Richard Tolimieri

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.

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.

Computing decimated finite cross-ambiguity functions

A fast algorithm computing decimated finite cross-ambiguity functions is designed, which is tailored for implementation on parallel computer architectures. Several standard methods carry out cross-ambiguity function computations in two stages, a coarse mode which estimates the location of peaks, and a fine mode relying on FFT algorithms which produces greater detail. The coarse mode computation is not required, but only decimated values ranging over the entire domain of the cross-ambiguity function are computed. >

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. >

FFT algorithms for prime transform sizes and their implementations on VAX, IBM3090VF, and IBM RS/6000

Variants of the Winograd fast Fourier transform (FFT) algorithm for prime transform size that offer options as to operational counts and arithmetic balance are derived. Their implementations on VAX, IBM 3090 VF, and IBM RS/6000 are discussed. For processors that perform floating-point addition, floating-point multiplication, and floating-point multiply-add with the same time delay, variants of the FFT algorithm have been designed such that all floating-point multiplications can be overlapped by using multiply-add. The use of a tensor product formulation, throughout, gives a means for producing variants of algorithms matching computer architectures. >

Factorization method for crystallographic Fourier transforms

Determining the structure of a crystal by X-ray methods requires repeated computation of the three-dimensional Fourier transform. Over the last few years, algorithms for computing finite Fourier transforms that take advantage of various crystal symmetries have been developed. These algorithms are efficient especially when the sampling space contains a prime number of points in each coordinate direction. In this work, we present a method of combining programs for two relatively prime integers p and q to obtain a program for sampling space containing p . q number of points in each coordinate direction.

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).