Abstract

One of the most fascinating problems addressed today is retrieving high-resolution data of blurred images obtained from biological objects. In most cases the research relays either on a priory knowledge of the image nature or a large number of images (either of the same object or of different objects obtained by the same imaging setup). If saturation is added to the blurring, most algorithms fail to sharpen the image and in some cases researchers decline to use such images as an input. In this work a single captured blurred and saturated image is given with no a priori knowledge except of the fact that the primary blurring is due to defocused imaging setup. The authors suggest a novel three-stage approach for retrieving higher resolution data from the intensity distribution of the blurred and saturated image. The core of the process is the phase retrieval algorithm suggested by Gerchberg and Saxton in 1972. The new method is explained in details and the algorithm is tested numerically and experimentally on several images to show the improvement in the sharpness of the spatial details.

© 2008 Optical Society of America

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References

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  1. S. Bellini, Blind deconvolution, S. Haykin, ed. (1994) Chap. 2, pp. 8-55.
  2. N. K. Bose, "Wavelet-based blind super resolution from video sequence and in MRI," Pennsylvania State University - Final Report (2005).
  3. W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
    [CrossRef]
  4. H. Chang, D-Y Yeung, and Y. Xiong, "Super-resolution through neighbor embedding," in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '04), vol. 1, pp. I-275-I-282, Washington, DC, USA, June-July 2004.
  5. M. Elad and A. Feuer, "Restoration of a single super resolution image from several blurred, noisy, and under sampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
    [CrossRef] [PubMed]
  6. A. Zomet and S. Peleg, "Multi-sensor super-resolution," IEEE workshop on Applications of Computer Vision (WACV02), 27-31 (2002).
  7. D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
    [CrossRef]
  8. M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
    [CrossRef]
  9. D. Capel and A. Zisserman, "Computer vision applied to super resolution," IEEE Signal Process. Mag. 20, 75- 86 (2003).
    [CrossRef]
  10. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for determination of phase from image and diffraction plane picture," Optik (Stuttgart) 35, 237-246 (1972).
  11. J. R. Feinup, "Phase retrieval algorithms - a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef]
  12. R. W. Gerchberg, "Super-resolution through error energy reduction," Optica Acta 21, 709-720 (1974).
    [CrossRef]
  13. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
    [CrossRef]
  14. E. Gur and Z. Zalevsky, "Iterative single-image digital super-resolution using partial high-resolution data," Lecture Notes in Engineering and Computer Science,  WCE2007, 630-634 (2007).
  15. S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
    [CrossRef] [PubMed]
  16. J. M. Rodenburg, "The phase problem, microdiffraction and wavelength-limited resolution," ltramicroscopy27, 413-422 (1989).
    [CrossRef]
  17. G. Haberlandt, "Vergleichende anatomie des assimilatorischen gewebesystems bei pflanzen," Jahrbuch der Wissenschaftlichen Botanik 13, 74-188 (1881).
  18. E. Williams, "Fine structure of vascular and epidermal plastids of the mature maize leaf," Protoplasma 79, 395-400 (1974).
    [CrossRef]

2007 (1)

E. Gur and Z. Zalevsky, "Iterative single-image digital super-resolution using partial high-resolution data," Lecture Notes in Engineering and Computer Science,  WCE2007, 630-634 (2007).

2003 (3)

D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
[CrossRef]

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

D. Capel and A. Zisserman, "Computer vision applied to super resolution," IEEE Signal Process. Mag. 20, 75- 86 (2003).
[CrossRef]

2002 (1)

W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
[CrossRef]

1997 (1)

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

1989 (1)

J. M. Rodenburg, "The phase problem, microdiffraction and wavelength-limited resolution," ltramicroscopy27, 413-422 (1989).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
[CrossRef] [PubMed]

1982 (1)

1975 (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

1974 (2)

R. W. Gerchberg, "Super-resolution through error energy reduction," Optica Acta 21, 709-720 (1974).
[CrossRef]

E. Williams, "Fine structure of vascular and epidermal plastids of the mature maize leaf," Protoplasma 79, 395-400 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for determination of phase from image and diffraction plane picture," Optik (Stuttgart) 35, 237-246 (1972).

1881 (1)

G. Haberlandt, "Vergleichende anatomie des assimilatorischen gewebesystems bei pflanzen," Jahrbuch der Wissenschaftlichen Botanik 13, 74-188 (1881).

Bose, N. K.

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

Capel, D.

D. Capel and A. Zisserman, "Computer vision applied to super resolution," IEEE Signal Process. Mag. 20, 75- 86 (2003).
[CrossRef]

Chaudhuri, S.

D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
[CrossRef]

Elad, M.

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

Feinup, J. R.

Feuer, A.

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

Freeman, W. T.

W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through error energy reduction," Optica Acta 21, 709-720 (1974).
[CrossRef]

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for determination of phase from image and diffraction plane picture," Optik (Stuttgart) 35, 237-246 (1972).

Gur, E.

E. Gur and Z. Zalevsky, "Iterative single-image digital super-resolution using partial high-resolution data," Lecture Notes in Engineering and Computer Science,  WCE2007, 630-634 (2007).

Haberlandt, G.

G. Haberlandt, "Vergleichende anatomie des assimilatorischen gewebesystems bei pflanzen," Jahrbuch der Wissenschaftlichen Botanik 13, 74-188 (1881).

Jones, T. R.

W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
[CrossRef]

Joshi, M. V.

D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
[CrossRef] [PubMed]

Ng, M. K.

