Thesis (Ph. D.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 125-129).

The ability to efficiently and sparsely represent seismic data is becoming an increasingly important problem in geophysics. Over the last thirty years many transforms such as wavelets, curvelets, contourlets, surfacelets, shearlets, and many other types of ‘x-lets’ have been developed. Such transform were leveraged to resolve this issue of sparse representations. In this work we compare the properties of four of these commonly used transforms, namely the shift-invariant wavelets, complex wavelets, curvelets and surfacelets. We also explore the performance of these transforms for the problem of recovering seismic wavefields from incomplete measurements. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis presents an algorithm for image reconstruction from projections intended for use in a new class of volume imaging PET scanners. The algorithm is based on the inversion of the three dimensional Radon Transform as it applies to the truncated cylindrical detector geometry and is derived from the X-ray Transform inversion given by the Orlov recovery operator. The algorithm is tested using Monte Carlo simulations of several phantom geometries and employs a single iterative step to include all detected events in the reconstruction. The reconstructed images are good representations of the original objects, however the iterative step is a source of some significant artefacts in the images. Also discussed is the extension of the Radon Transform technique to a non-iterative method for three dimensional image reconstruction using all detected events. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Yes / Two fractional Fourier transforms are used to define bi-fractional displacement operators, which interpolate between displacement operators and parity operators. They are used to define bi-fractional coherent states. They are also used to define the bi-fractional Wigner function, which is a two-parameter family of functions that interpolates between the Wigner function and the Weyl function. Links to the extended phase space formalism are also discussed.

No description available.

The fast Hartley transform (FHT) is a new tool for converting data between time and frequency domains. In this thesis, some speed-optimized software implementations of the radix-2 and split-radix FHT algorithms are presented initially, and then applied to the problems of convolution, computation of power spectra, image degradation, and image restoration. Subsequent work involved the development of a new bit-reversal algorithm. This algorithm is fast and efficient, and can be used to increase the throughput of the FHT. Finally, several hardware implementations are presented for the discrete Hartley transform (DHT) and the FHT with architectures using a single butterfly unit, pipelining and superparallelism. The advantages of each implementation are stressed. The data processing rates of these hardware implementations are analyzed. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

A review of the applications of the Radon transform is presented, with emphasis on emission computed tomography and transmission computed tomography. The theory of the 2D and 3D Radon transforms, and the effects of attenuation for emission computed tomography are presented. The algebraic iterative methods, their importance and limitations are reviewed. Analytic solutions of the 2D problem the convolution and frequency filtering methods based on linear shift invariant theory, and the solution of the circular harmonic decomposition by integral transform theory--are reviewed. The relation between the invisible kernels, the inverse circular harmonic transform, and the consistency conditions are demonstrated. The discussion and review are extended to the 3D problem-convolution, frequency filtering, spherical harmonic transform solutions, and consistency conditions. The Cormack algorithm based on reconstruction with Zernike polynomials is reviewed. An analogous algorithm and set of reconstruction polynomials is developed for the spherical harmonic transform. The relations between the consistency conditions, boundary conditions and orthogonal basis functions for the 2D projection harmonics are delineated and extended to the 3D case. The equivalence of the inverse circular harmonic transform, the inverse Radon transform, and the inverse Cormack transform is presented. The use of the number of nodes of a projection harmonic as a filter is discussed. Numerical methods for the efficient implementation of angular harmonic algorithms based on orthogonal functions and stable recursion are presented. The derivation of a lower bound for the signal-to-noise ratio of the Cormack algorithm is derived.

