Myoung An

Algorithms for discrete fourier transform and convolution

Contents: Introduction to Abstract Algebra.- Tensor Product and Stride Permutation.- Cooley-Tukey FFF Algorithms.- Variants of FFT Algorithms and Their Implementations.- Good-Thomas PFA.- Linear and Cyclic Convolutions.- Agarwal-Cooley Convolution Algorithm.- Introduction to Multiplicative Fourier Transform Algorithms (MFTA).- MFTA: The Prime Case.- MFTA: Product of Two Distinct Primes.- MFTA: Transform Size N = Mr. M-Composite Integer and r-Prime.- MFTA: Transform Size N = p2.- Periodization and Decimation.- Multiplicative Character and the FFT.- Rationality.- Index.

An introduction to abstract algebra

In this and the next chapters, we present several mathematical results needed to design the algorithms of the text. We assume that the reader has some knowledge of groups, rings and vector spaces but no extensive knowledge is required. Instead, we focus on those mathematical objects which will be used repeatedly in this text.

Self-sorting in-place FFT algorithm with minimum working space

Presents a modification of Tempertons (1991) self-sorting, in-place radix-p FFT algorithm. This modification reduces the required temporary working space from order of p/sup 2/ to p+1, providing a better match to the limited number of registers in a CPU. >

Mathematics of multidimensional Fourier transform algorithms

The main emphasis of this book is the development of algorithms for processing multi-dimensional digital signals, particularly algorithms for multi-dimensional Fourier transforms, in a form that is convenient for writing highly efficient code on a variety of vector and parallel computers. The rapidly increasing power of computing chips, the increased availability of vector and array processors, and the increased size of data sets to be analyzed make code-writing a difficult task. By emphasizing the unified basis for the various approaches to multidimensional Fourier transforms, this book also clarifies how to exploit the differences in optimizing implementations.

A new approach for computing multidimensional DFT's on parallel machines and its implementation on the iPSC/860 hypercube

Proposes a new approach for computing multidimensional DFTs that reduces interprocessor communications and is therefore suitable for efficient implementation on a variety of multiprocessor platforms including MIMD supercomputers and clusters of workstations. Group theoretic concepts are used to formulate a flexible computational strategy that hybrids the reduced transform algorithm (RTA) with the Good-Thomas factorization and can deal efficiently with non-power-of-two sizes without resorting to zero-padding. The RTA algorithm is employed not as a data processing but rather as a bookkeeping tool in order to decompose the problem into many smaller size subproblems (lines) that can be solved independently by the processors. Implementation issues on an Intel iPSC/i860 hypercube are discussed and timing results for large 2D and 3D DFTs with index sets in Z/MP/spl times/Z/KP and Z/N/spl times/Z/MP/spl times/Z/KP respectively are provided, where N, M, K are powers-of-two and P is a small prime number such as 3, 5, or 7. The nonoptimized realizations of the new hybrid RTA approach are shown to outperform by as much as 70% the optimized assembly coded realizations of the traditional row-column method on the iPSC/860. >

Group Filters and Image Processing

Abelian group DSP can be completely described in terms of a special class of signals, the characters, defined by their relationship to the translations defined by abelian group multiplication. The first problem to be faced in extending classical DSP theory is to decide on what is meant by a translation. We have selected certain classes of nonabelian groups and defined translations in terms of left nonabelian group multiplications.

Data restoration in chromotomographic hyperspectral imaging

Recently, a new approach to hyperspectral imaging, relying on the theory of computed tomography, was proposed by researchers at the Air Force Research Laboratory. The approach allows all photons to be recorded and therefore increases robustness of the imaging system to noise and focal plane array non-uniformities. However, as all computed tomography systems, the approach suffers form the limited angle problem, which obstructs reconstruction of the hyperspectral information. In this work we present a direct, one-step algorithm for reconstruction of the unknown information based on a priori knowledge about the hyperspectral image.

A Group Filter Algorithm for Sea Mine Detection

Automatic detection of sea mines in coastal regions is a difficult task due to the highly variable sea bottom conditions present in the underwater environment. Detection systems must be able to discriminate objects which vary in size, shape, and orientation from naturally occurring and man-made clutter. Additionally, these automated systems must be computationally efficient to be incorporated into unmanned underwater vehicle (UUV) sensor systems characterized by high sensor data rates and limited processing abilities. Using noncommutative group harmonic analysis, a fast, robust sea mine detection system is created. A family of unitary image transforms associated to noncommutative groups is generated and applied to side scan sonar image files supplied by Naval Surface Warfare Center Panama City (NSWC PC). These transforms project key image features, geometrically defined structures with orientations, and localized spectral information into distinct orthogonal components or feature subspaces of the image. The performance of the detection system is compared against the performance of an independent detection system in terms of probability of detection (Pd) and probability of false alarm (Pfa).

Pons, Reed-Muller Codes, and Group Algebras

In this work we develop the family of Prometheus orthonormal sets (PONS) in the framework of certain abelian group algebras. Classical PONS, introduced in 1991 by J. S. Byrnes, will be constructed using group algebra operations. This construction highlights the fundamental role played by group characters in the theory of PONS. In particular, we will relate classical PONS to idempotent systems in group algebras and show that signal expansions over classical PONS correspond to multiplications in the group algebra.

Poisson summation formula

The Poisson summation (PS) formula describes the fundamental duality between periodization and decimation operators under the Fourier transform. In this chapter, the finite abelian group version of the PS formula is derived as a simple application of the character formulas of Chapter 3. The general case is equally simple to prove, but special care must be taken to provide conditions for convergence. The power of the PS formula is not so much with the formula itself, but rather that it highlights a construction that is basic to many derivations and algorithms in pure and applied mathematics and engineering. Common to these applications is the importance of periodization and of computing the Fourier transform of periodizations. In algebraic and analytic number theory this computation results in closed form expressions, the transformation equations for theta and zeta functions, and other important number theoretic special functions.