Two-dimensional block transforms and their properties

Includes bibliographical references.For two-dimensional (2-D) digital filters implemented by a block recursive equation, explicit relations between their frequency characteristics and those of scalar filter are obtained. Specifically, these include the relation between the discrete-time Fourier transform (DTFT) of the block recursive equation and that of the scalar 2-D difference equation, and the relation between the block matrix transfer function of the block processor and the scalar transfer function. These relations that are independent of the type of realization of the block processor have been obtained using the eigenvalue properties of a special type of circulant matrix introduced in this correspondence

Sparse Fast Trigonometric Transforms

Trigonometric transforms like the Fourier transform or the discrete cosine transform (DCT) are of immense importance in signal and image processing, physics, engineering, and data processing. The research of past decades has provided us with runtime optimal algorithms for these transforms. Significant runtime improvements are only possible if there is additional a priori information about the sparsity of the signal. In the first part of this thesis we develop sublinear algorithms for the fast Fourier transform for frequency sparse periodic functions. We investigate three classes of sparsity: short frequency support, polynomially structured sparsity and block sparsity. For all three classes we present new deterministic, sublinear algorithms that recover the Fourier coefficients of periodic functions from samples. We prove theoretical runtime and sampling bounds for all algorithms and also investigate their performance in numerical experiments. In the second part of this thesis we focus on the reconstruction of vectors with short support from their DCT of type II. We present two different new, deterministic and sublinear algorithms for this problem. The first method is based on inverse discrete Fourier transforms and uses complex arithmetic, whereas the second one utilizes properties of the DCT and only employs real arithmetic. We show theoretical runtime and sampling bounds for both algorithms and compare them numerically in experiments. Furthermore, we generalize the techniques for recovering vectors with short support from their DCT of type II using only real arithmetic to the two-dimensional setting of recovering matrices with block support, also providing theoretical runtime and sampling complexities for the obtained new two-dimensional algorithm

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a $4\times 4$ Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.Comment: 13 pages plus 7 figures (PNG