OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23235–23252

Quantum limits of super-resolution of optical sparse objects via sparsity constraint

Hui Wang, Shensheng Han, and Mikhail I. Kolobov  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23235-23252 (2012)
http://dx.doi.org/10.1364/OE.20.023235


View Full Text Article

Enhanced HTML    Acrobat PDF (1162 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Sparsity constraint is a priori knowledge of the signal, which means that in some properly chosen basis only a small percentage of the total number of the signal components is nonzero. Sparsity constraint has been used in signal and image processing for a long time. Recent publications have shown that the Sparsity constraint can be used to achieve super-resolution of optical sparse objects beyond the diffraction limit. In this paper we present the quantum theory which establishes the quantum limits of super-resolution for the sparse objects. The key idea of our paper is to use the discrete prolate spheroidal sequences (DPSS) as the sensing basis. We demonstrate both analytically and numerically that this sensing basis gives superior performance of super-resolution over the Fourier basis conventionally used for sensing of sparse signals. The explanation of this phenomenon is in the fact that the DPSS are the eigenfunctions of the optical imaging system while the Fourier basis are not. We investigate the role of the quantum fluctuations of the light illuminating the object, in the performance of reconstruction algorithm. This analysis allows us to formulate the criteria for stable reconstruction of sparse objects with super-resolution. Our results imply that sparsity of the object is not the only parameter which describes super-resolution achievable via sparsity constraint.

© 2012 OSA

OCIS Codes
(100.6640) Image processing : Superresolution
(270.0270) Quantum optics : Quantum optics
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Imaging Systems

History
Original Manuscript: June 26, 2012
Revised Manuscript: September 15, 2012
Manuscript Accepted: September 16, 2012
Published: September 25, 2012

Citation
Hui Wang, Shensheng Han, and Mikhail I. Kolobov, "Quantum limits of super-resolution of optical sparse objects via sparsity constraint," Opt. Express 20, 23235-23252 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23235


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. I. Kolobov (Ed.), Quantum Imaging (Springer, NY, 2006).
  2. S. J. Bentley, Principles of Quantum Imaging: Ghost Imaging, Ghost Diffraction, and Quantum Lithography (Taylor & Francis, Boca Raton, 2010).
  3. M. I. Kolobov and C. Fabre, “Quantum limits on optical resolution,” Phys. Rev. Lett.85, 3789–3792 (2000). [CrossRef] [PubMed]
  4. V. N. Beskrovnyy and M. I. Kolobov, “Quantum limits of super-resolution in reconstruction of optical objects,” Phys. Rev. A71, 043802 (2005). [CrossRef]
  5. V. N. Beskrovnyy and M. I. Kolobov, “Quantum-statistical analysis of superresolution for optical systems with circular symmetry,” Phys. Rev. A78, 043824 (2008). [CrossRef]
  6. M. I. Kolobov, “Quantum limits of superresolution for imaging discrete subwavelength structures,” Opt. Express16, 58–66 (2008). [CrossRef] [PubMed]
  7. M. Elad, M. A. T. Figueiredo, and Y. Ma, “On the Role of Sparse and Redundant Representations in Image Processing,” Proc IEEE98, 972–982 (2010). [CrossRef]
  8. E. J. Candès, J. Romberg, and T. Tao,“Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006). [CrossRef]
  9. E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory52, 5406–5425 (2006). [CrossRef]
  10. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag.25, 21–30 (2008). [CrossRef]
  11. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52, 1289–1306 (2006). [CrossRef]
  12. S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express17, 23920–23946 (2009). [CrossRef]
  13. Y. Shechtman, S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse images carried by incoherent light,” Opt. Lett.35, 1148–1150 (2010). [CrossRef] [PubMed]
  14. W. Gong and S. Han, “Super-resolution ghost imaging via compressive sampling reconstruction,” arXiv [0910.4823v1] (2009).
  15. W. Gong and S. Han, “Super-resolution and reconstruction of far-field ghost imaging via sparcity constraints,” in the Proceedings of SPARS’11-Singal Processing with Adaptive Sparse Structured Representations, (Edinburgh, Scotland, 2011), p. 91. [PubMed]
  16. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: The discrete case,” Bell System Tech. J.57, 1371–1430 (1978).
  17. D. L. Donoho and X. Huo, “Uncertainty principles and ideal atomic decomposition,” IEEE Trans. Inf. Theory47, 2845–2862 (2001). [CrossRef]
  18. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review43, 129–159 (2001). [CrossRef]
  19. D. F. Walls and G. J. Milburn, 2nd ed.Quantum Optics (Springer, Berlin, 2008). [CrossRef]
  20. D. Slepian, “Some comments on Fourier analysis, uncerlainty and modeling,” SIAM Review25, 379–393 (1983). [CrossRef]
  21. D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math.49, 906–931 (1989). [CrossRef]
  22. E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” arXiv[1203.5871v1] (2012).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited