Nonuniform fast Fourier transforms using min-max interpolation

IEEE Transactions on Signal Processing

Volumn 51, Issue 2, 2003, Pages 560-574

  Fessler, Jeffrey A ^a   Sutton, Bradley P ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Discrete Fourier transform; Gridding; Imaging; Magnetic resonance; Min max interpolation; Tomography

Indexed keywords

BESSEL FUNCTIONS; DISCRETE FOURIER TRANSFORMS; FREQUENCY DOMAIN ANALYSIS; INTERPOLATION; MAGNETIC RESONANCE IMAGING; PARAMETER ESTIMATION; TOMOGRAPHY;

COMPUTATIONAL ELECTROMAGNETICS;

SIGNAL PROCESSING;

EID: 0037306521     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2002.807005     Document Type: Article

References (65)

1. A.V. Oppenheim, D. Johnson, and K. Steiglitz, "Computation of Spectra with unequal resolution using the fast Fourier transform," Proc. IEEE, vol. 59, pp. 299-301, Feb. 1971.
2. D. C. Munson, J. D. O'Brien, and W. K. Jenkins, "A tomographic formulation of spotlight mode synthetic aperture radar," Proc. IEEE, vol. 71, pp. 917-925, Aug. 1983.
3. D. C. Munson and J. L. Sanz, "Image reconstruction from frequency-offset Fourier data," Proc. IEEE, vol. 72, pp. 661-669, June 1984.
4. H. Choi and D. C. Munson, "Direct-Fourier reconstruction in tomography and synthetic aperture radar," Int. J. Imaging Syst. Tech., vol. 9, no. 1, pp. 1-13, 1998.
5. E. Larsson, P. Stoica, and J. Li, "SAR image construction from gapped phase-history data," in Proc. IEEE Int. Conf. Image Process., vol. 3, 2001, pp. 608-611.
6. Y. Wu and D. Munson, "Multistatic passive radar imaging using the smoothed pseudo Wigner-Ville distribution," in Proc. IEEE Int. Conf Image Process., vol. 3, 2001, pp. 604-607.
7. E. J. Candes and D. L. Donoho, "Ridgelets: A key to higher-dimensional intermittency?," Philos. Trans. R. Soc. Lond. A, Math. Phys. Eng. Sci., vol. 357, no. 1760, pp. 2495-2509, Sept. 1999.
8. Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs)," IEEE Microw. Guided Wave Lett., vol. 8, pp. 18-20, Jan. 1998.
9. Q. H. Liu, N. Nguyen, and X. Y. Tang, "Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications," in Proc. IEEE Geosci. Remote Sensing Symp., vol. 1, 1998, pp. 288-290.
10. Q. H. Liu and X. Y. Tang, "Iterative algorithm for nonuniform inverse fast Fourier transform," Electron. Lett., vol. 34, no. 20, pp. 1913-1914, Oct. 1998.
11. X. M. Xu and Q. H. Liu, "The conjugate-gradient nonuniform fast Fourier transform (CG-NUFFT) method for one- and two-dimensional media," Microw. Opt. Technol. Lett., vol. 24, no. 6, pp. 385-389, Mar. 2000.
12. Q. H. Liu, X. M. Xu, B. Tian, and Z. Q. Zhang, "Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations," IEEE Trans. Geosci. Remote Sensing, vol. 38, pp. 1551-1560, July 2000.
13. E. Angelidis and J. E. Diamessis, "A novel method for designing FIR digital filters with nonuniform frequency samples," IEEE Trans. Signal Processing, vol. 42, pp. 259-267, Feb. 1994.
14. S. Bagchi and S. K. Mitra, "The nonuniform discrete Fourier transform and its applications in filter design. I-1-D," IEEE Trans. Circuits Syst. II, vol. 43, pp. 422-433, June 1996.
15. A. Makur and S. K. Mitra, "Warped discrete-Fourier transform: Theory and applications," IEEE Trans. Circuits Syst. I, vol. 48, pp. 1086-1093, Sept. 2001.
16. S. Bagchi and S. Mitra, The Nonuniform Discrete Fourier Transform and its Applications in Signal Processing. Boston, MA: Kluwer, 1999.
17. K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, "SENSE: Sensitivity encoding for fast MRI," Magn. Resonance Med., vol. 42, pp. 952-962, 1999.
18. B. P. Sutton, J. A. Fessler, and D. Noll, "A min-max approach to the nonuniform N-D FFT for rapid iterative reconstruction of MR images," in Proc. Int. Soc. Mag. Res. Med., 2001, p. 763.
