## Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels

JOSA A, Vol. 15, Issue 11, pp. 2841-2848 (1998)

http://dx.doi.org/10.1364/JOSAA.15.002841

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### Abstract

We consider the problem of estimating one nonblurred and cleaned image from a sequence of <i>P</i> randomly translated images corrupted with Poisson noise. We develop a new algorithm based on maximum-likelihood (ML) estimation for two unknown parameters: the reconstructed image itself and the set of translations of the low-light-level images. We demonstrate that the ML reconstructed image is proportional to the sum of the low-light-level images after correcting for the unknown movement and that its entropy is minimal. The images of the sequence are matched together by means of an iterative minimum-entropy algorithm, where a systematic search under displacements for the images is performed. We develop a fast version of this algorithm, and we present results for simulated images and experimental data. The probability of good matching of a low-level image sequence is estimated numerically when the light level of the images in the sequence decreases, corresponding to small numbers of photons detected (down to 20) in each image of the sequence. We compare these results with those obtained when the low-light-level images are matched to a known reference, i.e., the linear correlation method, and with those from the optimal one, when the noise has a Poisson distribution. This approach is applied to astronomical images that are acquired by photocounting from a balloon-borne ultraviolet imaging telescope.

© 1998 Optical Society of America

**OCIS Codes**

(100.0100) Image processing : Image processing

**Citation**

Mireille Guillaume, Pierre Melon, Philippe Réfrégier, and Antoine Llebaria, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A **15**, 2841-2848 (1998)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-11-2841

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### References

- R. L. Gilliland, P. D. Edmonds, L. Petro, A. Saha, and M. M. Shara, “Stellar variability in the central populations of 47 Tucanae from WF/PC observations with the Hubble Space Telescope. I. Project overview, reduction techniques, and first results,” Astrophys. J. 447, 191–203 (1995).
- J. A. Morse, “A method for correcting aspect solution errors in ROSAT HRI observations of compact sources,” Publ. Astron. Soc. Pac. 106, 675–682 (1994).
- R. E. Sequeira, J. A. Gubner, and B. E. A. Sala, “Image detection using low-level illumination,” IEEE Trans. Image Process. 2, 18–26 (1994).
- D. L. Snyder and T. L. Schulz, “High resolution imaging at low-light levels through weak turbulence,” J. Opt. Soc. Am. A 7, 1251–1265 (1990).
- B. D. Hall, D. Reinhard, and R. Monot, “Optimum rate for a CCD detector,” Rev. Sci. Instrum. 66, 2668–2671 (1995).
- G. M. Morris, “Scene matching using photon-limited images,” J. Opt. Soc. Am. A 1, 482–488 (1984).
- D. S. Lalush and M. W. Tsui, “The importance of preconditioners in fast Poisson-based iterative reconstruction algorithms for SPECT,” in IEEE Nuclear Science Symposium and Medical Imaging Conference Record, P. A. Moonier, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N. Y., 1995), Vol. 3, pp. 1326–1330.
- J. M. Northcott, G. R. Ayers, and J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
- G. R. Ayers, M. J. Northcott, and J. C. Dainty, “Knox–Thomson and triple correlation through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
- A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
- A. Vander Lugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
- B. V. K. Vijaya Kumar, F. M. Dickey, and J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
- J. D. Downie and J. F. Walkup, “Optimal correlation filters for images with signal-dependent noise,” J. Opt. Soc. Am. A 11, 1599–1609 (1994).
- M. Guillaume, Th. Amouroux, Ph. Refregier, B. Milliard, and A. Llebaria, “Optimal correlation at low photon levels: study for astronomical images,” Opt. Lett. 22, 322–324 (1997).
- B. J. Slocumb and D. L. Snyder, “Maximum likelihood estimation applied to quantum limited optical position sensing,” in Acquisition, Tracking, and Pointing IV, S. Gowrinathan, ed., Proc. SPIE 1304, 165–176 (1990).
- T. J. Schulz and D. L. Snyder, “Imaging a randomly moving object from quantum-limited data: applications to image recovery from second- and third-order autocorrelations,” J. Opt. Soc. Am. A 8, 801–807 (1991).
- W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
- L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
- R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992–1998 (1996).
- A. Prades, J. Nunez, F. Perez, V. Pala, and R. Arbiol, “Aerial photography restoration using MLE algorithm,” in ISPRS Commission III Symposium: Spatial Information from Digital Photogrammetry and Computer Vision, H. Ebner, C. Helpke, and K. Eder, eds., Proc. SPIE 2357, 683–688 (1994).
- B. Milliard, J. Donas, and M. Laget, “A 40-cm UV (2000 Å) balloon-borne imaging telescope: results and current work,” Adv. Space Res. 11, 135–137 (1991).
- M. Laget, D. Burgarella, B. Milliard, and J. Donas, “UV-(2000 Å) imaging of globular clusters,” Astron. Astrophys. 259, 510–521 (1992).
- T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
- C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
- J. L. Nieto, A. Llebaria, and S. di Serego Alighieri, “Photo-counting detectors in time-resolved imaging mode: image recentering and selection algorithms,” Astron. Astrophys. 178, 301–306 (1987).
- C. W. Therrien, Decision Estimation and Classification (Wiley, New York, 1989).
- G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).

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