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Split-radix FFT algorithm
About: Split-radix FFT algorithm is a research topic. Over the lifetime, 1845 publications have been published within this topic receiving 41398 citations.
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TL;DR: This coherent detection algorithm can detect high-speed targets without the brute-force searching of unknown motion parameters and achieve a good balance between the computational cost and the antinoise performance.
Abstract: In this paper, by employing the symmetric autocorrelation function and the scaled inverse Fourier transform (SCIFT), a coherent detection algorithm is proposed for high-speed targets. This coherent detection algorithm is simple and can be easily implemented by using complex multiplications, the fast Fourier transform (FFT) and the inverse FFT (IFFT). Compared to the Hough transform and the keystone transform, this coherent detection algorithm can detect high-speed targets without the brute-force searching of unknown motion parameters and achieve a good balance between the computational cost and the antinoise performance. Through simulations and analyses for synthetic models and the real data, we verify the effectiveness of the proposed coherent detection algorithm.
102 citations
09 May 1977
TL;DR: A new radix-2 two-dimensional direct FFT developed by Rivard is generalized in this paper to include arbitrary radices and non-square arrays and it is shown that the Radix-4 version of this algorithm may require significantly fewer computations than conventional row-column transform methods.
Abstract: A new radix-2 two-dimensional direct FFT developed by Rivard is generalized in this paper to include arbitrary radices and non-square arrays. It is shown that the radix-4 version of this algorithm may require significantly fewer computations than conventional row-column transform methods. Also, the new algorithm eliminates the matrix transpose operation normally required when the array must reside on a bulk storage device. It requires the same number of passes over the array on bulk storage as efficient matrix transpose routines, but produces the transform in bit-reversed order. An additional pass over the data is necessary to sort the array if normal ordering is desired.
99 citations
TL;DR: A series-expansion approach and an operator framework are used to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction that is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small.
Abstract: We use a series-expansion approach and an operator framework to derive a new, fast, and accurate Fourier algorithm for iterative tomographic reconstruction. This algorithm is applicable for parallel-ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small. The conjugate gradient (CG) algorithm is used to minimize a regularized, spectrally weighted least-squares criterion, and we prove that the main step in each iteration is equivalent to a 2-D discrete convolution, which can be cheaply and exactly implemented via the fast Fourier transform (FFT). The proposed algorithm requires O(N/sup 2/logN) floating-point operations per iteration to reconstruct an N/spl times/N image from P view angles, as compared to O(N/sup 2/P) floating-point operations per iteration for iterative convolution-backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem. Numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse- and limited-angle tomography under realistic sampling scenarios. Although the proposed algorithm cannot explicitly account for noise with nonstationary statistics, additional simulations demonstrate that for low to moderate levels of nonstationary noise, the quality of reconstruction is almost unaffected by assuming that the noise is stationary.
97 citations
TL;DR: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest, but these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect.
Abstract: Fast Fourier transform (FFT)-based computations can be far more accurate than the slow transforms suggest. Discrete Fourier transforms computed through the FFT are far more accurate than slow transforms, and convolutions computed via FFT are far more accurate than the direct results. However, these results depend critically on the accuracy of the FFT software employed, which should generally be considered suspect. Popular recursions for fast computation of the sine/cosine table (or twiddle factors) are inaccurate due to inherent instability. Some analyses of these recursions that have appeared heretofore in print, suggesting stability, are incorrect. Even in higher dimensions, the FFT is remarkably stable.
96 citations
TL;DR: This paper deals with two's complement arithmetic with either rounding or chopping with eitherRoundoff errors for radix-2 FFT's and mixed-radix FFTs.
Abstract: A statistical model for roundoff errors is used to predict the output noise of the two common forms of the fast Fourier transform (FFT) algorithm, the decimations in-time and in-frequency. This paper deals with two's complement arithmetic with either rounding or chopping. The total mean-square errors and the mean-square errors for the individual points are derived for radix-2 FFT's. Results for mixed-radix FFT are also given.
93 citations