Abstract

Many cameras nowadays are equipped with an autofocus mechanism that attempts to take pictures at the best possible focus. However, there are situations in which the pictures are still out of focus, such as when the photographer has mistakenly focused at a wrong position or when the focusing region consists of objects of different depths and therefore confuses the autofocus system. With a digital camera, we can attempt to use digital image-restoration techniques to bring the pictures back into focus. We design an algorithm that restores the image by digitally enhancing the corresponding frequency bands. We employ the restoration in the wavelet domain so that this restoration scheme can be compliant with JPEG 2000, which is positioned to succeed JPEG as the next image-compression standard and has the potential to be widely adopted by the digital photography industry owing to its many advanced features.

© 2002 Optical Society of America

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References

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  1. W. Pennebaker, J. Mitchell, JPEG Still Image Data Compression Standard (Van Nostrand Reinhold, New York, 1992).
  2. D. Taubman, M. Marcellin, JPEG2000: Image Compression Fundamentals, Standards, and Practice (Kluwer Academic, Boston, 2001).
  3. E. Y. Lam, P. Y. Tam, X. Ouyang, “An efficient parallel implementation of the 2-D pyramidal discrete wavelet transform,” in 2001 International Conference on Imaging Science, Systems, and Technology (CSREA Press, Las Vegas, Nev., 2001), Vol. 1, pp. 178–183.
  4. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  5. E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
    [CrossRef]
  6. M. Vetterli, J. Kovac̆ević, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, N. J., 1995).
  7. G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).
  8. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  9. E. Y. Lam, J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process. 9, 1661–1666 (2000).
    [CrossRef]
  10. K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.
  11. A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, 1992).
    [CrossRef]
  12. E. Y. Lam, “Image restoration algorithms for monochrome digital photography,” Ph.D. dissertation, (Stanford University, Stanford, Calif., 2000).

2000 (1)

E. Y. Lam, J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process. 9, 1661–1666 (2000).
[CrossRef]

1998 (1)

1989 (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Bilgin, A.

K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.

Gersho, A.

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, 1992).
[CrossRef]

Goodman, J. W.

E. Y. Lam, J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process. 9, 1661–1666 (2000).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gray, R.

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, 1992).
[CrossRef]

Hunt, B. R.

K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.

Kovac?evic, J.

M. Vetterli, J. Kovac̆ević, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, N. J., 1995).

Lam, E. Y.

E. Y. Lam, J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process. 9, 1661–1666 (2000).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
[CrossRef]

E. Y. Lam, P. Y. Tam, X. Ouyang, “An efficient parallel implementation of the 2-D pyramidal discrete wavelet transform,” in 2001 International Conference on Imaging Science, Systems, and Technology (CSREA Press, Las Vegas, Nev., 2001), Vol. 1, pp. 178–183.

E. Y. Lam, “Image restoration algorithms for monochrome digital photography,” Ph.D. dissertation, (Stanford University, Stanford, Calif., 2000).

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Marcellin, M.

D. Taubman, M. Marcellin, JPEG2000: Image Compression Fundamentals, Standards, and Practice (Kluwer Academic, Boston, 2001).

Marcellin, M. W.

K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.

Mitchell, J.

W. Pennebaker, J. Mitchell, JPEG Still Image Data Compression Standard (Van Nostrand Reinhold, New York, 1992).

Nguyen, T.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

Ouyang, X.

E. Y. Lam, P. Y. Tam, X. Ouyang, “An efficient parallel implementation of the 2-D pyramidal discrete wavelet transform,” in 2001 International Conference on Imaging Science, Systems, and Technology (CSREA Press, Las Vegas, Nev., 2001), Vol. 1, pp. 178–183.

Panchapakesan, K.

K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.

Pennebaker, W.

W. Pennebaker, J. Mitchell, JPEG Still Image Data Compression Standard (Van Nostrand Reinhold, New York, 1992).

Strang, G.

