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Currently, many missile fire control systems use a three degree of freedom (3-DOF) trajectory model. The three degrees of freedom represent the linear motion of the missile in three dimensions. A 6-DOF model adds roll, pitch, and yaw, or angular motion in three dimensions to the first three degrees of freedom. Because more of the missile’s attributes are modeled, a 6-DOF model is more accurate than a 3-DOF model. For the same reason, a 3-DOF model is easier to develop and executes faster. Also, because a 3-DOF model ignores the seemingly random angular motion, the step sizes used to integrate 3-DOF models are larger. The goal of this project is to develop a 6-DOF re-entry model with the accuracy of a 6-DOF model with conventional equations of motion and computational speed at least comparable to the 3-DOF model. This can be achieved by transforming the equations that compute the effects of angular motion so that they are better conditioned. Essentially, this is done by fitting a sine wave to the oscillating state variables representing the orientation and angular rates, namely the quaternions and the angular velocity. This thesis shows the results of transforming the oscillating variables of the state vector. / Master of Science

M.S.

by Patrick Chi-man Fung. / Thesis (M.Sc.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 57-58). / LIST OF FIGURES / LIST OF IMAGES / LIST OF TABLES / NOTATIONS / Chapter 1 --- INTRODUCTION --- p.1 / Chapter 2 --- THEORY --- p.3 / Chapter 2.1 --- Matrix Representation of LOT --- p.3 / Chapter 2.2 --- Feasibility of LOT --- p.5 / Chapter 2.3 --- Properties of Good Feasible LOT --- p.6 / Chapter 2.4 --- An Optimal LOT --- p.7 / Chapter 2.5 --- Approximation of an Optimal LOT --- p.10 / Chapter 2.6 --- Representation of an Approximately Optimal LOT --- p.13 / Chapter 3 --- IMPLEMENTATION --- p.17 / Chapter 3.1 --- Mathematical Background --- p.17 / Chapter 3.2 --- Analysis of LOT Flowgraph --- p.17 / Chapter 3.2.1 --- The Fundamental LOT Building Block --- p.17 / Chapter 3.2.2 --- +1/-1 Butterflies --- p.19 / Chapter 3.3 --- Conclusion --- p.25 / Chapter 4 --- RESULTS --- p.27 / Chapter 4.1 --- Objective of Energy Packing --- p.27 / Chapter 4.2 --- Nature of Target Images --- p.27 / Chapter 4.3 --- Methodology of LOT Coefficient Selection --- p.28 / Chapter 4.4 --- dB RMS Error in Pixel Values --- p.29 / Chapter 4.5 --- Negative Pixel Values in Reverse LOT --- p.30 / Chapter 4.6 --- LOT Coefficient Energy Distribution --- p.30 / Chapter 4.7 --- Experimental Data --- p.32 / Chapter 5 --- DISCUSSION AND CONCLUSIONS --- p.46 / Chapter 5.1 --- RMS Error (dB) and LOT Coeffs. Drop Ratio --- p.46 / Chapter 5.1.1 --- Numeric Experimental Results --- p.46 / Chapter 5.1.2 --- Human Visual Response --- p.46 / Chapter 5.1.3 --- Conclusion --- p.49 / Chapter 5.2 --- Number of Negative Pixel Values in RLOT --- p.50 / Chapter 5.3 --- LOT Coefficient Energy Distribution --- p.51 / Chapter 5.4 --- Effect of Changing the Block Size --- p.54 / REFERENCES --- p.57 / APPENDIX / Tables of Experimental Data --- p.59

