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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11673–11689

Improvement of matrix condition of Hybrid, space variant optics by the means of Parallel Optics design

Iftach Klapp and David Mendlovic  »View Author Affiliations


Optics Express, Vol. 17, Issue 14, pp. 11673-11689 (2009)
http://dx.doi.org/10.1364/OE.17.011673


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Abstract

The problem of image restoration of space variant blur is common and important. One of the most useful descriptions of this problem is in its algebraic form I=H∗O, where O is the object represented as a column vector, I is the blur image represented as a column vector and H is the PSF matrix that represents the optical system. When inverting the problem to restore the geometric object from the blurred image and the known system matrix, restoration is limited in speed and quality by the system condition. Current optical design methods focus on image quality, therefore if additional image processing is needed the matrix condition is taken “as is”. In this paper we would like to present a new optical approach which aims to improve the system condition by proper optical design. In this new method we use Singular Value Decomposition (SVD) to define the weak parts of the matrix condition. We design a second optical system based on those weak SVD parts and then we add the second system parallel to the first one. The original and second systems together work as an improved parallel optics system. Following that, we present a method for designing such a “parallel filter” for systems with a spread SVD pattern. Finally we present a study case in which by using our new method we improve a space variant image system with an initial condition number of 8.76e4, down to a condition number of 2.29e3. We use matrix inversion to simulate image restoration. Results show that the new parallel optics immunity to Additive White Gaussian Noise (AWGN) is much better then that of the original simple lens. Comparing the original and the parallel optics systems, the parallel optics system crosses the MSEIF=0 [db] limit in SNR value which is more than 50db lower then the SNR value in the case of the original simple lens. The new parallel optics system performance is also compared to another method based on the MTF approach.

© 2009 Optical Society of America

OCIS Codes
(070.6110) Fourier optics and signal processing : Spatial filtering
(080.1010) Geometric optics : Aberrations (global)
(100.3190) Image processing : Inverse problems
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Image Processing

History
Original Manuscript: October 2, 2008
Revised Manuscript: April 1, 2009
Manuscript Accepted: April 12, 2009
Published: June 26, 2009

Citation
Iftach Klapp and David Mendlovic, "Improvement of matrix condition of Hybrid, space variant optics by the means of Parallel Optics design," Opt. Express 17, 11673-11689 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11673


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References

  1. G. M. Robbins, "Inverse Filtering for linear shift variant imaging system," Proc. IEEE,  60, 862-872 (1972). [CrossRef]
  2. H. Trussell and B. Hunt, "Image restoration of space-variant blurs by sectioned methods," IEEE.Trans. ASSP 26, 608-609 (1978). [CrossRef]
  3. H. Trussell and B. Hunt, "Image restoration of space-variant blurs by sectioned methods," Proc. IEEE. ASSP 3, 196-198 (1978).
  4. T.P. Costello and W.B. Mikhael, "Efficient restoration of space variant blurs from physical optics by sectioning with modified wiener filtering," Digital and Image Processing13, 1-22 (2003). [CrossRef]
  5. J. M. Varah, "On the numerical solution of ill-conditioned linear system with applications to ill posed problems," SIAM J. Numer. Anal  10, 257-265 (1972). [CrossRef]
  6. J. H. Wilkinson, " Rounding Errors in Algebraic Processes," 91-93 (Her Majesty’s stationery office, 1963).
  7. G. H. Golub and C. F. Van-loan, "Matrix Computation," 17-185 (North oxford academic, 1983).
  8. C. L. Lawson and R. J. Hanson, "Solving Least Squares Problems," SIAM J. Numer. Anal, 185-8 (1995)
  9. I. F. Gorodnitsky and D. Rao, "Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers, " 1, 25-29 (1994).
  10. C. M. Leung and W. S. Lu, "An L curve approach to optimal determination of regularization parameter in image restoration," Proc. IEEE 1021-1024 (1993).
  11. X. Wang, "Effect of small rank modification on the condition number of matrix," Computers and mathematics with applications 54, 819-825 (2007). [CrossRef]
  12. S. Twomay, "Information content in remote sensing, " Appl. Opt,  13, 942-945 (1974).
  13. M. Bertero and P. Boccacci, "Introduction to inverse problems in imaging," 86, 252 (IOP,1998).
  14. A. A. Sawchuk and M. J. Peyrovian, "Restoration of astigmatism and curvature of field, " J. Opt. Soc. Am. 65, 712-715 (1975). [CrossRef]
  15. H. C. Andrews and C.L. Paterson, "Singular value Decomposition and digital image processing," IEEE. Trans. ASSP. 24, 26-53 (1976). [CrossRef]
  16. J. W. Goodman, "Introduction to Fourier Optics," (Mcgraw-Hill, 1996).
  17. W. T. Welford, "Aberrations of Optical Systems," (Adam-Hilger, 1991).
  18. V. Shaoulov et al, " Model of wide angle optical field propagation using scalar diffraction theory," Opt. Eng,  43, 1561-1567 (2004). [CrossRef]
  19. A. W. Lohmann and W. T. Rhodes, "Two pupil synthesis of optical transfer function," Appl. Opt. 17, 1141-1146 (1978). [CrossRef] [PubMed]
  20. E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995). [CrossRef] [PubMed]
  21. W. Chi and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877, (2001). [CrossRef]
  22. S. Mezouari and A. R. Harvey, "Phase pupil function for reduction of defocus and spherical aberration," Opt. Lett. 28, 771-773, (2003). [CrossRef] [PubMed]
  23. S. Mezouari et al, "Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism," J, Opt, Soc. Am. A. 23, 1058-1062 (2006). [CrossRef]
  24. N. S. Kopeika, "A system Engineering approach to imaging," 517-520 (SPIE, 1998).

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