Abstract

We present a method for overcoming the pixel-limited resolution of digital imagers. Our method combines optical point-spread function engineering with subpixel image shifting. We place an optimized pseudorandom phase mask in the aperture stop of a conventional imager and demonstrate the improved performance that can be achieved by combining multiple subpixel shifted images. Simulation results show that the pseudorandom phase-enhanced lens (PRPEL) imager achieves as much as 50% resolution improvement over a conventional multiframe imager. The PRPEL imager also enhances reconstruction root-mean-squared error by as much as 20%. We present experimental results that validate the predicted PRPEL imager performance.

© 2007 Optical Society of America

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References

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  1. D. L. Litwiller, "CMOS vs. CCD: maturing technologies, maturing markets," Photonics Spectra (August 2005), pp. 54-59.
  2. L. Poletto and P. Nicolosi, "Enhancing the spatial resolution of a two-dimensional discrete array detector," Opt. Eng. 38, 1748-1757 (1999).
    [CrossRef]
  3. A. Papoulis, "Generalized sampling expansion," IEEE Trans. Circuits Syst. 24, 652-654 (1977).
    [CrossRef]
  4. S. Borman, "Topics in multiframe superresolution restoration," Ph.D. dissertation (University of Notre Dame, 2004).
  5. S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
    [CrossRef] [PubMed]
  6. N. Galatsanos and R. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. , Speech, Signal Process. 37, 415-421 (1989).
    [CrossRef]
  7. S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
    [CrossRef]
  8. H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," Comput. Vis. Graph. Image Process. 54, 181-186 (1992).
  9. M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy and undersampled images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
    [CrossRef] [PubMed]
  10. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, "Thin observation module by bound optics (TOMBO): concept and experimental verification," Appl. Opt. 40, 1806-1813 (2001).
    [CrossRef]
  11. Y. Kitamura, R. Shogenji, K. Yamada, S. Miyatake, M. Miyamoto, T. Morimoto, Y. Masaki, N. Kondou, D. Miyazaki, J. Tanida, and Y. Ichioka, "Reconstruction of a high-resolution image on a compound-eye image-capturing system," Appl. Opt. 43, 1719-1727 (2004).
    [CrossRef] [PubMed]
  12. P. M. Shankar, W. C. Hasenplaugh, R. L. Morrison, R. A. Stack, and M. A. Neifeld, "Multiaperture imaging," Appl. Opt. 45, 2871-2883 (2006).
    [CrossRef] [PubMed]
  13. M. A. Neifeld and A. Ashok, "Imaging using alternate point spread functions: lenslets with pseudo-random phase diversity," in Proceedings of the Optical Society of America Topical Meeting: Computational Optical Sensing and Imaging (COSI) (OSA, 2005), paper CMB1.
    [PubMed]
  14. A. Ashok and M. A. Neifeld, "Engineering the point spread function for superresolution from multiple low-resolution sub-pixel shifted frames," in Proceedings of the Optical Society of America Annual Meeting (OSA, 2005), paper FTHu4.
  15. Q. Tian and M. N. Huhns, "Algorithms for subpixel registration," Comput. Vis. Graph. Image Process. 35, 220-233 (1986).
    [CrossRef]
  16. S. Verdu, Multiuser Detection (Cambridge U. Press, 1998), Chap. 2.
  17. J. Solmon, Z. Zalevsky, and D. Mendlovicm, "Geometric superresolution by code division multiplexing," Appl. Opt. 44, 32-40 (2005).
  18. A. Ashok and M. A. Neifeld, "Information-based analysis of simple incoherent imaging systems," Opt. Express 11, 2153-2162 (2003).
    [CrossRef] [PubMed]
  19. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004), Chaps. 7 and 15.
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.
  21. E. Y. Lam, "Noise in superresolution reconstruction," Opt. Lett. 28, 2234-2236 (2003).
    [CrossRef] [PubMed]
  22. H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice Hall, 1977).
  23. D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
    [CrossRef] [PubMed]
  24. D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
    [CrossRef]
  25. D. J. Field and N. Brady, "Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes," Vision Res. 37, 3367-3383 (1997).
    [CrossRef]
  26. J. Burg, "Maximum entropy spectral analysis," Ph.D. dissertation (Stanford University, 1975).
  27. M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
    [CrossRef]
  28. A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).
  29. L. B. Lucy, "An iterative technique for the rectification of observed distribution," Astron. J. 79, 745-754 (1974).
    [CrossRef]
  30. W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. A 56, 1141-1142 (1972).
  31. A. Ashok and M. A. Neifeld, "Recent progress on multidomain optimization for ultrathin cameras," in Proc. SPIE 6232, 62320N (2006).
    [CrossRef]

