Mathematics of Multidimensional Fourier Transform Algorithms

1. Front Matter

   Pages i-xiv

   PDF

2. Tensor Product

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 1-28

3. Multidimensional Tensor Product and FFT

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 29-43

4. Finite Abelian Groups

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 45-62

5. Fourier Transform of Finite Abelian Groups

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 63-76

6. Cooley—Tukey and Good—Thomas

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 77-87

7. Lines

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 89-113

8. Duality of Lines and Planes

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 115-133

9. Reduced Transform Algorithms

   • Richard Tolimieri, Myoung An, Chao Lu

   Pages 135-157

10. Field Algorithm

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 159-177

11. Implementation on RISC Architectures

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 179-202

12. Implementation on Parallel Architectures

    • Richard Tolimieri, Myoung An, Chao Lu

    Pages 203-229

13. Back Matter

    Pages 231-233

    PDF

The Fourier transform of large multidimensional data sets is an essen­ tial computation in many scientific and engineering fields, including seismology, X-ray crystallography, radar, sonar and medical imaging. Such fields require multidimensional arrays for complete and faithful modelling. Classically, a set of data is processed one dimension at a time, permitting control over the size of the computation and calling on well-established I-dimensional programs. The rapidly increasing availability of powerful computing chips, vector processors, multinode boards and parallel machines has provided new tools for carrying out multidimensional computations. Multidimensional processing offers a wider range of possible implementations as compared to I-dimensional the greater flexibility of movement in the data in­ processing, due to dexing set. This increased freedom, along with the massive size data sets typically found in multidimensional applications, places intensive demands on the communication aspects of the computation. The writ­ ing of code that takes into account all the algorithmic possibilities and matches these possibilities to the communication capabilities of the tar­ get architecture is an extremely time-consuming task. A major goal of this text is to provide a sufficiently abstra

• algorithms
• fast Fourier transform (FFT)
• image processing
• algorithm analysis and problem complexity

• Aware, Inc., Cambridge, USA

  Richard Tolimieri, Myoung An

• Dept. of Computer and Information Sciences, Towson State University, Towson, USA

  Chao Lu

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