19. _, "Iterative MR image reconstruction using sensitivity and inhomogeneity field maps," in Proc. Int. Soc. Mag. Res. Med., 2001, p. 771.
20. B. P. Sutton, D. Noll, and J. A. Fessler, "Fast, iterative, field-corrected image reconstruction for MRI," IEEE Trans. Med. Imaging, 2002, submitted for publication.
21. C. R. Crawford and Elscint, "System for reprojecting images using transform techniques," US Patent 4616318, 1986.
22. C. R. Crawford, J. G. Colsher, N. J. Pelc, and A. H. R. Lonn, "High speed reprojection and its applications," in Proc. SPIE 914, Med. Imag. II: Image Formation, Detection, Process., Interpretation, 1988, pp. 311-318.
23. C. W. Steams, D. A. Chesler, and G. L. Brownell, "Three-dimensional image reconstruction in the Fourier domain," IEEE Trans. Nuclear Sci., vol. 34, pp. 374-378, Feb. 1987.
24. _, "Accelerated image reconstruction for a cylindrical positron tomograph using Fourier domain methods," IEEE Trans. Nuclear Sci., vol. 37, pp. 773-777, Apr. 1990.
25. H. Schomberg and J. Timmer, "The gridding method for image reconstruction by Fourier transformation," IEEE Trans. Med. Imaging, vol. 14, pp. 596-607, Sept. 1995.
26. S. Matej and R. M. Lewitt, "3-FRP: Direct Fourier reconstruction with Fourier reprojection for fully 3-D PET," IEEE Trans. Nuclear Sci., vol. 48, pp. 1378-1385, Aug. 2001.
27. J. A. Fessler and B. P. Sutton, "A min-max approach to the multidimensional nonuniform FFT: Application to tomographic image reconstruction," in Proc. IEEE Int. Conf. Image Process., vol. 1, 2001, pp. 706-709.
28. J. A. Fessler. (2001) Iterative tomographic image reconstruction using nonuniform fast Fourier transforms. Commun. and Signal Process. Lab, Dept. Elect. Eng. Comput. Sci., Univ. of Michigan, Ann Arbor, MI. [Online]. Available: http://www.eecs.umich.edu/̃fessler
29. T. Strohmer, "Numerical anaiysis of the nonuniform sampling problem," J. Comput. Appl. Math., vol. 122, no. 1-2, pp. 297-316, Oct. 2000.
30. G. Beylkin, "On the fast Fourier transform of functions with singularities," Appl. Comput. Harmonic Aha., vol. 2, no. 4, pp. 363-381, Oct. 1995.
31. S. Bagchi and S. K. Mitra, "The nonuniform discrete Fourier transform and its applications in filter design. II-2-D," IEEE Trans. Circuits Syst. II, vol. 43, pp. 434-444, June 1996.
32. G. Evangelista and S. Cavaliere, "Discrete frequency warped wavelets: Theory and applications," IEEE Trans. Signal Processing, vol. 46, pp. 874-885, Apr. 1998.
33. N. I. Cho and S. K. Mitra, "Warped discrete cosine transform and its application in image compression," IEEE Trans. Circuits Syst. Vid. Technol., vol. 10, pp. 1364-1373, Dec. 2000.
34. A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data," SIAM J. Sci. Comput., vol. 14, no. 6, pp. 1368-1393, Nov. 1993.
35. C. Anderson and M. D. Dahleh, "Rapid computation of the discrete Fourier transform," SIAM J. Sci. Comput., vol. 17, no. 4, pp. 913-919, July 1996.
36. A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data, II," Appl. Comput. Harmonic Anal., vol. 2, pp. 85-100, 1995.
37. G. Steidl, "A note on the fast Fourier transforms for nonequispaced grids," Adv. Comput. Math., vol. 9, no. 3, pp. 337-352, 1998.
38. A. F. Ware, "Fast approximate Fourier transforms for irregularly spaced data," SIAM Rev., vol. 40, no. 4, pp. 838-856, Dec. 1998.
39. A. J. W. Duijndam and M. A. Schonewille, "Nonuniform fast Fourier transform," Geophys., vol. 64, no. 2, pp. 539-551, Mar. 1999.
40. N. Nguyen and Q. H. Liu, "The regular Fourier matrices and nonuniform fast Fourier transforms," SIAM J. Sci. Comput., vol. 21, no. 1, pp. 283-293, 1999.
41. D. Potts, G. Steidl, and M. Tasche, "Fast Fourier transforms for nonequispaced data: A tutorial," in Modern Sampling Theory: Mathematics and Application, J. J. B. P. Ferreira, Ed. Boston, MA: Birkhauser, 2000, pp. 253-274.
