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This all is based on Fourier slice theorem: 1D FFT of projection taken at
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some angle equals the central radial slice at the same angle of the 2D FFT
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of the original object.
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The center of rotation is considered to be in the center of axis. Then we
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padding both ends with zeroes (the background color have to be zero). And
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exchanging left and right halves.
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- For DFT, we assume that signal is cyclic and we sample values from 0
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to N. But we actually have them from -N/2 to N/2 (because the center of
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rotation is in the center of axis). Therefore, we need to move first half
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of array after second half.
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The main problem is interpolation in Fourier space.
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- Operation on real data gives Hermetian (self-adjoint) matrix in Fourier space.
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A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex
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entries that is equal to its own conjugate transpose (A*). That is, the element
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in the i-th row and j-th column is equal to the complex conjugate of the element
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in the j-th row and i-th column (i.e. with sign of imaginary part changed).
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- cuFFT have implemented Cooley-Tukey algorith for radixes 2,3,5,7. The best
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performance still with large powers of two, but we should check balance
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between large padding and usage of sub-optimal radixes. Accroding to cuFFT
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documentation, the cuFFT performance is constrained by following restrictions
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in the order of decreasing importance:
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* Restrict the size along each dimension to use fewer distinct prime factors.
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* Radix2 part should be multiple of 256
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* Restrict the x dimension of single-precision transforms to be strictly a
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power of two either between 2 and 8192 for Fermi-class GPUs or between 2
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and 2048 for earlier architectures.
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* Use Native compatibility mode for in-place complex-to-real or
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real-to-complex transforms.
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- 2D FFT is series of independent 1D FFT transforms on rows and columns
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