Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.published_or_final_versio

Some fast algorithms in signal and image processing.

Kwok-po Ng.Thesis (Ph.D.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 138-139).AbstractsSummaryIntroduction --- p.1Summary of the papers A-F --- p.2Paper A --- p.15Paper B --- p.36Paper C --- p.63Paper D --- p.87Paper E --- p.109Paper F --- p.12

Image reconstruction with multisensors.

by Wun-Cheung Tang.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references.Abstract also in Chinese.Abstracts --- p.1Introduction --- p.3Toeplitz and Circulant Matrices --- p.3Conjugate Gradient Method --- p.6Cosine Transform Preconditioner --- p.7Regularization --- p.10Summary --- p.13Paper A --- p.19Paper B --- p.3

Discrete sine and cosine transforms on parallel processors

Starting point of this master thesis is Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) algorithms for signal processing. Based on the number system used in DCT and DST application, they can be categorized as fixed-point and floating-point DCT/DST. Floating-point numbers have large dynamic range to represent very large and small numbers. However, floating-point operation requires more clock cycles than fixed-point operation. Specialized hardware can be used for floating-point operations for high performance, but it also increases hardware cost. So, for general applications, use of fixed-point number system would be a good choice provided that an optimum accuracy is guaranteed. In this thesis, the existing floating–point DCT and DST of type-1 C-codes are first converted into fixed-point code. The fractional fixed-point representation is used for the fixed-point conversion for maximum possible accuracy. The choice of Q15 format provides highest precision for signed 16-bit fixed-point number. But in this format, the range of numbers has to be normalized between [-1, 1]. The conversion process introduces some error in the output which is calculated by signal to noise ratio (SNR). After designing the fixed-point DCT/DST code, the performance is evaluated in various Tensilica processor configurations. The configurations provided are generated for Tensilica’s Diamond Standard Processor cores in Tensilica Xtensa Environment. The clock cycle counts of both fixed-point and floating-point DCT/DST code on four different configurations are recorded. The results show that SNR of fixed-point DCT/DST is between (35-76dB) for different transform size of DCT/DST, which suggests that the fixed-point code is accurate enough. It is also observed that the fixed-point DCT/DST provides approximately 3 to 6 times performance improvement over floating-point code on Tensilica processors cores in terms of clock cycles. Furthermore, Tensilica’s Diamond Standard 570T parallel processor configuration provides the best performance among all configurations used for designed fixed-point code. Results have shown that the fixed-point DCT/DST code offers a large performance improvement over floating-point code provided that the floating-point code has no added hardware support

Symmetric Cauchy-like Preconditioners for the Regularized Solution of 1-D Ill-Posed Problems

The discretization of integral equations can lead to systems involving symmetric Toeplitz matrices. We describe a preconditioning technique for the regularized solution of the related discrete ill-posed problem. We use discrete sine transforms to transform the system to one involving a Cauchy-like matrix. Based on the approach of Kilmer and O'Leary, the preconditioner is a symmetric, rank $m^{*}$ approximation to the Cauchy-like matrix augmented by the identity. We shall show that if the kernel of the integral equation is smooth then the preconditioned matrix has two desirable properties; namely, the largest $m^{*}$ magnitude eigenvalues are clustered around and bounded below by one, and that small magnitude eigenvalues remain small. We also show that the initialization cost is less than the initialization cost for the preconditioner introduced by Kilmer and O'Leary. Further, we describe a method for applying the preconditioner in $O((n+1) \lg (n+1))$ operations when $n+1$ is a power of 2, and describe a variant of the MINRES algorithm to solve the symmetrically preconditioned problem. The preconditioned method is tested on two examples

Design and Optimization of Graph Transform for Image and Video Compression

The main contribution of this thesis is the introduction of new methods for designing adaptive transforms for image and video compression. Exploiting graph signal processing techniques, we develop new graph construction methods targeted for image and video compression applications. In this way, we obtain a graph that is, at the same time, a good representation of the image and easy to transmit to the decoder. To do so, we investigate different research directions. First, we propose a new method for graph construction that employs innovative edge metrics, quantization and edge prediction techniques. Then, we propose to use a graph learning approach and we introduce a new graph learning algorithm targeted for image compression that defines the connectivities between pixels by taking into consideration the coding of the image signal and the graph topology in rate-distortion term. Moreover, we also present a new superpixel-driven graph transform that uses clusters of superpixel as coding blocks and then computes the graph transform inside each region. In the second part of this work, we exploit graphs to design directional transforms. In fact, an efficient representation of the image directional information is extremely important in order to obtain high performance image and video coding. In this thesis, we present a new directional transform, called Steerable Discrete Cosine Transform (SDCT). This new transform can be obtained by steering the 2D-DCT basis in any chosen direction. Moreover, we can also use more complex steering patterns than a single pure rotation. In order to show the advantages of the SDCT, we present a few image and video compression methods based on this new directional transform. The obtained results show that the SDCT can be efficiently applied to image and video compression and it outperforms the classical DCT and other directional transforms. Along the same lines, we present also a new generalization of the DFT, called Steerable DFT (SDFT). Differently from the SDCT, the SDFT can be defined in one or two dimensions. The 1D-SDFT represents a rotation in the complex plane, instead the 2D-SDFT performs a rotation in the 2D Euclidean space