Abstract

In discrete-cosine-transform-based (DCT-based) compressions such as JPEG it is a common practice to use the same quantization matrix for both encoding and decoding. However, this need not be the case, and the flexibility of designing different matrices for encoding and decoding allows us to perform image restoration in the DCT domain. This is especially useful when we have severe limitations on the computational power, for instance, with in-camera image manipulation for programmable digital cameras. We provide an algorithm that compensates partially for a defocus error in image acquisition, and experimental results show that the restored image is closer to the in-focus image than is the defocused image.

© 1998 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. R. Brown, A. Boden, “A posteriori restoration of block transform-compressed data,” in 1995 Proceedings of the Data Compression Conference, J. A. Storer, M. Cohn, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1995), p. 426.
  3. Z. Fan, R. Eschbach, “JPEG decompression with reduced artifacts,” in Image and Video Compressions, M. Rabbani, R. J. Safranek, eds., Proc. SPIE2186, 50–55 (1994).
    [CrossRef]
  4. R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  6. V. Bhaskaran, K. Konstantinides, G. Beretta, “Text and image sharpening of scanned images in the JPEG domain,” in Proceedings of the Fourth IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 326–329.
    [CrossRef]
  7. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  8. K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).
  9. M. G. Kang, A. K. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
    [CrossRef] [PubMed]
  10. K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  11. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1989).
  12. R. Reininger, J. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. COM-31, 835–839 (1983).
    [CrossRef]

1997

R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
[CrossRef]

1995

M. G. Kang, A. K. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

1989

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

1983

R. Reininger, J. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. COM-31, 835–839 (1983).
[CrossRef]

1970

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Baskurt, A.

R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
[CrossRef]

Beretta, G.

V. Bhaskaran, K. Konstantinides, G. Beretta, “Text and image sharpening of scanned images in the JPEG domain,” in Proceedings of the Fourth IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 326–329.
[CrossRef]

Bhaskaran, V.

V. Bhaskaran, K. Konstantinides, G. Beretta, “Text and image sharpening of scanned images in the JPEG domain,” in Proceedings of the Fourth IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 326–329.
[CrossRef]

Boden, A.

R. Brown, A. Boden, “A posteriori restoration of block transform-compressed data,” in 1995 Proceedings of the Data Compression Conference, J. A. Storer, M. Cohn, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1995), p. 426.

Brown, R.

R. Brown, A. Boden, “A posteriori restoration of block transform-compressed data,” in 1995 Proceedings of the Data Compression Conference, J. A. Storer, M. Cohn, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1995), p. 426.

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Ding, Y.

R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
[CrossRef]

Eschbach, R.

Z. Fan, R. Eschbach, “JPEG decompression with reduced artifacts,” in Image and Video Compressions, M. Rabbani, R. J. Safranek, eds., Proc. SPIE2186, 50–55 (1994).
[CrossRef]

Fan, Z.

Z. Fan, R. Eschbach, “JPEG decompression with reduced artifacts,” in Image and Video Compressions, M. Rabbani, R. J. Safranek, eds., Proc. SPIE2186, 50–55 (1994).
[CrossRef]

Gibson, J.

R. Reininger, J. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. COM-31, 835–839 (1983).
[CrossRef]

Goodman, J. W.

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

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1989).

Kang, M. G.

M. G. Kang, A. K. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

Katsaggelos, A. K.

M. G. Kang, A. K. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

Konstantinides, K.

V. Bhaskaran, K. Konstantinides, G. Beretta, “Text and image sharpening of scanned images in the JPEG domain,” in Proceedings of the Fourth IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 326–329.
[CrossRef]

Miller, K.

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Mitchell, J.

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

Pennebaker, W.

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

Prost, R.

R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
[CrossRef]

Reininger, R.

R. Reininger, J. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. COM-31, 835–839 (1983).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

IEEE Trans. Commun.

R. Reininger, J. Gibson, “Distributions of the two-dimensional DCT coefficients for images,” IEEE Trans. Commun. COM-31, 835–839 (1983).
[CrossRef]

IEEE Trans. Image Process.