by Lai Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 68-70). / Abstract also in Chinese. / ACKNOWLEDGMENTS --- p.IV / ABSTRACT --- p.v / Chapter Chapter 1 --- Traditional Image and Video Coding --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Fundamental Principle of Compression --- p.1 / Chapter 1.3 --- Entropy - Value of Information --- p.2 / Chapter 1.4 --- Performance Measure --- p.3 / Chapter 1.5 --- Image Coding Overview --- p.4 / Chapter 1.5.1 --- Digital Image Formation --- p.4 / Chapter 1.5.2 --- Needs of Image Compression --- p.4 / Chapter 1.5.3 --- Classification of Image Compression --- p.5 / Chapter 1.5.4 --- Transform Coding --- p.6 / Chapter 1.6 --- Video Coding Overview --- p.8 / Chapter Chapter 2 --- Discrete Wavelets Transform (DWT) and Subband Coding --- p.11 / Chapter 2.1 --- Subband Coding --- p.11 / Chapter 2.1.1 --- Introduction --- p.11 / Chapter 2.1.2 --- Quadrature Mirror Filters (QMFs) --- p.12 / Chapter 2.1.3 --- Subband Coding for Image --- p.13 / Chapter 2.2 --- Discrete Wavelets Transformation (DWT) --- p.15 / Chapter 2.2.1 --- Introduction --- p.15 / Chapter 2.2.2 --- Wavelet Theory --- p.15 / Chapter 2.2.3 --- Comparison Between Fourier Transform and Wavelet Transform --- p.16 / Chapter Chapter 3 --- Non-expansive Symmetric Extension --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Types of extension scheme --- p.19 / Chapter 3.3 --- Non-expansive Symmetric Extension and Symmetric Sub-sampling --- p.21 / Chapter Chapter 4 --- Content-based Video Coding in MPEG-4 Purposed Standard --- p.24 / Chapter 4.1 --- Introduction --- p.24 / Chapter 4.2 --- Motivation of the new MPEG-4 standard --- p.25 / Chapter 4.2.1 --- Changes in the production of audio-visual material --- p.25 / Chapter 4.2.2 --- Changes in the consumption of multimedia information --- p.25 / Chapter 4.2.3 --- Reuse of audio-visual material --- p.26 / Chapter 4.2.4 --- Changes in mode of implementation --- p.26 / Chapter 4.3 --- Objective of MPEG-4 standard --- p.27 / Chapter 4.4 --- Technical Description of MPEG-4 --- p.28 / Chapter 4.4.1 --- Overview of MPEG-4 coding system --- p.28 / Chapter 4.4.2 --- Shape Coding --- p.29 / Chapter 4.4.3 --- Shape Adaptive Texture Coding --- p.33 / Chapter 4.4.4 --- Motion Estimation and Compensation (ME/MC) --- p.35 / Chapter Chapter 5 --- Shape Adaptive Wavelet Transformation Coding Scheme (SA WT) --- p.36 / Chapter 5.1 --- Shape Adaptive Wavelet Transformation --- p.36 / Chapter 5.1.1 --- Introduction --- p.36 / Chapter 5.1.2 --- Description of Transformation Scheme --- p.37 / Chapter 5.2 --- Quantization --- p.40 / Chapter 5.3 --- Entropy Coding --- p.42 / Chapter 5.3.1 --- Introduction --- p.42 / Chapter 5.3.2 --- Stack Run Algorithm --- p.42 / Chapter 5.3.3 --- ZeroTree Entropy (ZTE) Coding Algorithm --- p.45 / Chapter 5.4 --- Binary Shape Coding --- p.49 / Chapter Chapter 6 --- Simulation --- p.51 / Chapter 6.1 --- Introduction --- p.51 / Chapter 6.2 --- SSAWT-Stack Run --- p.52 / Chapter 6.3 --- SSAWT-ZTR --- p.53 / Chapter 6.4 --- Simulation Results --- p.55 / Chapter 6.4.1 --- SSAWT - STACK --- p.55 / Chapter 6.4.2 --- SSAWT ´ؤ ZTE --- p.56 / Chapter 6.4.3 --- Comparison Result - Cjpeg and Wave03. --- p.57 / Chapter 6.5 --- Shape Coding Result --- p.61 / Chapter 6.6 --- Analysis --- p.63 / Chapter Chapter 7 --- Conclusion --- p.64 / Appendix A: Image Segmentation --- p.65 / Reference --- p.68

