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Efficient Solutions to Toeplitz-Structured Linear Systems for Signal Processing

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Efficient Solutions to Toeplitz-Structured Linear Systems for Signal Processing EFFICIENT SOLUTIONS TO TOEPLITZ-STRUCTURED LINEAR SYSTEMS FOR SIGNAL PROCESSING A Dissertation Presented to The Academic Faculty by Christopher Kowalczyk Turnes In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Electrical and Computer Engineering Georgia Institute of Technology May 2014 Copyright © 2014 by Christopher Kowalczyk Turnes EFFICIENT SOLUTIONS TO TOEPLITZ-STRUCTURED LINEAR SYSTEMS FOR SIGNAL PROCESSING Approved by: Dr. James McClellan, Committee Chair Dr. Jack Poulson School of Electrical and Computer School of Computational Science and Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Justin Romberg, Advisor Dr. Chris Barnes School of Electrical and Computer School of Electrical and Computer Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Dr. Monson Hayes Dr. Doru Balcan School of Electrical and Computer Quantitative Finance Engineering Bank of America Georgia Institute of Technology Date Approved: May 2014 Effort is one of the things that gives meaning to life. Effort means you care about something, that something is important to you and you are willing to work for it. It would be an impoverished existence if you were not willing to value things and commit yourself to working toward them. – CAROL S. DWECK, SELF-THEORIES To my wonderful, amazing, and patient parents Cynthia and Patrick, without whom none of this would be possible. Thank you for everything (but mostly for life). ACKNOWLEDGEMENTS Foremost, I wish to express my sincere gratitude to my advisor, Dr. Justin Romberg, for his support of my work and for his patience, time, and immense knowledge. Under his direction, my technical and communication skills improved dramatically. During my time at Georgia Tech, he did a phenomenal job of cultivating an amiable and social research group and achieved the impossible: making graduate school fun. My research would not have been possible without the efforts of Dr. Doru Balcan. Doru took time out of a busy schedule to discuss my work, and often helped me to fully realize the significance and implications of my results. He has been a true mentor and friend. I also wish to thank my other thesis committee members, Drs. Jim McClellan, Monty Hayes, Jack Poulson, and Chris Barnes, for their time and insight. Many thanks to my brothers Jonathan and Walter for mastering that delicate balance between antagonism and encouragement over the course of my life, and to the rest of my extended family for their love and support. I would also like to acknowledge several friends, colleagues, and labmates. To Cathy Zanetti, Laura Hansen, Peter and Alex Tuuk, and Peter Siy, the camping, parties, barbe- cues, pizza festivals, and innumerable other social events have made my time in Atlanta unforgettable. To Steve Conover, I will never forget our collaborations on coding ven- tures, as well as the many sushi and movie nights with Jiun-Hong Lai. To Adam Charles and Diane Isaacson, Sam and Taylor Shapero, Becky Fong, Kyle and Steph Krueger, Sean Kelly, Jeff Bingham, Jeff Dugger, and the rest of team “Shake and Bake,” thank you for joining me on so many trivia nights. To all of the coordinators and participants of the 12- hour “outside-the-box creative research initiatives,” thank you for suffering along with me. Lastly, to the many, many others who have served as friends, labmates, collaborators, and v hockey teammates, I thank you for all of the wonderful memories. Finally, I am forever indebted to my wife and best friend, Auréle, without whose love and encouragement I would not have finished this dissertation. Merci, mon amour. vi TABLE OF CONTENTS ACKNOWLEDGEMENTS .............................. v LIST OF TABLES .................................. x LIST OF FIGURES .................................. xi NOMENCLATURE .................................. xii SUMMARY ...................................... xiv I INTRODUCTION AND FUNDAMENTALS .................. 1 1.1 Introduction ................................. 1 1.2 Structuredlinearalgebra. .. 1 1.2.1 Typesofstructure .......................... 2 1.2.2 Structureinsignalprocessing. 6 1.3 AshorthistoryofToeplitzalgorithms. ..... 8 1.4 Displacement ................................ 12 1.4.1 Motivationandhistory . 12 1.4.2 SylvesterandSteindisplacement . 13 1.4.3 Singularityandrecoverability. .. 15 1.4.4 Recoveryformulas . .. .. .. .. .. .. .. 17 1.4.5 Displacementoperatormatrices. 18 1.5 Multi-levellinearalgebra . .. 19 1.5.1 Basicdefinitions.. .. .. .. .. .. .. .. 19 1.5.2 Symmetryandpersymmetry . 22 II CHARACTERIZATION OF STRUCTURED SYSTEMS .......... 27 2.1 Displacement and generators of structured matrices . ......... 