M. G. Kang, A. K. Katsaggelos, “General choice of the regularization functional in regularized image restoration,” IEEE Trans. Image Process. 4, 594–602 (1995).
[CrossRef] [PubMed]

R. Prost, Y. Ding, A. Baskurt, “JPEG dequantization array for regularized decompression,” IEEE Trans. Image Process. 6, 883–888 (1997).
[CrossRef]

SIAM J. Math. Anal.

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Other

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1989).

K. R. Castleman, Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1996).

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

V. Bhaskaran, K. Konstantinides, G. Beretta, “Text and image sharpening of scanned images in the JPEG domain,” in Proceedings of the Fourth IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 2, pp. 326–329.
[CrossRef]

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

R. Brown, A. Boden, “A posteriori restoration of block transform-compressed data,” in 1995 Proceedings of the Data Compression Conference, J. A. Storer, M. Cohn, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1995), p. 426.

Z. Fan, R. Eschbach, “JPEG decompression with reduced artifacts,” in Image and Video Compressions, M. Rabbani, R. J. Safranek, eds., Proc. SPIE2186, 50–55 (1994).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Recommended JPEG quantization matrix.

Fig. 2
Fig. 2

Cross-sectional view of the OTF.

Fig. 3
Fig. 3

Laplacian convolution kernel.

Fig. 4
Fig. 4

(a) In-focus image. (b) Defocused image. (c) Image compressed and decompressed with the normal Q. (d) Image compressed and decompressed with the improved Q e and Q d .

Tables (1)

Tables Icon

Table 1 Values of the SNR for Various Images

Equations (22)

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X u ,   v Q e u ,   v = X q u ,   v + X n u ,   v .
X ˆ u ,   v = Q d u ,   v X q u ,   v ,
MSE = u = 0 7 v = 0 7 X u ,   v - X ˆ u ,   v 2 = u = 0 7 v = 0 7 Q e - Q d X q + Q e X n 2 ,
G i f x ,   f y = f x ,   f y G g f x ,   f y ,
W m = 1 2 1 z a - 1 z i r 2 ,
W x ,   y = W m x 2 + y 2 r 2 ,
f x ,   f y = A f x ,   f y exp jk W x + x 0 ,   y + y 0 - W x - x 0 ,   y - y 0 d x d y A 0,0 d x d y ,
x 0 = λ z i f x 2 ,     y 0 = λ z i f y 2 .
X out u ,   v a u ,   v X in u ,   v .
a u ,   v = X in u ,   v ,   X out u ,   v X out u ,   v ,   X out u ,   v ,
Q e u ,   v = Q u ,   v a u ,   v ,
G i = u = 0 7 v = 0 7 X in u ,   v L u ,   v 2
u = 0 7 v = 0 7 X ˆ out u ,   v L u ,   v 2 G = 1 B i = 1 B   G i .
u = 0 7 v = 0 7 X ˆ out u ,   v Q u ,   v - X q , out 2 R = 1 B i = 1 B u = 0 7 v = 0 7   X n , in 2 u ,   v .
J ν = u = 0 7 v = 0 7 X ˆ out u ,   v Q u ,   v - X q , out 2 + ν u = 0 7 v = 0 7 X ˆ out u ,   v L u ,   v 2 ,
J ν X ˆ out = 0 ,
X ˆ out u ,   v = Q u ,   v G G + Q 2 u ,   v L 2 u ,   v R   X q , out .
Q d u ,   v = Q u ,   v G G + Q 2 u ,   v L 2 u ,   v R .
SNR x ,   x ˆ = 10   log 10 j k x j ,   k 2 j k x j ,   k - x ˆ j ,   k 2 ,
a u ,   v = Var X in Var X out 1 / 2 = 1 n     X in 2 - 1 n     X in 2 1 n     X out 2 - 1 n     X out 2 1 / 2 ,
  X in   X out 2   X in 2     X out 2 ,
  X in   X out   X out 2 1 n     X in 2 1 n     X out 2 1 / 2 .

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