by Man-Ching Auyeung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 86-93). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Transform coding and the JPEG scheme --- p.2 / Chapter 1.2 --- Motivation --- p.5 / Chapter 1.3 --- Thesis outline --- p.6 / Chapter 2 --- MED and DC Coefficient Restoration scheme --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- MED and DC Coefficient Restoration scheme --- p.10 / Chapter 2.2.1 --- Definition --- p.10 / Chapter 2.2.2 --- Existing schemes --- p.11 / Chapter 2.3 --- DC Coefficient Restoration scheme using block selection scheme --- p.14 / Chapter 2.4 --- Joint optimization technique --- p.16 / Chapter 2.4.1 --- Lagrange multiplier method --- p.17 / Chapter 2.4.2 --- Algorithm description --- p.18 / Chapter 2.5 --- Experimental results --- p.20 / Chapter 2.6 --- Summary --- p.32 / Chapter 3 --- Low Frequency Walsh Transform Coefficient Restoration scheme --- p.34 / Chapter 3.1 --- Introduction --- p.34 / Chapter 3.2 --- Restoration of low frequency coefficient using Walsh transform --- p.35 / Chapter 3.3 --- Selection of quantization table optimized for Walsh transform --- p.37 / Chapter 3.3.1 --- Image model used --- p.39 / Chapter 3.3.2 --- Infinite uniform quantization --- p.40 / Chapter 3.3.3 --- Search for an optimized quantization matrix --- p.42 / Chapter 3.4 --- Walsh transform-based LFCR scheme --- p.44 / Chapter 3.5 --- Experimental results --- p.46 / Chapter 3.6 --- Summary --- p.56 / Chapter 4 --- Low Frequency DCT Coefficient Prediction --- p.57 / Chapter 4.1 --- Introduction --- p.57 / Chapter 4.2 --- Low Frequency Coefficient Prediction scheme with negligible side information --- p.58 / Chapter 4.2.1 --- Selection of threshold --- p.63 / Chapter 4.2.2 --- Representation of the AC component --- p.63 / Chapter 4.3 --- Experimental results --- p.67 / Chapter 4.4 --- Summary --- p.84 / Chapter 5 --- Conclusions --- p.86 / Appendix A --- p.89 / Bibliography --- p.90

by Tse, Fu Wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 155-[63]). / Acknowledgment --- p.iii / Abstract --- p.iv / Contents --- p.vi / List of Tables --- p.x / List of Figures --- p.xii / Notations --- p.xvii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- DC coefficient restoration --- p.1 / Chapter 1.2 --- Model based image compression --- p.5 / Chapter 1.3 --- The minimum edge difference criterion and the existing estima- tion schemes --- p.7 / Chapter 1.3.1 --- Fundamental definitions --- p.8 / Chapter 1.3.2 --- The minimum edge difference criterion --- p.9 / Chapter 1.3.3 --- The existing estimation schemes --- p.10 / Chapter 1.4 --- Thesis outline --- p.14 / Chapter 2 --- A mathematical description of the DC coefficient restoration problem --- p.17 / Chapter 2.1 --- Introduction --- p.17 / Chapter 2.2 --- Mathematical formulation --- p.18 / Chapter 2.3 --- Properties of H --- p.22 / Chapter 2.4 --- Analysis of the DC coefficient restoration problem --- p.22 / Chapter 2.5 --- The MED criterion as an image model --- p.25 / Chapter 2.6 --- Summary --- p.27 / Chapter 3 --- The global estimation scheme --- p.29 / Chapter 3.1 --- Introduction --- p.29 / Chapter 3.2 --- the global estimation scheme --- p.30 / Chapter 3.3 --- Theory of successive over-relaxation --- p.34 / Chapter 3.3.1 --- Introduction --- p.34 / Chapter 3.3.2 --- Gauss-Seidel iteration --- p.35 / Chapter 3.3.3 --- Theory of successive over-relaxation --- p.38 / Chapter 3.3.4 --- Estimation of optimal relaxation parameter --- p.41 / Chapter 3.4 --- Using successive over-relaxation in the global estimation scheme --- p.43 / Chapter 3.5 --- Experiments --- p.48 / Chapter 3.6 --- Summary --- p.49 / Chapter 4 --- The block selection scheme --- p.52 / Chapter 4.1 --- Introduction --- p.52 / Chapter 4.2 --- Failure of the minimum edge difference criterion --- p.53 / Chapter 4.3 --- The block selection scheme --- p.55 / Chapter 4.4 --- Using successive over-relaxation with the block selection scheme --- p.57 / Chapter 4.5 --- Practical considerations --- p.58 / Chapter 4.6 --- Experiments --- p.60 / Chapter 4.7 --- Summary --- p.61 / Chapter 5 --- The edge selection scheme --- p.65 / Chapter 5.1 --- Introduction --- p.65 / Chapter 5.2 --- Edge information and the MED criterion --- p.66 / Chapter 5.3 --- Mathematical formulation --- p.70 / Chapter 5.4 --- Practical Considerations --- p.74 / Chapter 5.5 --- Experiments --- p.76 / Chapter 5.6 --- Discussion of edge selection scheme --- p.78 / Chapter 5.7 --- Summary --- p.79 / Chapter 6 --- Performance Analysis --- p.81 / Chapter 6.1 --- Introduction --- p.81 / Chapter 6.2 --- Mathematical derivations --- p.82 / Chapter 6.3 --- Simulation results --- p.92 / Chapter 6.4 --- Summary --- p.96 / Chapter 7 --- The DC coefficient restoration scheme with baseline JPEG --- p.97 / Chapter 7.1 --- Introduction --- p.97 / Chapter 7.2 --- General specifications --- p.97 / Chapter 7.3 --- Simulation results --- p.101 / Chapter 7.3.1 --- The global estimation scheme with the block selection scheme --- p.101 / Chapter 7.3.2 --- The global estimation scheme with the edge selection scheme --- p.113 / Chapter 7.3.3 --- Performance comparison at the same bit rate --- p.121 / Chapter 7.4 --- Computation overhead using the DC coefficient restoration scheme --- p.134 / Chapter 7.5 --- Summary --- p.134 / Chapter 8 --- Conclusions and Discussions --- p.136 / Chapter A --- Fundamental definitions --- p.144 / Chapter B --- Irreducibility by associated directed graph --- p.146 / Chapter B.1 --- Irreducibility and associated directed graph --- p.146 / Chapter B.2 --- Derivation of irreducibility --- p.147 / Chapter B.3 --- Multiple blocks selection --- p.149 / Chapter B.4 --- Irreducibility with edge selection --- p.151 / Chapter C --- Sample images --- p.153 / Bibliography --- p.155

