Primary and Secondary Superresolution by Data Inversion (postprint)

Superresolution by data inversion is the extrapolation of measured Fourier data to regions outside the measurement bandwidth using postprocessing techniques. Here we characterize superresolution by data inversion for objects with finite support using the twin concepts of primary and secondary superresolution, where primary superresolution is the essentially unbiased portion of the superresolved spectra and secondary superresolution is the remainder. We show that this partition of superresolution into primary and secondary components can be used to explain why some researchers believe that meaningful superresolution is achievable with realistic signal-to-noise ratios, and other researchers do not. ©2005 Optical Society of America OCIS codes: (100.6640) Superresolution; (100.2980) Image enhancement; (100.3190); Inverse problems References and Links 1. M. Bertero and C. De Mol, “Super-resolution by data inversion,” in Progress in Optics XXXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1996), pp. 129-178. 2. S. Bhattacharjee and M. K. Sundareshan, “Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution,” J. Opt. Soc. Am. A 20, 1516-1527 (2003). 3. B. R. Hunt, “Super-resolution of images: algorithms, principles, and performance,” Int. J. Imaging Syst. Technol. 6, 297-304 (1995). 4. H. Liu, Y. Yan, Q. Tan, and G. Jin, “Theories for the design of diffractive superresolution elements and limits of optical superresolution,” J. Opt. Soc. Am. A 19, 2185-2193 (2002). 5. V. F. Canales, D. M. de Juana, and M. P. Cagigal, “Superresolution in compensated telescopes,” Opt. Lett. 29, 935-937 (2004). 6. C. K. Rushforth and R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539-545 (1968). 7. J. J. Green and B. R. Hunt, “Improved restoration of space object imagery,” J. Opt. Soc. Am. A 16, 2859-2865 (1999). 8. B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals based on the use of prolate functions,” in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 313-407. 9. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty – I,” Bell Syst. Tech. J. 40, 43-63 (1961). 10. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty – II,” Bell Syst. Tech. J. 40, 65-84 (1961). 11. M. Bertero and E. R. Pike, “Resolution in diffraction-limited imaging, a singular value analysis I. The case of coherent illumination,” Opt. Acta 29, 727-746 (1982). 12. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, 1996), p. 49. 13. W. P. Latham and M. L. Tilton, “Calculation of prolate functions for optical analysis,” Appl. Opt. 26, 2653-2658 (1987). 14. B. Porat, Digital Processing of Random Signals, Theory and Methods (Prentice-Hall, Englewood Cliffs, 1994), pp. 65-67. 15. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, 1992), ch. 5. 16. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN, 2 ed., (Cambridge Press, Cambridge, 1996), pp. 134-135. 17. C. L. Matson, “Variance reduction in Fourier spectra and their corresponding images with the use of support constraints,” J. Opt. Soc. Am. A 11, 97-106 (1994). 18. H. J. Landau and H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty – III: the dimension of the space of essentially time-and band-limited signals,” Bell Syst. Tech. J. 41, 1295-1336 (1962). 19. P. J. Sementilli, B. R. Hunt, and M. S. Nadar, “Analysis of the limit to superresolution in coherent imaging,” J. Opt. Soc. Am. A 10, 2265-2276 (1993). 20. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Sig. Process. 42, 156-163 (1994). 21. Y. L. Kosarev, “On the superresolution limit in signal reconstruction,” Sov. J. Commun. Technol. Electron. 35, 90-108 (1990). ___________________________________________________________________________