27 2.1.1 ScalarToeplitzmatricesandinverses . ... 27 2.1.2 Generalized Cauchy matrices and inverses . ... 37 2.2 Polynomial interpolationand structured systems . ......... 42 2.2.1 Generalizedpolynomialinterpolation. .... 42 vii 2.2.2 Structured systems and tangential interpolation . ....... 44 2.2.3 Solvingtangential-interpolationproblems . ...... 51 2.3 Schurrecursions............................... 60 2.3.1 SchurrecursionsforToeplitzinverses . ... 61 2.3.2 Schur recursions for generalized Cauchy inverses . ...... 62 III NEW ALGORITHMS FOR SCALAR STRUCTURED MATRICES .... 67 3.1 RegularizedleastsquaresforToeplitzsystems . ........ 67 3.1.1 Tikhonovregularization. 67 3.1.2 Regularizationalgorithm . 69 3.1.3 Displacementandinversegeneration . .. 72 3.1.4 Implementationissues. 75 3.1.5 Numericalresults . .. .. .. .. .. .. .. 82 3.2 Nonuniformresamplingofdigitalsignals . ...... 96 3.2.1 Structured matrices innon-uniformresampling . ..... 98 3.2.2 Superfast non-uniform-to-uniform resampling . .. ..100 3.2.3 Efficientpseudoinversionforresamplingmatrices . 102 3.2.4 Performancescaling. .105 3.2.5 Applicationsofinterest . .111 IV MULTI-LEVEL TOEPLITZ THEORY AND ALGORITHMS ....... 115 4.1 Inversionforspecialclasses . .115 4.1.1 CommutativeclassesofToeplitzmatrices . .115 4.1.2 Inversionformulasandalgorithms . 117 4.1.3 Numericalresults . .124 4.2 One-level formulas for two-level Toeplitz inverses . ..........128 4.2.1 Block-leveldisplacement . .129 4.2.2 Singularityofblock-leveldisplacement . .132 4.2.3 Inversionthroughblock-leveldisplacement . .133 4.2.4 Equivalenceofblock-leveldisplacements . .136 viii 4.2.5 Extensiontoblockwiseresults . 138 4.3 Miscellaneous results on two-level Toeplitz inverses . ...........138 4.3.1 On two-level displacement and displacement rank . .139 4.3.2 Ontwo-levelGohberg-Semenculformulas . 144 4.3.3 Ontwo-levelToeplitzinversegenerators . .151 4.3.4 On transformation into two-level Loewner matrices . .. ..156 V CONCLUSIONS ................................. 164 5.1 Summaryofresults .............................164 5.2 Futuredirections...............................166 APPENDIXA —PROOFS ............................ 169 APPENDIX B — TANGENTIAL-INTERPOLATION EXAMPLE ...... 202 APPENDIXC —SPIRALFFT .......................... 206 REFERENCES ..................................... 231 ix LIST OF TABLES 1 Tan.int.accuracy............................... 87 2 Tan. int. accuracy of artificial variables . ...... 88 3 Tan.int.accuracyformulti-columninputs . ..... 90 4 Tan.int.CGexecutiontimecomparison . .. 92 5 Tan.int.CGerrorcomparison .. .. .. .. .. .. .. 92 6 Tan. int. CG execution time comparison for multi-column inputs ...... 93 7 Tan. int. CG accuracy comparison for multi-columninputs . ........ 93 8 SCTLTinversiondeblurringaccuracy . 126 9 Equivalences for potential two-level inverse formulas . ..........151 10 Commonentriesintwo-levelgenerators . .154 11 SpiralFFTanglechoicecomparisons . .217 12 SpiralFFTperformancecomparison . 221 13 SpiralFFTvolumereconstructions . .229 x LIST OF FIGURES 1 Exampleofcirculantextension . 46 2 τ-degreeillustration. .. .. .. .. .. .. .. .. 53 3 Tikhonovbasisconstructionsubdivisionstrategy . ........ 77 4 Comparison of number of conditions for extension strategies ........ 81 5 Comparisonoftwonewextensionstrategies . .... 83 6 Tan.int.executiontimes ........................... 85 7 Tan. int. executiontimesformulti-columninputs . ....... 89 8 Tan.int.NUFFTreconstruction . 96 9 Tan. int. NUFFTreconstructionerrorcomparison . ...... 97 10 Resamplingalgorithmexecutiontimescaling . .. ..109 11 Resamplingalgorithmerrorplots. .110 12 ResamplingalgorithmNUFFTerrors. 113 13 Blurring filter for the imagedeblurring experiments . .........125 14 Deblurred images using the SCTLT inversion algorithm . ........125 15 Executiontimes for the SCTLT inversionalgorithm (1) . ........127 16 Executiontimes for the SCTLT inversionalgorithm (2) . ........127 17 Low-rank partitioningof thetwo-levelLoewner form . ........162 18 ExamplespiralsforSpiraFFT. 211 19 SpiralFFTangleselection. 216 20 SpiralFFTperformancecomparison . 224 21 SpiralFFTperformancecurves . 225 22 Syntheticandreal3-Dvolumes. 226 23 3-DMRIreconstructions .. .. .. .. .. .. .. ..228 24 Synthetc3-DMRIreconstructions . 228 xi NOMENCLATURE j2πkℓ/n F Fourier matrix with entries Fk,ℓ = e− [Sec. 1.2.1.2]. diag (a) Diagonal matrix formed from the entries of the vector a [Sec. 1.2.1.2]. ⊛c Circular convolution [Sec. 1.2.1.2]. ⊛ Linear convolution [Sec. 1.2.2.1]. rxy[n] Cross correlation of vectors x and y [Sec. 1.2.2.2]. ( ) Sylvester displacement operator [Sec. 1.4.2]. ∇A,B · ∆ ( ) Stein displacement operator [Sec. 1.4.2]. A,B · rank( ) Matrix rank [Sec. 1.4.2]. · I Identity matrix (size assumed from context) [Sec. 1.4.3]. vec ( ) Column-based vectorization of a matrix [Sec. 1.4.5]. ·
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