## Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution

JOSA A, Vol. 20, Issue 8, pp. 1516-1527 (2003)

http://dx.doi.org/10.1364/JOSAA.20.001516

Enhanced HTML Acrobat PDF (469 KB)

### Abstract

Several powerful iterative algorithms are being developed for the restoration and superresolution of diffraction-limited imagery data by use of diverse mathematical techniques. Notwithstanding the mathematical sophistication of the approaches used in their development and the potential for resolution enhancement possible with their implementation, the spectrum extrapolation that is central to superresolution comes in these algorithms only as a by-product and needs to be checked only after the completion of the processing steps to ensure that an expansion of the image bandwidth has indeed occurred. To overcome this limitation, a new approach of mathematically extrapolating the image spectrum and employing it to design constraint sets for implementing set-theoretic estimation procedures is described. Performance evaluation of a specific projection-onto-convex-sets algorithm by using this approach for the restoration and superresolution of degraded images is outlined. The primary goal of the method presented is to expand the power spectrum of the input image beyond the range of the sensor that captured the image.

© 2003 Optical Society of America

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.2980) Image processing : Image enhancement

(100.3020) Image processing : Image reconstruction-restoration

(100.6640) Image processing : Superresolution

**History**

Original Manuscript: October 27, 2002

Revised Manuscript: January 30, 2003

Manuscript Accepted: January 30, 2003

Published: August 1, 2003

**Citation**

Supratik Bhattacharjee and Malur K. Sundareshan, "Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution," J. Opt. Soc. Am. A **20**, 1516-1527 (2003)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-20-8-1516

Sort: Year | Journal | Reset

### References

- B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995). [CrossRef]
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
- C. K. Rushforth, J. L. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968). [CrossRef]
- D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
- D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–62 (1961). [CrossRef]
- B. R. Frieden, “Band-unlimited reconstruction of optical objects and spectral sources,” J. Opt. Soc. Am. 57, 1013–1019 (1967). [CrossRef]
- B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 51–55 (1972). [CrossRef]
- W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–60 (1972). [CrossRef]
- L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–759 (1974). [CrossRef]
- A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–38 (1977).
- L. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982). [CrossRef] [PubMed]
- B. R. Hunt, “Bayesian methods in digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977). [CrossRef]
- S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984). [CrossRef]
- R. K. Pina, R. C. Puetter, “Bayesian image reconstruction: the pixon and optimal image modeling,” Publ. Astron. Soc. Pac. 105, 630–637 (1993). [CrossRef]
- H. H. Barrett, D. W. Wilson, B. M. W. Tsui, “Noise properties of the EM algorithm: 1. Theory,” Phys. Med. Biol. 39, 833–846 (1994). [CrossRef] [PubMed]
- H. Y. Pang, M. K. Sundareshan, S. Amphay, “Optimized maximum-likelihood algorithms for superresolution of passive millimeter-wave imagery,” in, Passive Millimeter-Wave Imaging TechnologyII, R. Smith , ed., Proc. SPIE3378, 148–160 (1998). [CrossRef]
- H. Y. Pang, M. K. Sundareshan, S. Amphay, “Superresolution of millimeter-wave images by iterative blind maximum-likelihood restoration,” in Passive Millimeter-Wave Imaging Technology, R. Smith, ed., Proc. SPIE3064, 227–238 (1997). [CrossRef]
- R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974). [CrossRef]
- A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975). [CrossRef]
- L. G. Gubin, B. T. Polak, E. V. Raik, “The method of projection for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967). [CrossRef]
- A. Lent, H. Tuy, “An iterative method for extrapolation of band-limited functions,” J. Math. Anal. Appl. 83, 544–565 (1981). [CrossRef]
- D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978). [CrossRef]
- D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1. Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982). [CrossRef]
- M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: Part 2. Application and numerical results,” IEEE Trans. Med. Imaging MI-1, 95–101 (1982). [CrossRef]
- P. L. Combettes, “The foundations of set theoretic estimation,” Proc. IEEE 81, 182–208 (1993). [CrossRef]
- H. Stark, ed. Image Recovery: Theory and Application (Academic, San Diego, Calif., 1987).
- M. I. Sezan, “An overview of convex projections theory and its application to image recovery problems,” Ultramicroscopy 40, 55–67 (1992). [CrossRef]
- G. Crombez, “Image recovery by convex combinations of projections,” J. Math. Anal. Appl. 15, 413–419 (1991). [CrossRef]
- P. L. Combettes, H. Puh, “Iterations of parallel convex projections in Hilbert spaces,” Numer. Functi. Analy. Optim. 15, 225–243 (1994). [CrossRef]
- S. Bhattacharjee, M. K. Sundareshan, “Hybrid Bayesian and convex set projection algorithms for restoration and resolution enhancement of digital images,” in Applications of Digital Image Processing XXIII, A. G. Tescher, ed., Proc. SPIE4115, 12–22 (2000). [CrossRef]
- M. K. Sundareshan, S. Bhattacharjee, “Superresolution of passive millimeter-wave images using a combined maximum-likelihood optimization and projection-onto-convex-sets approach,” in Passive Millimeter-Wave Imaging Technology V, R. Smith, R. Appleby, eds., Proc. SPIE4373, 105–116 (2001). [CrossRef]
- D. Terzopoulos, “Regularization of the inverse visual problem involving discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 1, 413–424 (1986). [CrossRef]
- E. L. Kosarev, “Shannon’s super-resolution limit for signal recovery,” Inverse Probl. 6, 55–76 (1990). [CrossRef]
- M. K. Sundareshan, P. Zegers, “Role of over-sampled data in superresolution processing and a progressive up-sampling scheme for optimized implementations of iterative restoration algorithms,” in Passive Millimeter-Wave Imaging Technology III, R. Smith, ed., Proc. SPIE3703, 155–166 (1999). [CrossRef]
- P. J. Sementilli, B. R. Hunt, M. S. Nadar, “An analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993). [CrossRef]
- J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 372–381 (1986).

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.