Mathematics of Multidimensional Fourier Transform Algorithms

1. Front Matter

   Pages i-xi

   PDF

2. Tensor Product

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 1-23

3. Multidimensional Tensor Product and FFT

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 25-36

4. Finite Abelian Groups

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 37-50

5. Fourier Transform of Finite Abelian Groups

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 51-61

6. Cooley—Tukey and Good—Thomas

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 63-69

7. Lines

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 71-89

8. Duality of Lines and Planes

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 91-104

9. Reduced Transform Algorithms

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 105-124

10. Field Algorithm

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 125-140

11. Implementation on RISC Architectures

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 141-160

12. Implementation on Parallel Architectures

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 161-183

13. Back Matter

    Pages 185-187

    PDF

Fourier transforms of large multidimensional data sets arise in many fields --ranging from seismology to medical imaging. The rapidly increasing power of computer chips, the increased availability of vector and array processors, and the increasing size of the data sets to be analyzed make it both possible and necessary to analyze the data more than one dimension at a time. The increased freedom provided by multidimensional processing, however, also places intesive demands on the communication aspects of the computation, making it difficult to write code that takes all the algorithmic possiblities into account and matches these to the target architecture. This book develops algorithms for multi-dimensional Fourier transforms that yield highly efficient code on a variety of vector and parallel computers. By emphasizing the unified basis for the many approaches to one-dimensional and multidimensional Fourier transforms, this book not only clarifies the fundamental similarities, but also shows how to exploit the differences in optimizing implementations. This book will be of interest not only to applied mathematicians and computer scientists, but also to seismologists, high-energy physicists, crystallographers, and electrical engineers working on signal and image processing. Topics covered include: tensor products and the fast Fourier transform; finite Abelian groups and their Fourier transforms; Cooley- Tukey and Good-Thomas algorithms; lines and planes; reduced transform algorithms; field algorithms; implementation on Risc and parallel

• Fourier transform
• algorithms
• computer
• fast Fourier transform
• fast Fourier transform (FFT)
• image processing
• mathematics
• sets

• Department of Electrical and Computer Engineering, Rice University, Houston, USA

  C. S. Burrus

• Department of Electrical Engineering, City College of CUNY, New York, USA

  Richard Tolimieri

• A.J. Devaney Associates, Allston, USA

  Myoung An

• Department of Computer and Information Sciences, Towson State University, Towson, USA

  Chao Lu

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