2006 (2)

A. Ashok and M. A. Neifeld, "Recent progress on multidomain optimization for ultrathin cameras," in Proc. SPIE 6232, 62320N (2006).
[CrossRef]

P. M. Shankar, W. C. Hasenplaugh, R. L. Morrison, R. A. Stack, and M. A. Neifeld, "Multiaperture imaging," Appl. Opt. 45, 2871-2883 (2006).
[CrossRef] [PubMed]

2005 (2)

J. Solmon, Z. Zalevsky, and D. Mendlovicm, "Geometric superresolution by code division multiplexing," Appl. Opt. 44, 32-40 (2005).

D. L. Litwiller, "CMOS vs. CCD: maturing technologies, maturing markets," Photonics Spectra (August 2005), pp. 54-59.

2004 (2)

2003 (2)

2001 (1)

1999 (1)

L. Poletto and P. Nicolosi, "Enhancing the spatial resolution of a two-dimensional discrete array detector," Opt. Eng. 38, 1748-1757 (1999).
[CrossRef]

1997 (3)

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy and undersampled images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

D. J. Field and N. Brady, "Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes," Vision Res. 37, 3367-3383 (1997).
[CrossRef]

1992 (2)

D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," Comput. Vis. Graph. Image Process. 54, 181-186 (1992).

1991 (1)

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

1990 (1)

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

1989 (1)

N. Galatsanos and R. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. , Speech, Signal Process. 37, 415-421 (1989).
[CrossRef]

1986 (1)

Q. Tian and M. N. Huhns, "Algorithms for subpixel registration," Comput. Vis. Graph. Image Process. 35, 220-233 (1986).
[CrossRef]

1977 (2)

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).

A. Papoulis, "Generalized sampling expansion," IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

1974 (1)

L. B. Lucy, "An iterative technique for the rectification of observed distribution," Astron. J. 79, 745-754 (1974).
[CrossRef]

1972 (1)

W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. A 56, 1141-1142 (1972).

Andrews, H. C.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice Hall, 1977).

Ashok, A.

A. Ashok and M. A. Neifeld, "Recent progress on multidomain optimization for ultrathin cameras," in Proc. SPIE 6232, 62320N (2006).
[CrossRef]

A. Ashok and M. A. Neifeld, "Information-based analysis of simple incoherent imaging systems," Opt. Express 11, 2153-2162 (2003).
[CrossRef] [PubMed]

M. A. Neifeld and A. Ashok, "Imaging using alternate point spread functions: lenslets with pseudo-random phase diversity," in Proceedings of the Optical Society of America Topical Meeting: Computational Optical Sensing and Imaging (COSI) (OSA, 2005), paper CMB1.
[PubMed]

A. Ashok and M. A. Neifeld, "Engineering the point spread function for superresolution from multiple low-resolution sub-pixel shifted frames," in Proceedings of the Optical Society of America Annual Meeting (OSA, 2005), paper FTHu4.

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004), Chaps. 7 and 15.

Borman, S.

S. Borman, "Topics in multiframe superresolution restoration," Ph.D. dissertation (University of Notre Dame, 2004).

Bose, N. K.

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Brady, N.

D. J. Field and N. Brady, "Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes," Vision Res. 37, 3367-3383 (1997).
[CrossRef]

Burg, J.

J. Burg, "Maximum entropy spectral analysis," Ph.D. dissertation (Stanford University, 1975).

Chao, T.

D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Chin, R.

N. Galatsanos and R. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. , Speech, Signal Process. 37, 415-421 (1989).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).

