ 43 3.4.1.4 boundary nodes and the boundary cell...... 43 3.4.2 rays and ray-tracing....................... 43 3.4.3 Matrix particulars......................... 51 3.5 Error analysis............................... 57 3.5.1 Difference measures....................... 58 3.5.2 Diagonal cut............................ 60 3.6 Remarks.................................. 61 4 Phantoms in R 3 62 4.1 Shepp-Logan................................ 64 4.1.1 Reconstructions.......................... 67 4.1.2 Random noise........................... 71 4.1.3 Analysis.............................. 75 4.1.3.1 Difference measures.................. 75 4.1.3.2 Diagonal slice..................... 78 4.1.4 Remarks.............................. 79 4.2 Stark 36................................... 80 4.2.1 Reconstructions.......................... 83 4.2.2 Random noise........................... 85 4.2.3 Analysis.............................. 88 4.3 Stark 50................................... 93 4.3.1 Reconstructions.......................... 95 4.3.2 Analysis.............................. 96 4.3.3 Remarks.............................. 98 4.4 Summary comments........................... 99 5 Experimental results 100 5.1 hemisphere................................. 101
Contents vii 5.1.1 Reconstructions.......................... 104 5.1.2 Analysis.............................. 107 5.2 hyperboloid................................. 108 5.2.1 Reconstructions.......................... 112 5.2.2 Analysis.............................. 116 5.3 Remarks.................................. 117 6 Conclusion 119 A Computing details 122 Bibliography 124 Index 132
Symbols & abbreviations ANU : Australian National University APAC : Australian Partnership for Advanced Computing R n : φ : Γ : λ : η : x : n-dimensional space, ie. (x 1, x 2,..., x n ) space phase shift ray path wavelength refractive index vector in R n κ : Gladstone-Dale coefficient, page 9 ρ : f : R n R : density f is a scalar valued function of n variables h r,ω : hyperplane at distance r and angle ω, page 10 h R n : h is a subset of R n {x : conditions} : a subset of values x which adhere to the given conditions a b : x : x : the dot product of vectors a and b the absolute value of the scalar x the length of the vector x R : Radon transform, page 10 viii
Contents ix P : ray projection, page 11 P θ : projection at angle θ, page 11 θ : angle of projection, page 11 2 : Laplacian, page 12 b i=a q i : summation of values q i from i = a to i = b F : Fourier transform, page 12 DFM : Direct Fourier Method, page 14 B : Backprojection operator, page 15 p i, p : collection of ray projections, page 17 c j, c : e j : constants basis functions A, A i j : matrix with N rows and M columns x A : x / A : V : x is contained in the set A x is not contained in the set A volume A : Abel transform, page 23 FFT : fast Fourier transform, page 30 DFT : discrete Fourier transform, page 31 MI-RLS : Matrix Inversion using Ray-tracing and Least Squares Conjugate Gradient, page 38 MI-RLS s : MI-RLS with the smoothing regularisation, page 57 CFD : computational fluid dynamics, page 40 z : zero vector, page 44 s : slope vector, page 44
Contents x n : normal vector, page 48 c k : k-th attempt at iteratively finding a solution for c, page 53 N i : the number of neighbouring cells surrounding the i-th cell, page 56 x ink : the k-th neighbouring cell of the i-th cell, page 56 L 2 : an error measure, page 59 σ 2 : variance in the Gaussian error applied to a set of projections, page 71