Sun, Deqing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 86-91). / Abstracts in English and Chinese. / Abstract --- p.i / 摘要 --- p.iii / Contributions --- p.iv / Acknowledgement --- p.vi / Abbreviations --- p.xviii / Notations --- p.xxi / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Image compression and postprocessing --- p.1 / Chapter 1.2 --- A brief review of postprocessing --- p.3 / Chapter 1.3 --- Objective and methodology of the research --- p.7 / Chapter 1.4 --- Thesis organization --- p.8 / Chapter 1.5 --- A note on publication --- p.9 / Chapter 2 --- Background Study --- p.11 / Chapter 2.1 --- Image models --- p.11 / Chapter 2.1.1 --- Minimum edge difference (MED) criterion for block boundaries --- p.12 / Chapter 2.1.2 --- van Beek's edge model for an edge --- p.15 / Chapter 2.1.3 --- Fields of experts (FoE) for an image --- p.16 / Chapter 2.2 --- Degradation models --- p.20 / Chapter 2.2.1 --- Quantization constraint set (QCS) and uniform noise --- p.21 / Chapter 2.2.2 --- Narrow quantization constraint set (NQCS) --- p.22 / Chapter 2.2.3 --- Gaussian noise --- p.22 / Chapter 2.2.4 --- Edge width enlargement after quantization --- p.25 / Chapter 2.3 --- Use of these models for postprocessing --- p.27 / Chapter 2.3.1 --- MED and edge models --- p.27 / Chapter 2.3.2 --- The FoE prior model --- p.27 / Chapter 3 --- Postprocessing using MED and edge models --- p.28 / Chapter 3.1 --- Blocking artifacts suppression by coefficient restoration --- p.29 / Chapter 3.1.1 --- AC coefficient restoration by MED --- p.29 / Chapter 3.1.2 --- General derivation --- p.31 / Chapter 3.2 --- Detailed algorithm --- p.34 / Chapter 3.2.1 --- Edge identification --- p.36 / Chapter 3.2.2 --- Region classification --- p.36 / Chapter 3.2.3 --- Edge reconstruction --- p.37 / Chapter 3.2.4 --- Image reconstruction --- p.37 / Chapter 3.3 --- Experimental results --- p.38 / Chapter 3.3.1 --- Results of the proposed method --- p.38 / Chapter 3.3.2 --- Comparison with one wavelet-based method --- p.39 / Chapter 3.4 --- On the global minimum of the edge difference . . --- p.41 / Chapter 3.4.1 --- The constrained minimization problem . . --- p.41 / Chapter 3.4.2 --- Experimental examination --- p.42 / Chapter 3.4.3 --- Discussions --- p.43 / Chapter 3.5 --- Conclusions --- p.44 / Chapter 4 --- Postprocessing by the MAP criterion using FoE --- p.49 / Chapter 4.1 --- The proposed method --- p.49 / Chapter 4.1.1 --- The MAP criterion --- p.49 / Chapter 4.1.2 --- The optimization problem --- p.51 / Chapter 4.2 --- Experimental results --- p.52 / Chapter 4.2.1 --- Setting algorithm parameters --- p.53 / Chapter 4.2.2 --- Results --- p.56 / Chapter 4.3 --- Investigation on the quantization noise model . . --- p.58 / Chapter 4.4 --- Conclusions --- p.61 / Chapter 5 --- Conclusion --- p.71 / Chapter 5.1 --- Contributions --- p.71 / Chapter 5.1.1 --- Extension of the DCCR algorithm --- p.71 / Chapter 5.1.2 --- Examination of the MED criterion --- p.72 / Chapter 5.1.3 --- Use of the FoE prior in postprocessing . . --- p.72 / Chapter 5.1.4 --- Investigation on the quantization noise model --- p.73 / Chapter 5.2 --- Future work --- p.73 / Chapter 5.2.1 --- Degradation model --- p.73 / Chapter 5.2.2 --- Efficient implementation of the MAP method --- p.74 / Chapter 5.2.3 --- Postprocessing of compressed video --- p.75 / Chapter A --- Detailed derivation of coefficient restoration --- p.76 / Chapter B --- Implementation details of the FoE prior --- p.81 / Chapter B.1 --- The FoE prior model --- p.81 / Chapter B.2 --- Energy function and its gradient --- p.83 / Chapter B.3 --- Conjugate gradient descent method --- p.84 / Bibliography --- p.86

A Discrete Fourier Transform (DFT) changes the basis of a group algebra from the standard basis to a Fourier basis. An efficient application of a DFT is called a Fast Fourier Transform (FFT). This research pertains to a particular type of FFT called Decimation in Frequency (DIF). An efficient DIF has been established for commutative algebra; however, a successful analogue for non-commutative algebra has not been derived. However, we currently have a promising DIF algorithm for CSn called Orrison-DIF (ODIF). In this paper, I will formally introduce the ODIF and establish a bound on the operation count of the algorithm.

No description available.