Elad, M.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
[CrossRef] [PubMed]

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy and undersampled images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Farsiu, S.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
[CrossRef] [PubMed]

Feuer, A.

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy and undersampled images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Field, D. J.

D. J. Field and N. Brady, "Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes," Vision Res. 37, 3367-3383 (1997).
[CrossRef]

Galatsanos, N.

N. Galatsanos and R. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. , Speech, Signal Process. 37, 415-421 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

Gross, D.

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," Comput. Vis. Graph. Image Process. 54, 181-186 (1992).

Hasenplaugh, W. C.

Huhns, M. N.

Q. Tian and M. N. Huhns, "Algorithms for subpixel registration," Comput. Vis. Graph. Image Process. 35, 220-233 (1986).
[CrossRef]

Hunt, B. R.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice Hall, 1977).

Ichioka, Y.

Irani, M.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Ishida, K.

Kim, S. P.

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Kitamura, Y.

Kondou, N.

Kumagai, T.

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).

Lam, E. Y.

Litwiller, D. L.

D. L. Litwiller, "CMOS vs. CCD: maturing technologies, maturing markets," Photonics Spectra (August 2005), pp. 54-59.

Lucy, L. B.

L. B. Lucy, "An iterative technique for the rectification of observed distribution," Astron. J. 79, 745-754 (1974).
[CrossRef]

Masaki, Y.

Mendlovicm, D.

Milanfar, P.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
[CrossRef] [PubMed]

Miyamoto, M.

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Morrison, R. L.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004), Chaps. 7 and 15.

Neifeld, M. A.

A. Ashok and M. A. Neifeld, "Recent progress on multidomain optimization for ultrathin cameras," in Proc. SPIE 6232, 62320N (2006).
[CrossRef]

P. M. Shankar, W. C. Hasenplaugh, R. L. Morrison, R. A. Stack, and M. A. Neifeld, "Multiaperture imaging," Appl. Opt. 45, 2871-2883 (2006).
[CrossRef] [PubMed]

A. Ashok and M. A. Neifeld, "Information-based analysis of simple incoherent imaging systems," Opt. Express 11, 2153-2162 (2003).
[CrossRef] [PubMed]

A. Ashok and M. A. Neifeld, "Engineering the point spread function for superresolution from multiple low-resolution sub-pixel shifted frames," in Proceedings of the Optical Society of America Annual Meeting (OSA, 2005), paper FTHu4.

M. A. Neifeld and A. Ashok, "Imaging using alternate point spread functions: lenslets with pseudo-random phase diversity," in Proceedings of the Optical Society of America Topical Meeting: Computational Optical Sensing and Imaging (COSI) (OSA, 2005), paper CMB1.
[PubMed]

Nicolosi, P.

L. Poletto and P. Nicolosi, "Enhancing the spatial resolution of a two-dimensional discrete array detector," Opt. Eng. 38, 1748-1757 (1999).
[CrossRef]

Papoulis, A.

A. Papoulis, "Generalized sampling expansion," IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

Peleg, S.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Poletto, L.

L. Poletto and P. Nicolosi, "Enhancing the spatial resolution of a two-dimensional discrete array detector," Opt. Eng. 38, 1748-1757 (1999).
[CrossRef]

Richardson, W. H.

W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. A 56, 1141-1142 (1972).

Robinson, D.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
[CrossRef] [PubMed]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).

Ruderman, D. L.

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

Shankar, P. M.

Shogenji, R.

Solmon, J.

Stack, R. A.

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Tanida, J.

Tian, Q.

Q. Tian and M. N. Huhns, "Algorithms for subpixel registration," Comput. Vis. Graph. Image Process. 35, 220-233 (1986).
[CrossRef]

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Ur, H.

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," Comput. Vis. Graph. Image Process. 54, 181-186 (1992).

Valenzuela, H. M.

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Verdu, S.

S. Verdu, Multiuser Detection (Cambridge U. Press, 1998), Chap. 2.

Yamada, K.

Zalevsky, Z.

Appl. Opt. (4)

Astron. J. (1)

L. B. Lucy, "An iterative technique for the rectification of observed distribution," Astron. J. 79, 745-754 (1974).
[CrossRef]

Comput. Vis. Graph. Image Process. (2)

Q. Tian and M. N. Huhns, "Algorithms for subpixel registration," Comput. Vis. Graph. Image Process. 35, 220-233 (1986).
[CrossRef]

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," Comput. Vis. Graph. Image Process. 54, 181-186 (1992).

CVGIP: Graph. Models Image Process. (1)

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

IEEE Trans. Acoust. (2)

N. Galatsanos and R. Chin, "Digital restoration of multichannel images," IEEE Trans. Acoust. , Speech, Signal Process. 37, 415-421 (1989).
[CrossRef]

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. , Speech, Signal Process. 38, 1013-1027 (1990).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "Generalized sampling expansion," IEEE Trans. Circuits Syst. 24, 652-654 (1977).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy and undersampled images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Process. 13, 1327-1344 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. A 56, 1141-1142 (1972).

J. R. Stat. Soc. Ser. B Methodol. (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. Ser. B Methodol. 39, 1-38 (1977).

Ophthalmic. Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic. Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Opt. Eng. (1)

L. Poletto and P. Nicolosi, "Enhancing the spatial resolution of a two-dimensional discrete array detector," Opt. Eng. 38, 1748-1757 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Photonics Spectra (1)

D. L. Litwiller, "CMOS vs. CCD: maturing technologies, maturing markets," Photonics Spectra (August 2005), pp. 54-59.

Proc. SPIE (1)

A. Ashok and M. A. Neifeld, "Recent progress on multidomain optimization for ultrathin cameras," in Proc. SPIE 6232, 62320N (2006).
[CrossRef]

Vision Res. (2)

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

D. J. Field and N. Brady, "Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes," Vision Res. 37, 3367-3383 (1997).
[CrossRef]

Other (8)

J. Burg, "Maximum entropy spectral analysis," Ph.D. dissertation (Stanford University, 1975).

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice Hall, 1977).

S. Verdu, Multiuser Detection (Cambridge U. Press, 1998), Chap. 2.

S. Borman, "Topics in multiframe superresolution restoration," Ph.D. dissertation (University of Notre Dame, 2004).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004), Chaps. 7 and 15.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

M. A. Neifeld and A. Ashok, "Imaging using alternate point spread functions: lenslets with pseudo-random phase diversity," in Proceedings of the Optical Society of America Topical Meeting: Computational Optical Sensing and Imaging (COSI) (OSA, 2005), paper CMB1.
[PubMed]

A. Ashok and M. A. Neifeld, "Engineering the point spread function for superresolution from multiple low-resolution sub-pixel shifted frames," in Proceedings of the Optical Society of America Annual Meeting (OSA, 2005), paper FTHu4.

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Figures (27)

Fig. 1
Fig. 1

Schematic depicting the effect of pixel-limited resolution. (a) Optical PSF is impulse-like and (b) engineered optical PSF is extended.

Fig. 2
Fig. 2

Imaging system setup used in the simulation study.

Fig. 3
Fig. 3

Example simulated PSFs. (a) Conventional sinc2(⋅) PSF and (b) PSF obtained from PRPEL imager.

Fig. 4
Fig. 4

Reconstruction incorporates object priors. (a) Object class used for training and (b) PSD obtained from the object class and the best power-law fit used to define the LMMSE operator.

Fig. 5
Fig. 5

Rayleigh resolution estimation for multiframe imagers using a sinc2(⋅) fit to the postprocessed PSF.

Fig. 6
Fig. 6

RMSE versus number of frames for a conventional imager.

Fig. 7
Fig. 7

Rayleigh resolution versus number of frames for a conventional imager.

Fig. 8
Fig. 8

Example Rayleigh resolution versus mask roughness parameter Δ for ρ = 10λ c and K = 3.

Fig. 9
Fig. 9

Example RMSE versus mask roughness parameter Δ for ρ = 10λc and K = 3.

Fig. 10
Fig. 10

Rayleigh resolution versus number of frames for both PRPEL and conventional imagers.