The study of second order ordinary differential equations is vital given their proliferation in mechanics. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. However, many second order equations cannot be reduced to quadratures due to the lack of a sufficient number of point symmetries. We observe that increasing the order will result in a third order differential equation which, when reduced via an alternate symmetry, may result in a solvable second order equation. Thus the original second order equation can be solved. In this dissertation we will attempt to classify second order differential equations that can be solved in this manner. We also provide the nonlocal transformations between the original second order equations and the new solvable second order equations. Our starting point is third order differential equations. Here we concentrate on those invariant under two- and three-dimensional Lie algebras. / Thesis (M.Sc.)-University of Natal, Durban, 2000.

[Truncated abstract] One common task in Computer Vision is the estimation of three-dimensional surface shape from two-dimensional images. This task is important as a precursor to higher level tasks such as object recognition - since shape of an object gives clues to what the object is - and object modelling for graphics. Many visual cues have been suggested in the literature to provide shape information, including the shading of an object, its occluding contours (the outline of the object that slants away from the viewer) and its appearance from two or more views. If the image exhibits a significant amount of texture, then this too may be used as a shape cue. Here, ‘texture’ is taken to mean the pattern on the surface of the object, such as the dots on a pear, or the tartan pattern on a tablecloth. This problem of estimating the shape of an object based on its texture is referred to as shape-form-texture and it is the subject of this thesis . . . The work in this thesis is likely to impact in a number of ways. The second shape-form-texture algorithm provides one of the most general solutions to the problem. On the other hand, if the assumptions of the first shape-form-texture algorithm are met, this algorithm provides an extremely usable method, in that users should be able to input images of textured objects and click on the frontal texture to quickly reconstruct a fairly good estimation of the surface. And lastly, the algorithm for estimating the transformation between textures can be used as a part of many shape-form-texture algorithms, as well as being useful in other areas of Computer Vision. This thesis gives two examples of other applications for the method: re-texturing an object and placing objects in a scene.

Thesis (Ph.D.)--Ohio University, November, 2004. / Includes bibliographical references (p. 168-177)