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

Pasztora, E. C.

W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
[CrossRef]

Rajan, D.

D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
[CrossRef]

Rodenburg, J. M.

J. M. Rodenburg, "The phase problem, microdiffraction and wavelength-limited resolution," ltramicroscopy27, 413-422 (1989).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for determination of phase from image and diffraction plane picture," Optik (Stuttgart) 35, 237-246 (1972).

Vecchi, M. P.

S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
[CrossRef] [PubMed]

Williams, E.

E. Williams, "Fine structure of vascular and epidermal plastids of the mature maize leaf," Protoplasma 79, 395-400 (1974).
[CrossRef]

Zalevsky, Z.

E. Gur and Z. Zalevsky, "Iterative single-image digital super-resolution using partial high-resolution data," Lecture Notes in Engineering and Computer Science,  WCE2007, 630-634 (2007).

Zisserman, A.

D. Capel and A. Zisserman, "Computer vision applied to super resolution," IEEE Signal Process. Mag. 20, 75- 86 (2003).
[CrossRef]

Appl. Opt. (1)

IEEE Comput. Graphics Appl. (1)

W. T. Freeman, T. R. Jones, and E. C. Pasztora, "Example-based super-rsolution," IEEE Comput. Graphics Appl. 22, 56-65 (2002).
[CrossRef]

IEEE Signal Process. Mag. (3)

D. Rajan, S. Chaudhuri, and M. V. Joshi, "Multi-objective super resolution: concepts and examples," IEEE Signal Process. Mag. 20, 49-61 (2003).
[CrossRef]

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

D. Capel and A. Zisserman, "Computer vision applied to super resolution," IEEE Signal Process. Mag. 20, 75- 86 (2003).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

IEEE Trans. Image Process. (1)

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

Jahrbuch der Wissenschaftlichen Botanik (1)

G. Haberlandt, "Vergleichende anatomie des assimilatorischen gewebesystems bei pflanzen," Jahrbuch der Wissenschaftlichen Botanik 13, 74-188 (1881).

Lecture Notes in Engineering and Computer Science (1)

E. Gur and Z. Zalevsky, "Iterative single-image digital super-resolution using partial high-resolution data," Lecture Notes in Engineering and Computer Science,  WCE2007, 630-634 (2007).

ltramicroscopy (1)

J. M. Rodenburg, "The phase problem, microdiffraction and wavelength-limited resolution," ltramicroscopy27, 413-422 (1989).
[CrossRef]

Optica Acta (1)

R. W. Gerchberg, "Super-resolution through error energy reduction," Optica Acta 21, 709-720 (1974).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for determination of phase from image and diffraction plane picture," Optik (Stuttgart) 35, 237-246 (1972).

Protoplasma (1)

E. Williams, "Fine structure of vascular and epidermal plastids of the mature maize leaf," Protoplasma 79, 395-400 (1974).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. GelattJr., and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-679 (1983).
[CrossRef] [PubMed]

Other (4)

A. Zomet and S. Peleg, "Multi-sensor super-resolution," IEEE workshop on Applications of Computer Vision (WACV02), 27-31 (2002).

H. Chang, D-Y Yeung, and Y. Xiong, "Super-resolution through neighbor embedding," in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '04), vol. 1, pp. I-275-I-282, Washington, DC, USA, June-July 2004.

S. Bellini, Blind deconvolution, S. Haykin, ed. (1994) Chap. 2, pp. 8-55.

N. K. Bose, "Wavelet-based blind super resolution from video sequence and in MRI," Pennsylvania State University - Final Report (2005).

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Figures (10)

Fig. 1.
Fig. 1.

The Gerchberg-Saxton algorithm. Ain, φin are the input plane amplitude and phase respectively, Aout, φout are the Fourier plane amplitude and phase respectively.

Fig. 2.
Fig. 2.

Frequency regions: Central area contains required frequencies, both LF and HF recently generated. Intermediate area contains undesired by-product of optimization. Exterior contains region to be padded with zeros.

Fig. 3.
Fig. 3.

(a). Original 128 by 128 gray-scale object. (b). Blurred out-of-focus image obtained by free space propagation simulation.

Fig. 4.
Fig. 4.

(a). Reconstructed image without noise reduction. (b). Reconstructed image after noise reduction.

Fig. 5.
Fig. 5.

Original 128 by 128 blurred and saturated gray-scale image as captured by a freezing fluorescence microscope.

Fig. 6.
Fig. 6.

(a). Gaussian magnitude is imposed. The image is 256 by 256 pixels and it includes zero padding. (b). The obtained magnitude distribution.

Fig. 7.
Fig. 7.

(a). Image intensity obtained at optimal focal plane, after second stage of algorithm. (b). Final intensity distribution of the required image.

Fig. 8.
Fig. 8.

(a). Intensity image that is obtained by Matlab® built-in blind deconvolution function. (b). The final intensity distribution of the required image when using data from the deconvolved image.

Fig. 9.
Fig. 9.

(a). The original image containing very small amount of spatial details. (b). The reconstructed image that contains significantly larger amount of spatial information.

Fig. 10.
Fig. 10.

(a). Original image with very small amount of spatial details (b). The reconstructed image with more spatial details.

Equations (2)

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P ~ ( p , q ) = P ( p , q ) · exp [ jk ( 1 d i + 1 d o 1 f ) ( p 2 + q 2 ) ]
x = max x X { Energy ( HF + LF ) Energy ( VHF ) }

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