Phase Retrieval is an inference problem where one seeks to recover an unknown complex-valued 𝓃-dimensional signal vector from the magnitudes of 𝓶 linear measurements. The linear measurements are specified using a 𝓶 × 𝓃 sensing matrix. This problem is a mathematical model for imaging systems arising in X-ray crystallography and other applications where it is infeasible to acquire the phase of the measurements. This dissertation presents some results regarding the analysis of this problem in the high-dimensional asymptotic regime where the number of measurements and the signal dimension diverge proportionally so that their ratio remains fixed. A limitation of existing high-dimensional analyses of this problem is that they model the sensing matrix as a random matrix with independent and identically (i.i.d.) distributed Gaussian entries. In practice, this matrix is highly structured with limited randomness. This work studies a correction to the i.i.d. Gaussian sensing model, known as the sub-sampled Haar sensing model which faithfully captures a crucial orthogonality property of realistic sensing matrices. The first result of this thesis provides a precise asymptotic characterization of the performance of commonly used spectral estimators for phase retrieval in the sub-sampled Haar sensing model. This result can be leveraged to tune certain parameters involved in the spectral estimator optimally. The second part of this dissertation studies the information-theoretic limits for better-than-random (or weak) recovery in the sub-sampled Haar sensing model. The main result in this part shows that appropriately tuned spectral methods achieve weak recovery with the information-theoretically optimal number of measurements. Simulations indicate that the performance curves derived for the sub-sampled Haar sensing model accurately describe the empirical performance curves for realistic sensing matrices such as randomly sub-sampled Fourier sensing matrices and Coded Diffraction Pattern (CDP) sensing matrices. The final part of this dissertation tries to provide a mathematical understanding of this empirical universality phenomenon: For the real-valued version of the phase retrieval problem, the main result of the final part proves that the dynamics of a class of iterative algorithms, called Linearized Approximate Message Passing schemes, are asymptotically identical in the sub-sampled Haar sensing model and a real-valued analog of the sub-sampled Fourier sensing model.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980. / Includes bibliographical references. / by Seshadri Ramaswamy. / Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980.

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published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation laws of some partial di erence equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation. Symmetries and the more recent ideas and notions of characteristics (multipliers) for di erence equations are also discussed. We then determine the symmetry generators of some ordinary di erence equations and proceed to nd the rst integral and reduce the order of the di erence equations. We show that, in some cases, the symmetry generator and rst integral are associated via the `invariance condition'. That is, the rst integral may be invariant under the symmetry of the original di erence equation. We proceed to carry out double reduction of the di erence equation in these cases. We then consider discrete versions of the Painlev e equations. We assume that the characteristics depend on n and un only and we obtain a number of symmetries. These symmetries are used to construct exact solutions in some cases. Finally, we discuss symmetries of linear iterative equations and their transformation properties. We characterize coe cients of linear iterative equations for order less than or equal to ten, although our approach of characterization is valid for any order. Furthermore, a list of coe cients of linear iterative equations of order up to 10, in normal reduced form is given.