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

Tam, P. Y.

E. Y. Lam, P. Y. Tam, X. Ouyang, “An efficient parallel implementation of the 2-D pyramidal discrete wavelet transform,” in 2001 International Conference on Imaging Science, Systems, and Technology (CSREA Press, Las Vegas, Nev., 2001), Vol. 1, pp. 178–183.

Taubman, D.

D. Taubman, M. Marcellin, JPEG2000: Image Compression Fundamentals, Standards, and Practice (Kluwer Academic, Boston, 2001).

Vetterli, M.

M. Vetterli, J. Kovac̆ević, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, N. J., 1995).

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

E. Y. Lam, J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,” IEEE Trans. Image Process. 9, 1661–1666 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Other (9)

K. Panchapakesan, A. Bilgin, M. W. Marcellin, B. R. Hunt, “Joint compression and restoration of images using wavelets and non-linear interpolative vector quantization,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), Vol. 5, pp. 2649–2652.

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, 1992).
[CrossRef]

E. Y. Lam, “Image restoration algorithms for monochrome digital photography,” Ph.D. dissertation, (Stanford University, Stanford, Calif., 2000).

W. Pennebaker, J. Mitchell, JPEG Still Image Data Compression Standard (Van Nostrand Reinhold, New York, 1992).

D. Taubman, M. Marcellin, JPEG2000: Image Compression Fundamentals, Standards, and Practice (Kluwer Academic, Boston, 2001).

E. Y. Lam, P. Y. Tam, X. Ouyang, “An efficient parallel implementation of the 2-D pyramidal discrete wavelet transform,” in 2001 International Conference on Imaging Science, Systems, and Technology (CSREA Press, Las Vegas, Nev., 2001), Vol. 1, pp. 178–183.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. Vetterli, J. Kovac̆ević, Wavelets and Subband Coding (Prentice Hall, Englewood Cliffs, N. J., 1995).

G. Strang, T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge, Wellesley, Mass., 1996).

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Figures (7)

Fig. 1
Fig. 1

Image formation with the out-of-focus aberration.

Fig. 2
Fig. 2

Cross-sectional view of the OTFs for a circular pupil with different amounts of defocus.

Fig. 3
Fig. 3

Block diagram of the JPEG 2000 compression and decompression system.

Fig. 4
Fig. 4

Quantization of wavelet coefficients with a dead zone.

Fig. 5
Fig. 5

Wavelet coefficient distribution of the bridge image by use of the reversible wavelet basis. The number below each distribution refers to the subband level, and the letters L and H denote whether they are for the low frequency subband or the high frequency subband, respectively. The first letter is for the horizontal decomposition, while the second letter is for the vertical decomposition.

Fig. 6
Fig. 6

In-focus and out-of-focus images for simulation.

Fig. 7
Fig. 7

Simulation results for the bridge image by use of the reversible wavelet basis.

Tables (1)

Tables Icon

Table 1 Improvement in SNR by Use of New Quantization Levels in the Wavelet Domain

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Gifx, fy=fx, fyGgfx, fy,
Wm=121za-1zir2,
Wx, y=Wmx2+y2r2,
fx, fy=Afx,fy expjkWx+x0, y+y0-Wx-x0, y-y0dxdyA0,0 dxdy,
x0=λzifx2, y0=λzify2.
q=IQeif I0IQeif I<0,
Î=q+α sgnqQd,
JoutbabJinb.
ab=Jinb, JoutbJoutb, Joutb,
Qeb=Qdbab,
pI=μ2exp-μ|I|,
μb=2v2b1/2.
Î=qQdq+1Qd IpIdIqQdq+1Qd pIdI,
Î=q+0.5Qd-cothμQd2Qd2+1μ.
α=0.5-12cothμQd2-1μQd.
qQdΠpIdI=Îq+1Qd pIdI;
Î=q+0.5Qd-1μln coshμQd2.
α=0.5-1μQdln coshμQd2.

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