Forward and inverse wave propagation using bandlimited functions and a fast reconstruction algorithm for electron microscopy Kristian Sandberg Department of Applied Mathematics University of Colorado at Boulder1 July 25, 20032 [email protected] 2Re-formatted version of the PhD-thesis with the same name submitted by the author to Grad- uate School at University of Colorado on July 10, 2003. Abstract The thesis consists of three major parts. First, we develop an algorithm for solving the wave equation in two dimensions with spatially varying coefficients. In what is a new approach, we use the basis of approximate prolate spheroidal wavefunctions and construct derivative operators that incorporate boundary and interface conditions. Writing the wave equation as a first-order system, we evolve the equation in time using the matrix exponen- tial. Computation of the matrix exponential requires efficient representation of operators in two dimensions and for this purpose we use short sums of one-dimensional operators. We also use a partitioned low-rank representation in one dimension to further speed up the algorithm. We demonstrate that the method significantly reduces numerical dispersion and the computational time in comparison with a fourth-order finite difference scheme in space with the explicit fourth-order Runge-Kutta solver in time. Second, using efficient representations of spectral projectors, we develop a stable solver for initial value problems on space-like surfaces. By writing the Helmholtz equation in two dimensions as an initial value problem in space and using spectral projectors, we construct a numerically stable scheme for propagating the solution. This solver is a first step toward a fast algorithm for solving inverse scattering problems in two and higher dimensions. Finally, we implement a fast Fourier summation algorithm for tomographic reconstruction of biological data sets obtained via transmission electron microscopy. For two-dimensional images, the new algorithm scales as O(NθM log M)+O(MN log N) operations, compared to O(NθMN) for the standard filtered back projection, where Nθ is the number of projection angles and M N is the size of the reconstructed image. For typical data sets, the new algorithm is 1.5-2.5× times faster than computing the filtered backprojection using the direct summation. The speed advantage is greater as the size of the data sets grow. The new algorithm also allows us to use higher order spline interpolation of the data without additional computational cost. Acknowledgments This thesis could not have been completed without the help and support from several people. First, let me thank Gregory Beylkin for supervising the thesis, and for all the encourage- ment and help. I would also like to thank my committee members Bradley Alpert, James Curry, Keith Julien, and Richard McIntosh for their continuous interest in my research and valuable suggestions for improvements. I would like to thank Lucas Monz´on for providing valuable help with some of the mathematical background for this thesis, Fernando P´erez for help with various computer issues, and our department secretary Laurie Conway for all the administrative help. I would like to thank my dear parents Christina and Ulf Sandberg for all the encour- agement, interest, support, and good advice over the years. My sisters Sofia, Katarina, and Ulrika have always been a source of support. Let me also mention my grandfather Bert Sandberg who taught me how to multiply at an early age. Finally, my warmest thanks to Dawn Boiani for her endless support, encouragement, and inspiration throughout the completion of this thesis. ii Contents 1 Introduction 1 1.1 Wave propagation . 1 1.1.1 Bandlimited functions . 2 1.1.2 Derivative operators with respect to bandlimited functions . 3 1.1.3 Time evolution and operator representations . 3 1.1.4 Wave propagation on space-like surfaces . 4 1.2 Biological imaging . 5 1.3 New results . 6 1.4 Speed comparisons . 7 1.5 Outline of the thesis . 7 2 Bandlimited functions 8 2.1 Preliminaries . 9 2.2 The space of bandlimited functions . 10 2.3 The Prolate Spheroidal Wave Functions . 12 2.4 Bandlimited functions on an interval . 14 2.5 Approximation of bandlimited functions on an interval . 16 2.6 Bases for bandlimited functions on an interval . 19 2.6.1 Exponentials . 20 2.6.2 Approximate prolate spheroidal wave functions . 21 2.6.3 Interpolating functions . 27 2.6.4 Transformation matrices . 28 2.7 Numerical results . 31 2.7.1 Approximations of trigonometric functions by bandlimited functions on an interval . 32 2.7.2 Approximations of polynomials by bandlimited functions on an interval 35 2.7.3 Approximations of Gaussians by bandlimited functions on an interval 37 iii 3 Derivative matrices with boundary and interface conditions 39 3.1 Derivative matrices on an interval . 40 3.1.1 Example . 43 3.1.2 Properties . 44 3.2 Derivative matrix over a subdivided interval . 