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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 29, Iss. 11 — Nov. 1, 2012
  • pp: 2263–2271

Point-spread function reconstruction in ground-based astronomy by l1-lp model

Raymond H. Chan, Xiaoming Yuan, and Wenxing Zhang  »View Author Affiliations

JOSA A, Vol. 29, Issue 11, pp. 2263-2271 (2012)

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In ground-based astronomy, images of objects in outer space are acquired via ground-based telescopes. However, the imaging system is generally interfered by atmospheric turbulence, and hence images so acquired are blurred with unknown point-spread function (PSF). To restore the observed images, the wavefront of light at the telescope’s aperture is utilized to derive the PSF. A model with the Tikhonov regularization has been proposed to find the high-resolution phase gradients by solving a least-squares system. Here we propose the l 1 - l p ( p = 1 , 2) model for reconstructing the phase gradients. This model can provide sharper edges in the gradients while removing noise. The minimization models can easily be solved by the Douglas–Rachford alternating direction method of a multiplier, and the convergence rate is readily established. Numerical results are given to illustrate that the model can give better phase gradients and hence a more accurate PSF. As a result, the restored images are much more accurate when compared to the traditional Tikhonov regularization model.

© 2012 Optical Society of America

OCIS Codes
(100.1830) Image processing : Deconvolution
(100.3020) Image processing : Image reconstruction-restoration
(100.5070) Image processing : Phase retrieval
(110.4155) Imaging systems : Multiframe image processing

ToC Category:
Imaging Systems

Original Manuscript: June 4, 2012
Revised Manuscript: August 22, 2012
Manuscript Accepted: September 12, 2012
Published: October 9, 2012

Raymond H. Chan, Xiaoming Yuan, and Wenxing Zhang, "Point-spread function reconstruction in ground-based astronomy by l1-lp model," J. Opt. Soc. Am. A 29, 2263-2271 (2012)

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  1. T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet and Stochastic Methods (SIAM, 2005).
  2. L. Rudin, S. Osher, and F. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992). [CrossRef]
  3. A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems(V. H. Winston, 1997).
  4. T. F. Chan and C. K. Wong, “Total variation blind deconvolution,” IEEE. Trans. Image Process. 7, 370–375 (1998). [CrossRef]
  5. E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A. 17, 1177–1184 (2000). [CrossRef]
  6. J. M. Bardsley, “Wavefront reconstruction methods for adaptive optics systems on ground-based telescopes,” SIAM J. Matrix Anal. Appl. 30, 67–83 (2008). [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  8. S. M. Jefferies, M. L. Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. 41, 2095–2102 (2002). [CrossRef]
  9. L. M. Mugnier, C. Robert, J. M. Conan, V. Michau, and S. Salem, “Myopic deconvolution from wave-front sensing,” J. Opt. Soc. Am. 18, 862–872 (2001). [CrossRef]
  10. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977). [CrossRef]
  11. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977). [CrossRef]
  12. S. M. Jefferies and M. Hart, “Deconvolution from wave front sensing using the frozen flow hypothesis,” Opt. Express 19, 1975–1984 (2011). [CrossRef]
  13. J. Nagy, S. Jefferies, and Q. Chu, “Fast PSF reconstruction using the frozen flow hypothesis,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference (2010).
  14. N. K. Bose and K. Boo, “High-resolution image reconstruction with multisensors,” Int. J. Imaging Syst. Technol. 9, 294–304 (1998). [CrossRef]
  15. R. H. Chan, T. F. Chan, L. X. Shen, and Z. W. Shen, “Wavelet algorithms for high-resolution image reconstruction,” SIAM J. Sci. Comput. 24, 1408–1432 (2003). [CrossRef]
  16. R. H. Chan, Z. W. Shen, and T. Xia, “A framelet algorithm for enhancing video stills,” Appl. Comput. Harmon. Anal. 23, 153–170 (2007). [CrossRef]
  17. R. Y. Tsai and T. S. Huang, “Multiframe image restoration and registration,” in Advances in Computer Vision and Image Processing (1984), Vol. 1, pp. 317–339.
  18. S. Aliney, “A property of the minimum vectors of a regularizing functional defined by means of the absolute norm,” IEEE Trans. Signal Process. 45, 913–917 (1997). [CrossRef]
  19. M. Nikolova, “A variational approach to remove outliers and impulse noise,” J. Math. Imaging. Vis. 20, 99–120 (2004). [CrossRef]
  20. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Comput. Math. Appl. 2, 17–40 (1976). [CrossRef]
  21. R. Glowinski and A. Marocco, “Approximation par éléments finis d’ordre un et résolution par pénalisation dualité d’une classe de problèmes non linéaires,” RAIRO R2, 41–76 (1975).
  22. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-Verlag, 1984).
  23. B. S. He and X. M. Yuan, “On the O(1/n) convergence rate of Douglas-Rachford alternating direction method,” SIAM J. Numer. Anal. 50, 700–709 (2012). [CrossRef]
  24. J. Nocedaland and S. J. Wright, Numerical Optimization(Springer-Verlag, 2006).
  25. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010). [CrossRef]
  26. M. Afonso, J. Bioucas-Dias, and M. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010). [CrossRef]
  27. R. H. Chan, J. F. Yang, and X. M. Yuan, “Alternating direction method for image inpainting in wavelet domain,” SIAM J. Imaging Sci. 4, 807–826 (2011). [CrossRef]
  28. M. Ng, P. A. Weiss, and X. M. Yuan, “Solving constrained total-variation problems via alternating direction methods,” SIAM J. Sci. Comput. 32, 2710–2736 (2010). [CrossRef]
  29. D. Donoho, “De-noising by soft-thresholding,” IEEE Trans. Inform. Theory 41, 613–627 (1995). [CrossRef]
  30. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University, 1996).
  31. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979). [CrossRef]
  32. V. A. Morozov, Methods for Solving Incorrectly Posed Problems. (Springer-Verlag, 1984).
  33. Y. W. Wen and R. H. Chan, “Parameter selection for total variation based image restoration using discrepancy principle,” IEEE Trans. Image Process. 21, 1770–1781 (2012). [CrossRef]
  34. B. S. He, H. Yang, and S. L. Wang, “Alternating direction method with self-Adaptive penalty parameters for monotone variational inequalities,” J. Opt. Theory Appl. 106, 337–356 (2000). [CrossRef]
  35. B. S. He, L. Z. Liao, D. R. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Math. Program. 92, 103–118 (2002). [CrossRef]
  36. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999). [CrossRef]
  37. B. S. He and X. M. Yuan, “Convergence analysis of primal-dual algorithms for total variation image restoration,” SIAM J. Imaging Sci. 5, 119–149 (2012). [CrossRef]
  38. H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. 35, 1561–1565 (1996). [CrossRef]

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