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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 26, Iss. 3 — Mar. 1, 2009
  • pp: 566–575

Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem

Leonid P. Yaroslavsky, Gil Shabat, Benjamin G. Salomon, Ianir A. Ideses, and Barak Fishbain  »View Author Affiliations


JOSA A, Vol. 26, Issue 3, pp. 566-575 (2009)
http://dx.doi.org/10.1364/JOSAA.26.000566


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Abstract

In many applications, sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases, it is necessary to convert irregularly sampled signals to regularly sampled ones or to restore missing data. We address this problem in the framework of a discrete sampling theorem for band-limited discrete signals that have a limited number of nonzero transform coefficients in a certain transform domain. Conditions for the image unique recovery, from sparse samples, are formulated and then analyzed for various transforms. Applications are demonstrated on examples of image superresolution and image reconstruction from sparse projections.

© 2009 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(100.2000) Image processing : Digital image processing
(100.3020) Image processing : Image reconstruction-restoration
(070.2025) Fourier optics and signal processing : Discrete optical signal processing
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Image Processing

History
Original Manuscript: August 18, 2008
Manuscript Accepted: November 11, 2008
Published: February 19, 2009

Citation
Leonid P. Yaroslavsky, Gil Shabat, Benjamin G. Salomon, Ianir A. Ideses, and Barak Fishbain, "Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem," J. Opt. Soc. Am. A 26, 566-575 (2009)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-3-566


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References

  1. D. Shepard, “A two-dimensional interpolation function for irregularly-spaced data,” in Proceedings of the 1968 ACM 23rd National Conference (ACM, 1968), pp. 517-523. [CrossRef]
  2. S. K. Lodha and R. Franke, “Scattered data techniques for surfaces,” in Proceedings of the IEEE Conference on Scientific Visualization (IEEE, 1997), Vol. 38, No. 157, pp. 181-200.
  3. H. Landau, “Necessary density conditions for sampling and interpolation of certain entire functions,” Acta Math. 117, 37-52 (1967). [CrossRef]
  4. A. Aldroubi and K. Grochenig, “Non-uniform sampling and reconstruction in shift-invariant spaces,” SIAM Rev. 43, 585-620 (2001). [CrossRef]
  5. F.Marvasti, ed., Nonuniform Sampling (Kluwer Academic/Plenum, 2001). [CrossRef]
  6. M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. June 1999, pp. 22-38. [CrossRef]
  7. S. Lee, G. Wolberg, and S. Y. Shin, “Scattered data interpolation with multilevel B-splines,” IEEE Trans. Visualization. Comput. Graphics 3, 228-244 (1997). [CrossRef]
  8. E. Margolis and Y. C. Eldar, “Interpolation with non-uniform B-splines,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2004), Vol. 2, pp. 577-580.
  9. P. J. S. G. Ferreira, “Iterative and noniterative recovery of missing samples for 1-D band-limited signals,” in Nonuniform Sampling, F.Marvasti, ed. (Kluwer Academic/Plenum, 2001), pp. 235-278. [CrossRef]
  10. M. Hasan and F. Marvasti, “Application of nonuniform sampling to error concealment,” in Nonuniform Sampling, F.Marvasti, ed. (Kluwer Academic/Plenum, 2001), pp. 619-646.
  11. A. Averbuch, R. Coifman, M. Israeli, I. Sidelnikov, and Y. Shkolinsky, “Irregular sampling for multi-dimensional polar processing of integral transforms,” in Advances in Signal Transforms: Theory and Applications, J.Astola and L.Yaroslavsky, eds. (Hindawi, 2007), pp. 143-198.
  12. A. Averbuch and V. Zheludev, “Wavelet and frame transforms originated from continuous and discrete splines,” in Advances in Signal Transforms: Theory and Applications, J.Astola and L.Yaroslavsky, eds. (Hindawi, 2007), pp. 1-54.
  13. Compressed sensing resources: http://www.dsp.ece.rice.edu/cs/. Last accessed December 18, 2008.
  14. Y. Katiyi and L. Yaroslavsky, “Regular matrix methods for synthesis of fast transforms: general pruned and integer-to-integer transforms,” in Proceedings of IEEE International Workshop on Spectral Methods and Multirate Signal Processing (IEEE, 2001), pp. 17-24.
  15. Y. Katiyi and L. Yaroslavsky, “V/HS structure for transforms and their fast algorithms,” in Proceedings of the IEEE 3rd International Symposium on Signal Processing and Analysis (IEEE, 2003), Vol. 1, pp. 482--487.
  16. L. Yaroslavsky, Digital Holography and Digital Signal Processing (Kluwer Academic, 2004).
  17. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, pp. 735-742 (1975). [CrossRef]
  18. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991). [CrossRef]
  19. L. Yaroslavsky, “Fast discrete sinc-interpolation: a gold standard for image resampling,” in Advances in Signal Transforms: Theory and Application, J.Astola and L.Yaroslavsky, eds., EURASIP Book Series on Signal Processing and Communications (Hindawi, 2007), pp. 337-405.
  20. L. P. Yaroslavsky, B. Fishbain, G. Shabat, and I. Ideses, “Super-resolution in turbulent videos: making profit from damage,” Opt. Lett. 32, 3038-3040 (2007). [CrossRef] [PubMed]
  21. A. Averbuch, R. Coifman, D. Donoho, M. Israeli, and Y. Shkolinsky, “A framework for discrete integral transformations II--the 2D discrete Radon transform,” SIAM J. Sci. Comput. (USA) 30, 764-784 (2008). [CrossRef]
  22. Standford University, Statistics Department, David Donoho's homepage,http://www.stat.stanford.edu/~donoho/.

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