45 3.2.1 Conditions for the end intervals . 48 3.2.2 Interior intervals . 49 3.2.3 Periodic boundary conditions . 50 3.2.4 Construction of the derivative matrix . 50 3.2.5 Properties of the derivative matrix . 54 4 Algorithms and numerical results for differentiation and integration of bandlimited functions 56 4.1 The derivative operator . 56 4.1.1 Construction of derivative matrices . 57 4.1.2 Accuracy of the derivative matrix for varying bandwidth . 57 4.1.3 Comparison with pseudo-spectral methods and finite differences . 61 4.1.4 Differentiation of piecewise differentiable functions . 63 4.2 Using spectral projectors to improve properties of the derivative matrix with periodic boundary conditions . 64 4.2.1 Properties of the derivative matrix . 64 4.2.2 Construction of the projected derivative matrix . 65 4.3 The second derivative operator with zero boundary conditions . 66 4.3.1 Properties of the derivative matrix . 67 4.3.2 Construction of the second derivative matrix . 67 4.3.3 Numerical results . 68 4.4 The integration operator . 69 4.4.1 Construction of the integration matrix . 72 4.4.2 Numerical results . 72 5 Low rank representations of operators 75 5.1 The separated representation . 76 5.1.1 Operator calculus using the separated rank representation . 78 5.1.2 Separated representation of operators for point wise multiplication . 79 5.2 The Partitioned Low Rank (PLR) representation . 80 5.2.1 Compression of matrices using the PLR representation . 81 5.2.2 Operator calculus using the PLR representation . 84 iv 6 Numerical solutions to the acoustic equation in two dimensions 89 6.1 The acoustic equation in two dimensions as a first order system . 90 6.2 The semigroup approach . 91 6.3 Numerical results . 92 6.3.1 The bandlimited semigroup method . 92 6.3.2 Comparison of the results . 94 6.3.3 Numerical results for constant coefficients . 94 6.3.4 Numerical results for variable coefficients . 102 7 Wave propagation on space-like surfaces 105 7.1 Spectral projectors for wave propagation problems on space-like surfaces . 106 7.2 Computation of spectral projectors . 108 7.2.1 The spectral decomposition of a diagonalizable matrix . 109 7.2.2 The sign function . 109 7.3 A numerical scheme for wave propagation on space-like surfaces . 111 7.3.1 Wave propagation on space-like surfaces as a first order system . 111 7.3.2 Numerical results . 112 8 A fast reconstruction algorithm for electron microscopy 118 8.1 Preliminaries . 119 8.2 Inversion in the Fourier domain . 120 8.3 Implementation . 120 8.4 Results . 121 A Proof of Corollary 10 127 B Algorithms for low rank representations of operators 129 B.1 Rank reduction . 129 B.2 PLR matrix-vector multiplication . 132 B.3 PLR products . 132 C The acoustic equation in two dimensions 136 C.1 The derivative operator in two dimensions on a subdivided domain . 136 C.2 Construction of derivative matrices for the acoustic equation . 137 D Inverse problems 139 D.1 Proof of Proposition 26 . 139 D.2 The inverse problem for acoustics in the time domain . 140 v E A fast reconstruction algorithm for electron microscopy [6] 141 E.1 Introduction . 142 E.2 Preliminaries . 143 E.2.1 Formulation of the problem . 143 E.2.2 Inversion of the Radon transform . 145 E.2.3 Filtered backprojection using direct summation . 145 E.3 Inversion algorithm in the Fourier domain . 146 E.4 Implementation . 149 E.4.1 Discretization . 149 E.4.2 Unequally spaced FFT (USFFT) . 150 E.4.3 Oversampling . 151 E.4.4 Numerical algorithm . 153 E.4.5 Interpolation . 156 E.5 Results . 157 E.5.1 Incorporation of the Fast Fourier Summation into IMOD . 157 E.5.2 Tests . 158 vi Chapter 1 Introduction This thesis covers two main topics: numerical algorithms for wave propagation and a fast tomographic reconstruction algorithm for transmission electron microscopy. The main con- tributions of this thesis are: using bases for bandlimited functions as a numerical tool in algorithms of wave prop- • agation, developing and using efficient representations of operators in two dimensions for pur- • poses of wave propagation, and constructing a fast algorithm for tomographic reconstruction of thick biological speci- • mens for the transmission electron microscopy. Using bases for bandlimited functions allows us to achieve a low oversampling rate while significantly reducing numerical dispersion. By using the operator representations introduced in this thesis for wave propagation, application of the matrix exponential for large time steps becomes feasible as a numerical propagation scheme. 1.1 Wave propagation There are numerous applications of wave propagation in acoustics, elasticity and electro magnetics, including medical imaging, sonar, seismology, radar, and noise modeling in civil engineering. Few problems can be solved analytically, and almost all wave propagation problems in engineering and geophysics involve numerical solutions of equations of acoustics, elasticity, and other hyperbolic systems.