Direct Inversion of the Three-Dimensional Pseudo-polar Fourier Transform

Abstract

The pseudo-polar Fourier transform is a specialized non-equally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid.The advantage of the pseudo-polar grid over other non-uniform sampling geometries is that the transformation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative.The convergence speed of these algorithms as well as their accuracy are difficult to control, as they depend both on the sampling geometry as well as on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations, and is shown to be significantly faster than existing iterative inversion algorithms.

Low-dose cone-beam computed tomography reconstruction through a fast three-dimensional compressed sensing method based on the three-dimensional pseudo-polar fourier transform

“…PPFT has three significant properties making it an ideal substitute to conventional Fourier methods: (1) PPFT provides an exact PP-based FFT and its inverse, setting the gridding error to zero, (2) PPFT has a fast-forward and a fast-backward calculation algorithm,[ 23 28 45 ] enabling our proposed algorithm to avoid the regridding step used in iterative non-Cartesian Fourier-based reconstruction methods, and (3) PPFT has an analytical inverse function. [ 45 ]…”

“…In each iteration, there are three operators with the highest computational cost: the PPFT , its inverse operator, T and the FCSA solver. The computational complexity of [ 23 28 ] and T [ 28 45 ] has been proven to be O(N 3 logN).…”

Direct inversion of the nonequispaced fast Fourier transform

“…Another approach for computing an inverse NFFT in the setting N ≥ M can be obtained by using the fact that A * A is of Toeplitz structure. To this end, the Gohberg-Semencul formula, see [18], can be used to solve the normal equations A * A f = A * f exactly by analogy with [3]. Therefore, the inverse NFFT consists of two steps: an adjoint NFFT applied to f and the multiplication with the inverse of A * A, which can be realized by means of 8 FFTs.…”

“…However, even if this algorithm is exact and therefore yields better results than our approach from Section 3.2.2 there is no exact generalization to higher dimensions since there is no generalization of the Gohberg-Semencul formula to dimensions > 1. This approach can only be used for approximations on very special grids, such as linogramm grids, utilizing the given specific structure, see [2,3]. second experiment we fix N = 2048 and M = 512 and the cut-off parameter m shall be chosen m = c with c = 4, .…”

“…Based on this, in [28] another direct method was deduced for the same setting which also uses Lagrange interpolation but additionally incorporates an imaginary shift for the fast evaluation of occurring sums. Since A * A is a Toeplitz matrix another direct method for the overdetermined setting can be derived using this special structure, see [18], analogously to [3]. In addition, also a frame-theoretical approach is known from [15] which provides a link between the adjoint NFFT and frame approximation and could therefore be seen as a method to invert the NFFT.…”