Fung Yiu-fai Phineas. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 142-148). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Model-based Pose Estimation --- p.3 / Chapter 1.1.1 --- Application - 3D Motion Tracking --- p.4 / Chapter 1.2 --- Image-based View Synthesis --- p.4 / Chapter 1.3 --- Thesis Contribution --- p.7 / Chapter 1.4 --- Thesis Outline --- p.8 / Chapter 2 --- General Background --- p.9 / Chapter 2.1 --- Notations --- p.9 / Chapter 2.2 --- Camera Models --- p.10 / Chapter 2.2.1 --- Generic Camera Model --- p.10 / Chapter 2.2.2 --- Full-perspective Camera Model --- p.11 / Chapter 2.2.3 --- Affine Camera Model --- p.12 / Chapter 2.2.4 --- Weak-perspective Camera Model --- p.13 / Chapter 2.2.5 --- Paraperspective Camera Model --- p.14 / Chapter 2.3 --- Model-based Motion Analysis --- p.15 / Chapter 2.3.1 --- Point Correspondences --- p.16 / Chapter 2.3.2 --- Line Correspondences --- p.18 / Chapter 2.3.3 --- Angle Correspondences --- p.19 / Chapter 2.4 --- Panoramic Representation --- p.20 / Chapter 2.4.1 --- Static Mosaic --- p.21 / Chapter 2.4.2 --- Dynamic Mosaic --- p.22 / Chapter 2.4.3 --- Temporal Pyramid --- p.23 / Chapter 2.4.4 --- Spatial Pyramid --- p.23 / Chapter 2.5 --- Image Pre-processing --- p.24 / Chapter 2.5.1 --- Feature Extraction --- p.24 / Chapter 2.5.2 --- Spatial Filtering --- p.27 / Chapter 2.5.3 --- Local Enhancement --- p.31 / Chapter 2.5.4 --- Dynamic Range Stretching or Compression --- p.32 / Chapter 2.5.5 --- YIQ Color Model --- p.33 / Chapter 3 --- Model-based Pose Estimation --- p.35 / Chapter 3.1 --- Previous Work --- p.35 / Chapter 3.1.1 --- Estimation from Established Correspondences --- p.36 / Chapter 3.1.2 --- Direct Estimation from Image Intensities --- p.49 / Chapter 3.1.3 --- Perspective-3-Point Problem --- p.51 / Chapter 3.2 --- Our Iterative P3P Algorithm --- p.58 / Chapter 3.2.1 --- Gauss-Newton Method --- p.60 / Chapter 3.2.2 --- Dealing with Ambiguity --- p.61 / Chapter 3.2.3 --- 3D-to-3D Motion Estimation --- p.66 / Chapter 3.3 --- Experimental Results --- p.68 / Chapter 3.3.1 --- Synthetic Data --- p.68 / Chapter 3.3.2 --- Real Images --- p.72 / Chapter 3.4 --- Discussions --- p.73 / Chapter 4 --- Panoramic View Analysis --- p.76 / Chapter 4.1 --- Advanced Mosaic Representation --- p.76 / Chapter 4.1.1 --- Frame Alignment Policy --- p.77 / Chapter 4.1.2 --- Multi-resolution Representation --- p.77 / Chapter 4.1.3 --- Parallax-based Representation --- p.78 / Chapter 4.1.4 --- Multiple Moving Objects --- p.79 / Chapter 4.1.5 --- Layers and Tiles --- p.79 / Chapter 4.2 --- Panorama Construction --- p.79 / Chapter 4.2.1 --- Image Acquisition --- p.80 / Chapter 4.2.2 --- Image Alignment --- p.82 / Chapter 4.2.3 --- Image Integration --- p.88 / Chapter 4.2.4 --- Significant Residual Estimation --- p.89 / Chapter 4.3 --- Advanced Alignment Algorithms --- p.90 / Chapter 4.3.1 --- Patch-based Alignment --- p.91 / Chapter 4.3.2 --- Global Alignment (Block Adjustment) --- p.92 / Chapter 4.3.3 --- Local Alignment (Deghosting) --- p.93 / Chapter 4.4 --- Mosaic Application --- p.94 / Chapter 4.4.1 --- Visualization Tool --- p.94 / Chapter 4.4.2 --- Video Manipulation --- p.95 / Chapter 4.5 --- Experimental Results --- p.96 / Chapter 5 --- Panoramic Walkthrough --- p.99 / Chapter 5.1 --- Problem Statement and Notations --- p.100 / Chapter 5.2 --- Previous Work --- p.101 / Chapter 5.2.1 --- 3D Modeling and Rendering --- p.102 / Chapter 5.2.2 --- Branching Movies --- p.103 / Chapter 5.2.3 --- Texture Window Scaling --- p.104 / Chapter 5.2.4 --- Problems with Simple Texture Window Scaling --- p.105 / Chapter 5.3 --- Our Walkthrough Approach --- p.106 / Chapter 5.3.1 --- Cylindrical