Fig. 11
Fig. 11

RMSE versus number of frames for both PRPEL and conventional imagers.

Fig. 12
Fig. 12

Schematic of the optical setup used for experimental validation of the PRPEL imager.

Fig. 13
Fig. 13

Experimentally measured PSFs obtained from the (a) conventional imager, (b) PRPEL imager, and (c) simulated PRPEL PSF with phase-mask parameters Δ = 2.0 λ c and ρ = 175 λ c .

Fig. 14
Fig. 14

Experimentally measured Rayleigh resolution versus number of frames for both the PRPEL and conventional imagers.

Fig. 15
Fig. 15

USAF resolution target (a) group 0 element 1 and (b) group 0 elements 2 and 3.

Fig. 16
Fig. 16

Raw detector measurements obtained using USAF group 0 element 1 from (a) the conventional imager and (b) the PRPEL imager.

Fig. 17
Fig. 17

LMMSE reconstructions of USAF group 0 element 1 with left column for PRPEL imager and right column for conventional imager. Top row, K = 1 ; middle row, K = 4 ; bottom row, K = 9 .

Fig. 18
Fig. 18

(Color online) Horizontal line scans through the USAF target and its LMMSE reconstruction for conventional and PRPEL imagers for K = 4 . (a) Group 0 elements 1 and (b) group 0 elements 2 and 3.

Fig. 19
Fig. 19

LMMSE reconstructions of USAF group 0 element 2 and 3 with left column for PRPEL imager and right column for conventional imager. Top row, K = 1 ; middle row, K = 4 ; bottom row, K = 9 .

Fig. 20
Fig. 20

RL reconstructions of USAF group 0 element 1 with left column for PRPEL imager and right column for conventional imager. Top row, K = 1 ; middle row, K = 4 ; bottom row, K = 9 .

Fig. 21
Fig. 21

RL reconstructions of USAF group 0 element 2 and 3 with left column for PRPEL imager and right column for conventional imager. Top row, K = 1 ; middle row, K = 4 ; bottom row, K = 9 .

Fig. 22
Fig. 22

(Color online) Horizontal line scans through the USAF target and its RL reconstruction for conventional and PRPEL imagers for K = 4 . (a) Group 0 elements 1 and (b) group 0 elements 2 and 3.

Fig. 23
Fig. 23

Rayleigh resolution versus number of frames for multiframe imagers that employ smaller pixels and lower measurement SNR.

Fig. 24
Fig. 24

RMSE versus the number of frames for multiframe imagers that employ smaller pixels and lower measurement SNR.

Fig. 25
Fig. 25

Optical PSF obtained using PRPEL with both narrowband ( 10   nm ) and broadband ( 150   nm ) illumination.

Fig. 26
Fig. 26

Rayleigh resolution versus the number of frames for broadband PRPEL and conventional imagers.

Fig. 27
Fig. 27

RMSE versus the number of frames for broadband PRPEL and conventional imagers.

Equations (16)

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g = H c d f c + n ,
f i = S Φ i f c ( r ) ϕ i ( r ) d r 2 ,
f a ( r ) = i = 1 N f i ψ i ( r ) ,
ϕ i ( r ) = 1 Ω r   rect ( r i Ω r Ω r ) ,
Φ i Φ j ϕ i ( r ) ϕ j ( r ) d r 2 = δ i j ,
g = H f + n ,
g i = H i f + n i ,
g = H c f + n ,
R Φ Φ ( r ) = Δ 2 12 exp [ r 2 4 ρ 2 ] .
p s f ( r ) = A c ( λ f ) 4 | T pupil ( r λ f ) | 2 ,
T pupil ( ω ) = F { exp [ j 2 π ( n 1 ) Φ ( r ) / λ ] t a p ( r ) } ,
f ^ = W g ,
W = R f H c T ( H c R f H c T + R n ) 1 ,
RMSE = f ^ f 2 255 × 100 % ,
f ^ n ( k + 1 ) = f ^ n ( k ) 1 s n m = 1 K M g m ( H c f ^ ( k ) ) m ( H c ) m n ,
s n = m = 1 K M ( H c ) m n ,

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