Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction

[1]  A. Nuttall.  Some windows with very good sidelobe behavior , 1981 .

[2]  P.-E. Danielsson,et al.  Backprojection in O(N/sup 2/logN) time [X-ray CT image reconstruction] , 1997, 1997 IEEE Nuclear Science Symposium Conference Record.

[3]  Jeffrey A. Fessler,et al.  A min-max approach to the multidimensional nonuniform FFT: application to tomographic image reconstruction , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[4]  D. C. Noll,et al.  Multishot rosette trajectories for spectrally selective MR imaging , 1997, IEEE Transactions on Medical Imaging.

[5]  R. Hingorani,et al.  An Investigation of Computerized Tomography by Direct Fourier Inversion and Optimum Interpolation , 1981 .

[6]  Charles W. Stearns,et al.  Accelerated image reconstruction for a cylindrical positron tomograph using Fourier domain methods , 1990 .

[7]  Jan Timmer,et al.  The gridding method for image reconstruction by Fourier transformation , 1995, IEEE Trans. Medical Imaging.

[8]  J. D. O'Sullivan,et al.  A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography , 1985, IEEE Transactions on Medical Imaging.

[9]  Samuel Matej,et al.  3D-FRP: direct Fourier reconstruction with Fourier reprojection for fully 3-D PET , 2000 .

[10]  Yoram Bresler,et al.  O(N2log2N) filtered backprojection reconstruction algorithm for tomography , 2000, IEEE Trans. Image Process..

[11]  Jeffrey A. Fessler,et al.  Nonuniform fast Fourier transforms using min-max interpolation , 2003, IEEE Trans. Signal Process..

[12]  I Bajla,et al.  A high-speed reconstruction from projections using direct Fourier method with optimized parameters-an experimental analysis. , 1990, IEEE transactions on medical imaging.

[13]  B. De Man,et al.  Distance-driven projection and backprojection in three dimensions. , 2004, Physics in medicine and biology.

[14]  Dwight G. Nishimura,et al.  Rapid gridding reconstruction with a minimal oversampling ratio , 2005, IEEE Transactions on Medical Imaging.

[15]  Pawel A Penczek,et al.  Gridding-based direct Fourier inversion of the three-dimensional ray transform. , 2004, Journal of the Optical Society of America. A, Optics, image science, and vision.

[16]  Hossein Sedarat,et al.  On the optimality of the gridding reconstruction algorithm , 2000, IEEE Transactions on Medical Imaging.

[17]  J. Colsher,et al.  Fully-three-dimensional positron emission tomography , 1980, Physics in medicine and biology.

[18]  T K Narayan,et al.  A methodology for testing for statistically significant differences between fully 3D PET reconstruction algorithms. , 1994, Physics in medicine and biology.

[19]  Brian K. Rutt,et al.  Fast reprojection of volume data , 1990, [1990] Proceedings of the First Conference on Visualization in Biomedical Computing.

[20]  Maria Magnusson,et al.  Artefacts and remedies in direct Fourier tomographic reconstruction , 1992, IEEE Conference on Nuclear Science Symposium and Medical Imaging.

[21]  Robert M. Lewitt,et al.  Practical considerations for 3-D image reconstruction using spherically symmetric volume elements , 1996, IEEE Trans. Medical Imaging.

[22]  Robert M. Lewitt,et al.  Overview of methods for image reconstruction from projections in emission computed tomography , 2003, Proc. IEEE.

[23]  Makoto Tabei,et al.  Backprojection by upsampled Fourier series expansion and interpolated FFT , 1992, IEEE Trans. Image Process..

[24]  Martin L. Brady,et al.  A Fast Discrete Approximation Algorithm for the Radon Transform , 1998, SIAM J. Comput..

[25]  R M Lewitt,et al.  Multidimensional digital image representations using generalized Kaiser-Bessel window functions. , 1990, Journal of the Optical Society of America. A, Optics and image science.

[26]  Thomas Malzbender,et al.  Fourier volume rendering , 1993, TOGS.

[27]  Salvatore Lanzavecchia,et al.  Electron tomography in conical tilt geometry. The accuracy of a direct Fourier method (DFM) and the suppression of non-tomographic noise , 1996 .

[28]  C. Michel,et al.  Fast PET EM reconstruction from linograms , 2002, 2002 IEEE Nuclear Science Symposium Conference Record.

[29]  David Brasse,et al.  Fast fully 3-D image reconstruction in PET using planograms , 2004, IEEE Transactions on Medical Imaging.

[30]  Jeffrey A. Fessler,et al.  Iterative tomographic image reconstruction using Fourier-based forward and back-projectors , 2004, IEEE Transactions on Medical Imaging.

[31]  Avinash C. Kak,et al.  Principles of computerized tomographic imaging , 2001, Classics in applied mathematics.

[32]  Marc Levoy,et al.  Frequency domain volume rendering , 1993, SIGGRAPH.

[33]  Bob S. Hu,et al.  Fast Spiral Coronary Artery Imaging , 1992, Magnetic resonance in medicine.

[34]  A. Macovski,et al.  Selection of a convolution function for Fourier inversion using gridding [computerised tomography application]. , 1991, IEEE transactions on medical imaging.

[35]  J. Pipe,et al.  Sampling density compensation in MRI: Rationale and an iterative numerical solution , 1999, Magnetic resonance in medicine.

[36]  R. Lewitt.  Reconstruction algorithms: Transform methods , 1983, Proceedings of the IEEE.

[37]  Rolf Clackdoyle,et al.  Analytical reconstruction for multi-segment slant hole SPECT , 2002, 2002 IEEE Nuclear Science Symposium Conference Record.