Abstract In this paper we present a parallelization of the Cooley- Tukey FFT algorithm that is implemented on a shared-memory MIMD (non-vector) machine that was built in the Dept. of Computer Science, Tel Aviv University. A parallel algorithm is presented for one dimension Fourier transform with performance analysis. For a large array of complex numbers to be transformed, an almost linear speed-up is demonstrated. This algorithm can be executed by any number of processors, but generally the number is much less than the length of the input data.

A Parallel FFT on an MIMD Machine.

In this paper we present a parallelization of the Cooley- Tukey FFT algorithm that is implemented on a shared-memory MIMD (non-vector) machine that was built in the Dept. of Computer Science, Tel Aviv University. A parallel algorithm is presented for one dimension Fourier transform with performance analysis. For a large array of complex numbers to be transformed, an almost linear speed-up is demonstrated. This algorithm can be executed by any number of processors, but generally the number is much less than the length of the input data.

A parallel FFT on an MIMD machine

Key references in distributed computer systems 1959–1989

Nuclear Magnetic Resonance (NMR) imaging, also known as Magnetic Resonance Imaging (MRI) is a non-invasive diagnostic technique used in the medical field for body scanning and is a sensitive method to look at the soft tissue. MRI is also used in the imaging of solids in order to study their internal structure. The non-ionising nature of the radiation used in this imaging technique allows the objects to be scanned frequently. One consequence is the generation of large amounts of data to be processed within given time constraints. Once the data is generated from a scanner, it has to be manipulated to recover the original information in the form of a reconstructed image. The developments in the field of Very Large Scale Integrated (VLSI) circuit design and in computing science in the last two to three decades have made significant contributions to the data processing part of imaging. Avenues have yet to be explored as many of the problems posed by physical systems are complex and are not computable in real time, because of their complexity or technological constraints. The past decade has seen many developments in faster processors, specialised processors and low and high level concurrent and distributed systems. One of the major advances has been the development of the transputer and within the last decade many applications have been implemented on transputer based systems. The objective of this thesis is to present one such application with respect to the image reconstruction problem. Issues addressed are the implementation of the algorithm on a concurrent system, flexibility of the algorithm as a function of computing nodes and the development of multi-dimensional reconstruction. A distributed Radix-2 Fast Fourier Transform has been developed and applied in a transputer based environment to the problem of reconstruction of images generated from a full body Nuclear Magnetic Resonance scanner. A transputer ring network of one host and four slaves was used for the development The algorithm was also tested on a sixteen processor system and the scalability of the algorithm assessed. The reconstruction times and memory usage were measured as functions of processor nodes. It is shown that the performance scales as expected theoretically. The interprocessor communications time on the network was measured and the resulting limitations investigated. The limitations imposed by the algorithm are discussed. The reconstruction algorithm was extended to handle three spatial dimensional data. Two methods were used, a three dimensional reconstruction by stacking of two dimensional slices and a three dimensional Fourier Transform technique. These algorithms were implemented on the multi-transputer systems. The limitations of the methods are discussed and the advantages are presented. Finally, a general overview of the constraints of the reconstruction schemes that were developed is given and methods of removing them are suggested. The problem of slice motion in the form of translation and rotation is discussed with reference to the three dimensional stacking reconstruction. Some methods for solving this problem in the concurrent environment are discussed and techniques for their implementation in the present context suggested. The advances in the practical implementation of three dimensional scanning techniques are discussed and their relevance to the concurrent computer systems and the developed algorithms are presented, along with other developments in the field in of parallel computing.

S. Mai

Papers

A parallel algorithm for the Fast Fourier Transform