OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 28, Iss. 6 — Mar. 15, 1989
  • pp: 1235–1243

Image recovery by minimum discrimination from a template

B. Roy Frieden  »View Author Affiliations

Applied Optics, Vol. 28, Issue 6, pp. 1235-1243 (1989)

View Full Text Article

Enhanced HTML    Acrobat PDF (1525 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A longstanding problem of estimation is extraction of the signal image from a given noisy image. Here, prior knowledge in the form of a second, template image is also assumed to be present. Examples of templates are a blurred version of the signal image, the reference image used in cross-entropy minimization, or the signal power spectrum. We propose and test a method of optimally combining the data and template images to form an improved output. The output is biased, respectively, toward the template or the data image by trading off two goals: (a) minimum output probability of being successfully distinguished from the template as predicted by standard maximum likelihood theory, and (b) maximum output probability of having formed the image data. For additive Gaussian noise the estimation approach is least-squares; for Poisson noise the approach is a compromise between maximum Shannon cross entropy and maximum Burg-type entropy; and for exponential noise the approach includes maximum Burg entropy.

© 1989 Optical Society of America

Original Manuscript: May 23, 1988
Published: March 15, 1989

B. Roy Frieden, "Image recovery by minimum discrimination from a template," Appl. Opt. 28, 1235-1243 (1989)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. See, e.g., H. Stark, Ed. Image Recovery, Theory and Applications (Academic, Orlando, 1987).
  2. The classic treatment of Wiener-Kolmogoroff filtering is in H.W. Bode, C. E. Shannon, “A Simplified Derivation of Linear Least-Square Smoothing and Prediction Theory,” Proc. IRE 38, 417 (1950). [CrossRef]
  3. L. A. Wainstein, V. D. Zubakov, Extraction of Signals from Noise (Prentice-Hall, Englewood Cliffs, NJ, 1962).
  4. R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” J. Basic Eng., ASME Trans. 82D, 35 (1960). [CrossRef]
  5. G. U. Yule, “On the Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer’s Sun-Spot Numbers,” Philos. Trans. R. Soc. London Ser. A 226, 267 (1927). This is the original reference on autoregressive modeling. A more recent review is in A. S.Willsky, Digital Signal Processing and Control, and Estimation Theory (MIT Press, Cambridge, 1979). [CrossRef]
  6. A. C. Schell, “Enhancing the Angular Resolution of Incoherent Sources,” Radio Electron. Eng. 29, 21 (1965). [CrossRef]
  7. H. Wolter, “On Basic Analogies and Principal Differences Between Optical and Electronic Information,” Prog. Opt. 1, 155 (1961). [CrossRef]
  8. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. ACM 9, 8–97 (1962). [CrossRef]
  9. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Spectral Resolution Enhancement,” J. Opt. Soc. Am. 60, 596 (1970). [CrossRef]
  10. J. E. Shore, R. W. Johnson, “Properties of Cross-Entropy Minimization,” IEEE Trans. Inf. Theory IT-27, 472 (1981). [CrossRef]
  11. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1 (Wiley, New York, 1968).
  12. B. R. Frieden, Probability, Statistical Optics and Data Testing (Springer-Verlag, New York, 1983). [CrossRef]
  13. J. L. Harris, “Resolving Power and Decision Theory,” J. Opt. Soc. Am. 54, 606 (1964). [CrossRef]
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  15. J. P. Burg, “Maximum Entropy Spectral Analysis,” paper presented at Thirty-Seventh Meeting of the Society of Exploration Geophysicists, Oklahoma City (1960);J. P. Burg, “Maximum Entropy Spectral Analysis,” Ph.D. Dissertation, Geophysics Department, Stanford U. (May1975).
  16. J. W. Goodman, “Some Effects of Target-Induced Scintillation on Optical Radar Performance,” Proc. IEEE 53, 1688 (1965). [CrossRef]
  17. V. S. Frost, J. A. Stiles, K. S. Shanmugan, J. C. Holtzman, “A Model for Radar Images and Its Application to Adaptive Digital Filtering of Multiplicative Noise,” in IEEE Trans. Pattern Anal. Machine Intell. PAMI-4, 157 (1982). [CrossRef]
  18. J. C. Dainty, “Stellar Speckle Interferometry” (Springer-Verlag, New York, 1984).
  19. J. W. Tukey, Exploratory Data Analysis (Addison-Wesley, Reading, MA, 1977).

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.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited