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The Fast Octa-Polar Fourier Transform and its expansion to an accurate discrete radon transform

2. Outline. Main result and the relation to Radon TransformRectilinear DFT: General Definition and Properties The Pseudo-Polar FFT (PPFFT)The FFFT and it's relation to structured matricesThe Octa-Polar FFTSummary and on-going work . 3. 2D Fast Fourier Transforms. The computational cost for a si

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The Fast Octa-Polar Fourier Transform and its expansion to an accurate discrete radon transform

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    1. The Fast Octa-Polar Fourier Transform and its expansion to an accurate discrete radon transform Ofer Levi

    2. 2 Outline Main result and the relation to Radon Transform Rectilinear DFT: General Definition and Properties The Pseudo-Polar FFT (PPFFT) The FFFT and its relation to structured matrices The Octa-Polar FFT Summary and on-going work

    3. 3 2D Fast Fourier Transforms

    4. 4 Main Result a new, almost-polar FFT The Octa-Polar FFT

    5. 5 Importance of Polar DFT

    6. 6 Relation between Radon and Fourier Transforms

    7. 7 Relation between Radon and Fourier Transforms

    8. 8 Approximated Polar DFTs

    9. 9 Approximated Polar DFTs

    10. 10 1D DFT: General Definition and Properties

    11. 11 1D DFT: Computability Direct evaluation of the 1D DFT costs o(n2)

    12. 12 Example Spectral Decomposition

    13. 13 Example Spectral Decomposition

    14. 14 Example - Denoising

    15. 15 2D DFT Cartesian Grid

    16. 16 2D complex exponents

    17. 17 2D FFT

    18. 18 Applications of rectilinear 2D FFT Spectral Analysis

    19. 19 Polar DFT

    20. 20 Polar DFT Direct Polar DFT is impractical o(n4) and no direct inverse

    21. 21 The Pseudo-Polar FFT (PPFFT) (Donoho et. al.)

    22. 22 The Pseudo Polar FFT

    23. 23 Fractional FFT Algorithm (D. Bailey and P. Swarztrauber 1990)

    24. 24 Some basic facts about Toeplitz Matrices

    25. 25 FFFT and Structured Matrices

    26. 26 The PPFFT Algorithm

    27. 27 The PPFFT Matrix notation A can be implicitly applied in O(Nlog(N)) operations

    28. 28 Inverse PPFFT

    29. 29 Weighted PPFFT

    30. 30 The Slow Slant-Stack Transform

    31. 31 The Fast Slant-Stack Algorithm

    32. 32 The Fast Slant-Stack transform

    33. 33

    34. 34 Treating The NW/SW and NE/SE grid points sets

    35. 35 Summary and Future research

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