42. W. H. Press and G. B. Rybicki, "Fast algorithm for spectral analysis of unevenly sampled data," Astrophys. J., vol. 338, pp. 227-280, Mar. 1989.
43. T. W. Parks and D. P. Kolba, "Interpolation minimizing maximum normalized error for band-limited signals," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-26, pp. 381-384, Aug. 1978.
44. D. S. Chen and J. P. Allebach, "Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 173-180, Feb. 1987.
45. R. G. Shenoy and T. W. Parks, "An optimal recovery approach to interpolation," IEEE Trans. Signal Processing, vol. 40, pp. 1987-1992, Aug. 1992.
46. N. P. Willis and Y. Bresler, "Norm invariance of minimax interpolation," IEEE Trans. Inform. Theory, vol. 38, pp. 1177-1181, May 1992.
47. G. Calvagno and D. C. Munson, "A frequency-domain approach to interpolation from a nonuniform grid," Signal Process., vol. 52, no. 1, pp. 1-21, July 1996.
48. H. Choi and D. C. Munson, "Analysis and design of minimax-optimal interpolators," IEEE Trans. Signal Processing, vol. 46, pp. 1571-1579, June 1998.
49. D. D. Muresan and T. W. Parks, "Optimal recovery approach to image interpolation," in Proc. IEEE Int. Conf. Image Process., vol. 3, 2001, pp. 848-851.
50. T. W. Parks and J. H. McClellan, "Chebyshev approximation for nonrecursive digital filters with linear phase," IEEE Trans. Circuit Theory, vol. 19, no. CT-2, pp. 189-199, Mar. 1972.
51. R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983.
52. S. K. Mitra, Digital Signal Procesing: A Computer-Based Approach, 2nd ed. New York: McGraw-Hill, 2001.
53. M. Tabei and M. Ueda, "Backprojection by upsampled Fourier series expansion and interpolated FFT," IEEE Trans. Image Processing, vol. 1, pp. 77-87, Jan. 1992.
54. A. J. W. Duijndam, M. A. Schonewille, and C. O. H. Hindriks, "Reconstruction of band-limited signals, irregularly sampled along one spatial direction," Geophys., vol. 64, no. 2, pp. 524-538, Mar. 1999.
55. B. Lee and A. E. Yagle, "A sensitivity measure for image reconstruction from irregular 2-D DTFT samples," in Proc. IEEE Conf. Acoust. Speech Signal Process., vol. 4, 2002, pp. 3245-3248.
56. K. P. Pruessmann, M. Weiger, P. Börnert, and P. Boesiger, "Advances in sensitivity encoding with arbitrary k-space trajectories," Magn. Resonance Med., vol. 46, no. 4, pp. 638-651, Oct. 2001.
57. M. Magnusson, P.-E. Danielsson, and P. Edholm, "Artefacts and remedies in direct Fourier tomographic reconstruction," in Proc. IEEE Nuc. Sci. Symp. Med. Imag. Conf., vol. 2, 1992, pp. 1138-1340.
58. S. Lanzavecchia and P. L. B., "A bevy of novel interpolating kernels for the Shannon reconstruction of high-bandpass images," J. Visual Commun. Imag. Represent., vol. 6, no. 2, pp. 122-131, June 1995.
59. R. M. Lewitt, "Multidimensional digital image representations using generalized Kaiser-Bessel window functions," J. Opt. Soc. Amer. A, vol. 7, no. 10, pp. 1834-1846, Oct. 1990.
60. _, "Alternatives to voxels for image representation in iterative reconstruction algorithms," Phys. Med. Biol., vol. 37, no. 3, pp. 705-716, 1992.
61. J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, "Selection of a convolution function for Fourier inversion using gridding," IEEE Trans. Med. Imaging, vol. 10, pp. 473-478, Sept. 1991.
62. L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nuclear Sci., vol. NS-21, pp. 21-43, June 1974.
63. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging. New York: IEEE Press, 1988.
64. H. Sedarat and D. G. Nishimura, "On the optimality of the gridding reconstruction algorithm," IEEE Trans. Med. Imaging, vol. 19, pp. 306-317, Apr. 2000.
65. Q. H. Lin and Z. Q. Zhang, "Nonuniform fast hankel transform (NUFHT) algorithm," Appl. Opt., vol. 38, no. 32, pp. 6705-6708, Nov. 1999.