Optimized least-square nonuniform fast Fourier transform

IEEE Transactions on Signal Processing

Volumn 57, Issue 6, 2009, Pages 2165-2177

  Jacob, Mathews ^a  

^a University of Rochester   (United States)

Author keywords

Fourier transform; Interpolation; Nonuniform; Sampling; Shift invariant

Indexed keywords

APPROXIMATION ERRORS; CONVENTIONAL SCHEMES; ERROR COMPONENTS; EXPERIMENTAL COMPARISON; ITERATIVE SIGNALS; LEAST SQUARES; LOOK UP TABLE; MEAN SQUARE; MEMORY EFFICIENT; MEMORY REQUIREMENTS; NON-UNIFORM FAST FOURIER TRANSFORMS; NONUNIFORM; OPTIMAL SCALE; RECONSTRUCTION ALGORITHMS; SHIFT-INVARIANT;

INTERPOLATION; MICROSTRIP DEVICES; TABLE LOOKUP;

FAST FOURIER TRANSFORMS;

EID: 66849141890     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2009.2014809     Document Type: Article

References (32)

1. S. Matej, J. Fessler, and I. Kazantsev, "Iterative tomographic image reconstruction using Fourier-based forward and back-projectors," IEEE Trans. Med. Imag., vol. 23, no. 4, pp. 401-412, Apr. 2004.
2. Y. O'Connor and J. Fessler, "Fourier-based forward and back-projectors in iterative fan-beam tomographic image reconstruction," IEEE Trans. Med. Imag., vol. 25, no. 5, pp. 582-589, May 2006.
3. K. 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. 638-51, Oct. 2001.
4. H. Eggers, T. Knopp, and D. Potts, "Field inhomogeneity correction based on gridding reconstruction for magnetic resonance imaging," IEEE Trans. Med. Imag., vol. 26, no. 3, pp. 374-384, Mar. 2007.
5. B. Sutton, D. Noll, and J. Fessler, "Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities," IEEE Trans. Med. Imag., vol. 22, no. 2, pp. 178-188, Feb. 2003.
6. D. C. Munson, Jr. and J. L. C. Sanz, "Image reconstruction from frequency-offset Fourier data," Proc. IEEE, vol. 72, no. 6, pp. 661-669, Jun. 1984.
7. T. Strohmer, "Computationally attractive reconstruction of bandlimited images from irregular samples," IEEE Trans. Image Process., vol. 6, no. 4, pp. 540-548, Apr. 1997.
8. H. Feichtinger and T. Strohmer, "Efficient numerical methods in nonuniform sampling theory," Numer. Math., vol. 69, no. 4, pp. 423-440, Jan. 1995.
9. S. Bagchi and S. K. Mitra, The Nonuniform Discrete Fourier Transform and Its Applications in Signal Processing, ser. Springer Int. Series in Engineering and Computer Science. New York: Springer, Nov. 1998.
10. A. Makur and S. Mitra, "Warped discrete-Fourier transform: Theory and applications," IEEE Trans. Circuits Syst. I, vol. 48, no. 9, pp. 1086-1093, Sep. 2001.
11. E. Candes and D. Donoho, "Ridgelets: A key to higher-dimensional intermittency?," R. Soc. Lond. Philos. Trans. A, Math. Phys. Eng. Sci., vol. 357, no. 1760, pp. 2495-2509, 1999.
12. N. Nguyen and Q. Liu, "The regular Fourier matrices and nonuniform fast Fourier transforms," SIAM J. Sci. Comput., vol. 21, no. 1, pp. 283-293, 1999.
13. J. Fessler and B. Sutton, "Nonuniform fast Fourier transforms using min-max interpolation," IEEE Trans. Signal Process., vol. 51, no. 2, pp. 560-574, Feb. 2003.
14. A. Dutt and V. Rokhlin, "Fast Fourier transforms for nonequispaced data," SIAM J. Sci. Comput., vol. 14, no. 6, pp. 1368-1393, 1993.
15. D. Potts, G. Steidl, and M. Tasche, Fast Fourier Transforms for Nonequispaced Data: A Tutorial, J. J. Benedetto and P. J. S. G. Ferreira, Eds. Boston, MA: Birkhäuser, 2001, pp. 247-270, "Fast Fourier transforms for nonequispaced data: A tutorial," Modern Sampling Theory: Mathematics and Applications, Jan. 2000.
16. G. Strang and T. Nguyen, Wavelets and Filter Banks. Cambridge, MA: Wellesley-Cambridge, Jan. 1996, p. 490.
17. M. Unser, "Sampling-50 years after Shannon," Proc. IEEE, vol. 88, no. 4, pp. 569-587, Apr. 2000.
18. T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques. I. Interpolators and projectors," IEEE Trans. Signal Process., vol. 47, no. 10, pp. 2783-2795, Oct. 1999.
19. M. Jacob, T. Blu, and M. Unser, "Sampling of periodic signals:A quantitative error analysis," IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1153-1159, May 2002.
20. A. Nieslony and G. Steidl, "Approximate factorizations of fourier matrices with nonequispaced knots," Linear Algebra Appl., vol. 366, pp. 337-351, 2003.
21. G. Steidl, "A note on fast Fourier transforms for nonequispaced grids," Adv. Comput. Math., vol. 9, no. 3-4, pp. 337-352, 1998.
22. J. Jackson, C. Meyer, D. Nishimura, and A. Macovski, "Selection of a convolution function for fourier inversion using gridding [computerised tomography application]," IEEE Trans. Med. Imag., vol. 10, no. 3, pp. 473-478, 1991.
23. P. Beatty, D. Nishimura, and J. Pauly, "Rapid gridding reconstruction with a minimal oversampling ratio," IEEE Trans. Med. Imag., vol. 24, no. 6, pp. 799-808, Jun. 2005.
24. G. Chuang and C. Kuo, "Wavelet descriptor of planar curves: Theory and applications," IEEE Trans. Image Process., vol. 5, no. 1, pp. 56-70, Jan. 1996.
25. T. Strohmer, "Numerical analysis of the non-uniform sampling problem," J. Comput. Appl. Math. (Special Issue on Numerical Analysis 2000, Vol. II: Interpolation and Extrapolation), vol. 122, no. 1-2, pp. 297-316, 2000.
26. M. Unser and I. Daubechies, "On the approximation power of convolution-based least squares versus interpolation," IEEE Trans. Signal Process., vol. 45, no. 7, pp. 1697-1711, Jul. 1997.
27. C. Lee, M. Eden, and M. Unser, "High-quality image resizing using oblique projection operators," IEEE Trans. Image Process., vol. 7, no. 5, pp. 679-692, May 1998.
28. D. Slepian, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V-the discrete case," Bell Syst. Tech. J., vol. 57, pp. 1371-1430, May-Jun. 1978.
29. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, Jan. 1989.
30. M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag., vol. 16, no. 6, pp. 22-38, Nov. 1999.
31. G. Strang, Introduction to Linear Algebra. Cambridge, MA: Wellesley-Cambridge, Apr. 2003.
32. T. SØrensen, T. Schaeffter, K. Noe, and M. Hansen, "Accelerating the nonequispaced fast Fourier transform on commodity graphics hardware," IEEE Trans. Med. Imag., vol. 27, no. 4, pp. 538-547, Apr. 2008.