## B. Non-Uniform Fast Fourier Transform C. Fourier-Based Iterative Reconstruction D. Emulation of Image Representation Using Basis Functions (2003)

### BibTeX

@MISC{Matej03b.non-uniform,

author = {Samuel Matej and Jeffrey A. Fessler and Ivan G. Kazantsev},

title = {B. Non-Uniform Fast Fourier Transform C. Fourier-Based Iterative Reconstruction D. Emulation of Image Representation Using Basis Functions},

year = {2003}

}

### OpenURL

### Abstract

### Citations

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Citation Context ... �¥��������� ��� � � � ���¥��������� ��� ��� ��� ����� � � ����� ��������� � ��� (9) for and value zero for , where is the distance from the KB kernel center, � denotes the modified � Bessel function =-=[32]-=- of order � , � is the size of the KB window and � is a parameter controlling the KB window shape and frequency characteristics [29, 30] (see examples of KB window functions and of their spectra in Fi... |

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Citation Context ...as , where . Let denote the - dimensional image whose projections (the ray transform) we wish to compute. Let be the image spectrum, obtained by Fourier transform (2) Then the central section theorem =-=[11]-=- is given by (3) where � ������� � ������� ��������� � � ������� � � ��� � ����� � � � � � ��� � � ����� ��������� � ����� � ��� ������� � ��������� ��� � � ����� � (4) � image spectrum , , distribute... |

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Citation Context ...e crucial step influencing the reconstruction quality and speed is the interpolation between polar and Cartesian rasters in frequency space. Gridding interpolation [18, 19], with proper interpolating =-=[20]-=- and data weighting functions, as investigated in the MRI literature [21–23], brought improvement in the direct Fourier reconstruction. Recently, the Fourier-based reprojection has been applied for (n... |

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Citation Context ...nsform (NUFFT) [26] used in this paper is related to gridding methods for interpolation in frequency space. The KB interpolation kernels used in this work have been optimized using a min-max approach =-=[27]-=-, thus providing substantial improvement of the interpolation accuracy. In the previous works on gridding, the focus was on using the interpolation to find a non-iterative approximate solution to an i... |

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Citation Context ...errors, which are about 6.5-times higher than in the optimum case. For comparison, we show also alpha s representing typical KB window having desirable properties for the spatial image representation =-=[30]-=-, but poor performance as the interpolation kernel. It is interesting that although all of them have similar shape, they provide quite dramatic difference in the NUFFT performance. E.1 Shift-invariant... |

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Citation Context ...ction operation (e.g., [16, 17]). The crucial step influencing the reconstruction quality and speed is the interpolation between polar and Cartesian rasters in frequency space. Gridding interpolation =-=[18, 19]-=-, with proper interpolating [20] and data weighting functions, as investigated in the MRI literature [21–23], brought improvement in the direct Fourier reconstruction. Recently, the Fourier-based repr... |

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Citation Context ...ntation Using Basis Functions In the conventional space-domain iterative algorithms, the reconstructed image is usually represented by a set of coefficients of basis functions (e.g., pixels, or blobs =-=[29]-=-), rather than by the set of image samples. The values of continuous image ������� , � ����� , are then obtained from coefficients ��� , where � represents the discrete set of locations � ��� � � by �... |

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Citation Context ...essel (KB) windows were used for interpolation, which are known to be reasonably accurate but without explicitly evaluating the accuracy. The concept of the non-uniform Fast Fourier transform (NUFFT) =-=[26]-=- used in this paper is related to gridding methods for interpolation in frequency space. The KB interpolation kernels used in this work have been optimized using a min-max approach [27], thus providin... |

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Citation Context ...ect Fourier reconstruction. Recently, the Fourier-based reprojection has been applied for (non-iterative) fully 3D PET reconstruction [24] and for calculation of attenuation correction factors in PET =-=[25]-=-. In these works, Kaiser-Bessel (KB) windows were used for interpolation, which are known to be reasonably accurate but without explicitly evaluating the accuracy. The concept of the non-uniform Fast ... |

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