Projection onto Image Plane --- p.106 / Chapter 5.3.2 --- Generating Intermediate Frames --- p.108 / Chapter 5.3.3 --- Occlusion Handling --- p.114 / Chapter 5.4 --- Experimental Results --- p.116 / Chapter 5.5 --- Discussions --- p.116 / Chapter 6 --- Conclusion --- p.121 / Chapter A --- Formulation of Fischler and Bolles' Method for P3P Problems --- p.123 / Chapter B --- Derivation of z1 and z3 in terms of z2 --- p.127 / Chapter C --- Derivation of e1 and e2 --- p.129 / Chapter D --- Derivation of the Update Rule for Gauss-Newton Method --- p.130 / Chapter E --- Proof of (λ1λ2-λ 4）>〉0 --- p.132 / Chapter F --- Derivation of φ and hi --- p.133 / Chapter G --- Derivation of w1j to w4j --- p.134 / Chapter H --- More Experimental Results on Panoramic Stitching Algorithms --- p.138 / Bibliography --- p.148

by Cheng Sze Wan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 72-[76]). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Traffic Scheduling in Input-buffered Switches .。 --- p.3 / Chapter 1.3 --- Organization of Thesis --- p.7 / Chapter 2 --- Principle of Enchanced PIM Algorithm --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.1.1 --- Switch Model --- p.9 / Chapter 2.2 --- Enhanced Parallel Iterative Matching Algorithm (EPIM) --- p.10 / Chapter 2.2.1 --- Motivation --- p.10 / Chapter 2.2.2 --- Algorithm --- p.12 / Chapter 2.3 --- Performance Evaluation --- p.16 / Chapter 2.3.1 --- Simulation --- p.16 / Chapter 2.3.2 --- Delay Analysis --- p.18 / Chapter 3 --- Providing Bandwidth Guarantee in Input-Buffered Switches --- p.25 / Chapter 3.1 --- Introduction --- p.25 / Chapter 3.2 --- Bandwidth Reservation in Static Scheduling Algorithm --- p.26 / Chapter 3.3 --- Incorporation of Dynamic and Static Scheduling Algorithms .。 --- p.32 / Chapter 3.4 --- Simulation --- p.34 / Chapter 3.4.1 --- Switch Model --- p.35 / Chapter 3.4.2 --- Simulation Results --- p.36 / Chapter 3.5 --- Comparison with Existing Schemes --- p.42 / Chapter 3.5.1 --- Statistical Matching --- p.42 / Chapter 3.5.2 --- Weighted Probabilistic Iterative Matching --- p.45 / Chapter 4 --- EPIM and Cross-Path Switch --- p.50 / Chapter 4.1 --- Introduction --- p.50 / Chapter 4.2 --- Concept of Cross-Path Switching --- p.51 / Chapter 4.2.1 --- Principle --- p.51 / Chapter 4.2.2 --- Supporting Performance Guarantee in Cross-Path Switch --- p.52 / Chapter 4.3 --- Implication of EPIM on Cross-Path switch --- p.55 / Chapter 4.3.1 --- Problem Re-definition --- p.55 / Chapter 4.3.2 --- Scheduling in Input Modules with EPIM --- p.58 / Chapter 4.4 --- Simulation --- p.63 / Chapter 5 --- Conclusion --- p.70 / Bibliography --- p.72

Wai-shing Lee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 114). / Abstracts in English and Chinese. / Contents --- p.i / List of Figures --- p.iv / List of Tables --- p.v / Abstract --- p.vi / Acknowledgement --- p.vii / Chapter Chapter 1. --- Layout of the thesis --- p.1 / Chapter Chapter 2. --- Introduction --- p.3 / Chapter 2.1 --- What is the reasoning problem? --- p.3 / Chapter 2.2 --- Fundamental nature of Knowledge --- p.4 / Chapter 2.3 --- Fundamental methodology of Reasoning --- p.7 / Chapter 2.4 --- Our intended approach --- p.9 / Chapter Chapter 3. --- Probabilistic reasoning networks --- p.11 / Chapter 3.1 --- Overview --- p.11 / Chapter 3.2 --- Causality and influence diagrams --- p.11 / Chapter 3.3 --- Bayesian networks - influence diagrams endowed with a probability interpretation --- p.13 / Chapter 3.3.1 --- A detour to the interpretations of probability --- p.13 / Chapter 3.3.2 --- Bayesian networks --- p.15 / Chapter 3.3.3 --- Acyclicity and global probability --- p.17 / Chapter 3.4 --- Reasoning on probabilistic reasoning networks I - local updating formulae --- p.17 / Chapter 3.4.1 --- Rationale of the intended reasoning strategy --- p.18 / Chapter 3.4.2 --- Construction of the local updating formula --- p.19 / Chapter 3.5 --- Cluster graphs - another perspective to reasoning problems --- p.23 / Chapter 3.6 --- Semi-lattices - another representation of Cluster graphs --- p.26 / Chapter 3.6.1 --- Construction of semi-lattices --- p.26 / Chapter 3.7 --- Bayesian networks and semi-lattices --- p.28 / Chapter 3.7.1 --- Bayesian networks to acyclic semi-lattices --- p.29 / Chapter 3.8 --- Reasoning on (acyclic) probabilistic reasoning networks II - global updating schedules --- p.29 / Chapter 3.9 --- Conclusion --- p.30 / Chapter Chapter 4. --- Cyclic reasoning networks - a possibility? --- p.32 / Chapter 4.1 --- Overview --- p.32 / Chapter 4.2 --- A meaningful cyclic structure - derivation of the ideal gas law --- p.32 / Chapter 4.3 --- "What's ""wrong"" to be in a cyclic world" --- p.35 / Chapter 4.4 --- Communication - Dynamics - Complexity --- p.39 / Chapter 4.4.1 --- Communication as dynamics; dynamics to complexity --- p.42 / Chapter 4.5 --- Conclusion --- p.42 / Chapter Chapter 5. --- Cyclic reasoning networks ´ؤ error-control application --- p.43 / Chapter 5.1 --- Overview --- p.43 / Chapter 5.2 --- Communication schemes on cyclic reasoning networks directed to error-control applications --- p.43 / Chapter 5.2.1 --- Part I ´ؤ Local updating formulae --- p.44 / Chapter 5.2.2 --- Part II - Global updating schedules across the network --- p.46 / Chapter 5.3 --- Probabilistic reasoning based error-control schemes --- p.47 / Chapter 5.3.1 --- Local sub-universes and global universe underlying the error- control structure --- p.47 / Chapter 5.4 --- Error-control structure I --- p.48 / Chapter 5.4.1 --- Decoding algorithm - Communication between local sub- universes in compliance with the global topology --- p.51 / Chapter 5.4.2 --- Decoding rationales --- p.55 / Chapter 5.4.3 --- Computational results --- p.55 / Chapter 5.5 --- Error-control structure II --- p.57 / Chapter 5.5.1 --- Structure of the code and the corresponding decoding algorithm --- p.57 / Chapter 5.5.2 --- Computational results --- p.63 / Chapter 5.6 --- Error-control structure III --- p.66 / Chapter 5.6.1 --- Computational results --- p.70 / Chapter 5.7 --- Error-control structure IV --- p.71 / Chapter 5.7.1 --- Computational results --- p.73 / Chapter 5.8 --- Conclusion --- p.74 / Chapter Chapter 6. --- Dynamics on cyclic probabilistic reasoning networks --- p.75 / Chapter 6.1 --- Overview --- p.75 / Chapter 6.2 --- Decoding rationales --- p.76 / Chapter 6.3 --- Error-control structure I - exact solutions --- p.77 / Chapter 6.3.1 --- Dynamical invariant - a key to tackle many dynamical problems --- p.77 / Chapter 6.3.2 --- Dynamical invariant for error-control structure I --- p.78 / Chapter 6.3.3 --- Iteration dynamics --- p.79 / Chapter 6.3.4 --- Structure preserving property and the maximum a posteriori solutions --- p.86 / Chapter 6.4 --- Error-control structures III & IV - exact solutions --- p.92 / Chapter 6.4.1 --- Error-control structure III --- p.92 / Chapter 6.4.1.1 --- Dynamical invariants for error-control structure III --- p.92 / Chapter 6.4.1.2 --- Iteration dynamics --- p.93 / Chapter 6.4.2 --- Error-control structure IV --- p.96 / Chapter 6.4.3 --- Structure preserving property and the maximum a posteriori solutions --- p.98 / Chapter 6.5 --- Error-control structure II - exact solutions --- p.101 / Chapter 6.5.1 --- Iteration dynamics --- p.102 / Chapter 6.5.2 --- Structure preserving property and the maximum a posteriori solutions --- p.105 / Chapter 6.6 --- A comparison on the four error-control structures --- p.106 / Chapter 6.7 --- Conclusion --- p.108 / Chapter Chapter 7. --- Conclusion --- p.109 / Chapter 7.1 --- Our thesis --- p.109 / Chapter 7.2 --- Hind-sights and foresights --- p.110 / Chapter 7.3 --- Concluding remark --- p.111 / Appendix A. An alternative derivation of the local updating formula --- p.112 / Bibliography --- p.114