## Physically constrained Fourier transform deconvolution method

JOSA A, Vol. 26, Issue 5, pp. 1191-1194 (2009)

http://dx.doi.org/10.1364/JOSAA.26.001191

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### Abstract

An iterative Fourier-transform-based deconvolution method for resolution enhancement is presented. This method makes use of the *a priori* information that the data are real and positive. The method is robust in the presence of noise and is efficient especially for large data sets, since the fast Fourier transform can be employed.

© 2009 Optical Society of America

## 1. INTRODUCTION

*a priori*information that in many cases of interest the measured quantities are real and positive, which is employed in some of the better known methods [1, 2

2. P. A. Jansson, R. H. Hunt, and E. K. Plyler, “Resolution enhancement of spectra,” J. Opt. Soc. Am. **60**, 596–599 (1970). [CrossRef]

3. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. **62**, 511–517 (1972). [CrossRef] [PubMed]

4. B. R. Frieden and J. J. Burke, “Restoring with maximum entropy, II: superresolution of photographs of diffraction-blurred impulses,” J. Opt. Soc. Am. **62**, 1202–1210 (1972). [CrossRef]

## 2. THEORY

*N*data points and

*a priori*knowledge that the original data do not have negative values to restore missing frequency coefficients. Let

*M*be the minimum magnitude of

*a priori*information about the data, in particular the positive value of many physically measurable quantities such as, for example, the intensity of spectral lines. This can be accomplished by minimizing the sum of the squares of the negative portion of the deconvolved data. Proceeding in a manner similar to the approach of Howard [6

6. S. J. Howard, “Continuation of discrete Fourier spectra using a minimum-negativity constraint,” J. Opt. Soc. Am. **71**, 819–824 (1981). [CrossRef]

*K*in this expression is an arbitrary parameter that in the limit that

*N*for a particular unknown frequency

*m*subscript denotes a

*v*missing from the measured data, the FT expansion gives

*m*refers to the coefficients of the missing frequencies, Eq. (8) and the orthogonality of the summation eliminates

## 3. DATA AND ANALYSIS

7. P. J. Tadrous, BiaQIm image processing software, Version 21.01, 2008, URL http://www.bialith.com.

## 4. CONCLUSIONS

*a priori*information can be incorporated. Conditions such as finite extent [8] (also known as compact support) and bounded upper limits can be handled in exactly the same way.

## ACKNOWLEDGMENTS

1. | Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. |

2. | P. A. Jansson, R. H. Hunt, and E. K. Plyler, “Resolution enhancement of spectra,” J. Opt. Soc. Am. |

3. | B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. |

4. | B. R. Frieden and J. J. Burke, “Restoring with maximum entropy, II: superresolution of photographs of diffraction-blurred impulses,” J. Opt. Soc. Am. |

5. | P. A. Jansson, |

6. | S. J. Howard, “Continuation of discrete Fourier spectra using a minimum-negativity constraint,” J. Opt. Soc. Am. |

7. | P. J. Tadrous, BiaQIm image processing software, Version 21.01, 2008, URL http://www.bialith.com. |

8. | S. J. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. |

**OCIS Codes**

(070.4790) Fourier optics and signal processing : Spectrum analysis

(100.1830) Image processing : Deconvolution

(100.2980) Image processing : Image enhancement

(300.6170) Spectroscopy : Spectra

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 11, 2008

Revised Manuscript: March 5, 2009

Manuscript Accepted: March 16, 2009

Published: April 15, 2009

**Citation**

Francis A. Flaherty, "Physically constrained Fourier transform deconvolution method," J. Opt. Soc. Am. A **26**, 1191-1194 (2009)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-5-1191

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### References

- Y. Biraud, “A new approach for increasing the resolving power by data processing,” Astron. Astrophys. 1, 124-127 (1969).
- P. A. Jansson, R. H. Hunt, and E. K. Plyler, “Resolution enhancement of spectra,” J. Opt. Soc. Am. 60, 596-599 (1970). [CrossRef]
- B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511-517 (1972). [CrossRef] [PubMed]
- B. R. Frieden, and J. J. Burke, “Restoring with maximum entropy, II: superresolution of photographs of diffraction-blurred impulses,” J. Opt. Soc. Am. 62, 1202-1210 (1972). [CrossRef]
- P. A. Jansson, Deconvolution of Images and Spectra (Academic, 1997).
- S. J. Howard, “Continuation of discrete Fourier spectra using a minimum-negativity constraint,” J. Opt. Soc. Am. 71, 819-824 (1981). [CrossRef]
- P. J. Tadrous, BiaQIm image processing software, Version 21.01, 2008, URL http://www.bialith.com.
- S. J. Howard, “Method for continuing Fourier spectra given by the fast Fourier transform,” J. Opt. Soc. Am. 71, 95-98 (1981).

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