Abstract

An optical parallel architecture for the random-iteration algorithm to decode a fractal image by use of iterated-function system (IFS) codes is proposed. The code value is first converted into transmittance in film or a spatial light modulator in the optical part of the system. With an optical-to-electrical converter, electrical-to-optical converter, and some electronic circuits for addition and delay, we can perform the contractive affine transformation (CAT) denoted in IFS codes. In the proposed decoding architecture all CAT’s generate points (image pixels) in parallel, and these points then are joined for display purposes. Therefore the decoding speed is improved greatly compared with existing serial-decoding architectures. In addition, an error and stability analysis that considers nonperfect elements is presented for the proposed optical system. Finally, simulation results are given to validate the proposed architecture.

© 1998 Optical Society of America

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References

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  1. M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988), Chap. 3.
  2. M. F. Barnsley, A. Sloan, “A better way to compress image,” BYTE 13, 215–223 (Jan.1988).
  3. M. Barnsley, L. Hurd, Fractal Image Compression (Peters, Wellesley, Mass., 1993).
  4. M. Kawamata, H. Kanbada, T. Higuchi, “Determination of IFS codes using scale–space correlation functions,” Proceedings of the IEEE Workshop on Intelligent Signal Processing Communication Systems (Institution of Electrical and Electronics Engineers, New York, 1992), pp. 219–233.
  5. S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
    [CrossRef] [PubMed]
  6. R. Rinaldo, A. Zakhor, “Inverse and approximation problem for two-dimensional fractal sets,” IEEE Trans. Image Process. 3, 802–820 (1994).
    [CrossRef] [PubMed]
  7. H. A. Cohen, “Deterministic scanning and hybrid algorithms for fast decoding of IFS encoded image sets,” in Proceedings of the IEEE 1992 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1992), Vol. 3, pp. 509–512.
    [CrossRef]
  8. S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
    [CrossRef] [PubMed]
  9. J. Tanida, A. Uemoto, Y. Ichioka, “Optical fractal synthesizer: concept and experimental verification,” Appl. Opt. 32, 653–658 (1993).
    [CrossRef] [PubMed]
  10. H. T. Chang, C. J. Kuo, “An optical decoding architecture for the random iteration algorithm of iterated function system codes,” Opt. Rev. 1, 146–149 (1994).
    [CrossRef]
  11. H. T. Chang, C. J. Kuo, “A fully parallel algorithm for fractal image decoding using IFS codes,” IEEE Trans. Image Process. (to be published).
  12. A. Yariv, P. Yeh, Optical Wave in Crystals (Wiley, New York, 1994), pp. 241–243.
  13. 1995/96 OptoSigma Catalog: Optics Opto-Mechanics (OptoSigma Corporation, Santa Ana, Calif., 1995).
  14. The ranges of the mirror loss and the beam-splitter unbalance are chosen according to the data given in Ref. 13.

1996 (1)

S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
[CrossRef] [PubMed]

1995 (1)

S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
[CrossRef] [PubMed]

1994 (2)

R. Rinaldo, A. Zakhor, “Inverse and approximation problem for two-dimensional fractal sets,” IEEE Trans. Image Process. 3, 802–820 (1994).
[CrossRef] [PubMed]

H. T. Chang, C. J. Kuo, “An optical decoding architecture for the random iteration algorithm of iterated function system codes,” Opt. Rev. 1, 146–149 (1994).
[CrossRef]

1993 (1)

1988 (1)

M. F. Barnsley, A. Sloan, “A better way to compress image,” BYTE 13, 215–223 (Jan.1988).

Barnsley, M.

M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988), Chap. 3.

M. Barnsley, L. Hurd, Fractal Image Compression (Peters, Wellesley, Mass., 1993).

Barnsley, M. F.

M. F. Barnsley, A. Sloan, “A better way to compress image,” BYTE 13, 215–223 (Jan.1988).

Chang, H. T.

H. T. Chang, C. J. Kuo, “An optical decoding architecture for the random iteration algorithm of iterated function system codes,” Opt. Rev. 1, 146–149 (1994).
[CrossRef]

H. T. Chang, C. J. Kuo, “A fully parallel algorithm for fractal image decoding using IFS codes,” IEEE Trans. Image Process. (to be published).

Cohen, H. A.

H. A. Cohen, “Deterministic scanning and hybrid algorithms for fast decoding of IFS encoded image sets,” in Proceedings of the IEEE 1992 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1992), Vol. 3, pp. 509–512.
[CrossRef]

Higuchi, T.

M. Kawamata, H. Kanbada, T. Higuchi, “Determination of IFS codes using scale–space correlation functions,” Proceedings of the IEEE Workshop on Intelligent Signal Processing Communication Systems (Institution of Electrical and Electronics Engineers, New York, 1992), pp. 219–233.

Hurd, L.

M. Barnsley, L. Hurd, Fractal Image Compression (Peters, Wellesley, Mass., 1993).

Ichioka, Y.

Kanbada, H.

M. Kawamata, H. Kanbada, T. Higuchi, “Determination of IFS codes using scale–space correlation functions,” Proceedings of the IEEE Workshop on Intelligent Signal Processing Communication Systems (Institution of Electrical and Electronics Engineers, New York, 1992), pp. 219–233.

Kawamata, M.

M. Kawamata, H. Kanbada, T. Higuchi, “Determination of IFS codes using scale–space correlation functions,” Proceedings of the IEEE Workshop on Intelligent Signal Processing Communication Systems (Institution of Electrical and Electronics Engineers, New York, 1992), pp. 219–233.

Kuo, C. J.

H. T. Chang, C. J. Kuo, “An optical decoding architecture for the random iteration algorithm of iterated function system codes,” Opt. Rev. 1, 146–149 (1994).
[CrossRef]

H. T. Chang, C. J. Kuo, “A fully parallel algorithm for fractal image decoding using IFS codes,” IEEE Trans. Image Process. (to be published).

Lin, C.

S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
[CrossRef] [PubMed]

S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
[CrossRef] [PubMed]

Pei, S.

S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
[CrossRef] [PubMed]

S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
[CrossRef] [PubMed]

Rinaldo, R.

R. Rinaldo, A. Zakhor, “Inverse and approximation problem for two-dimensional fractal sets,” IEEE Trans. Image Process. 3, 802–820 (1994).
[CrossRef] [PubMed]

Sloan, A.

M. F. Barnsley, A. Sloan, “A better way to compress image,” BYTE 13, 215–223 (Jan.1988).

Tanida, J.

Tseng, C.

S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
[CrossRef] [PubMed]

S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
[CrossRef] [PubMed]

Uemoto, A.

Yariv, A.

A. Yariv, P. Yeh, Optical Wave in Crystals (Wiley, New York, 1994), pp. 241–243.

Yeh, P.

A. Yariv, P. Yeh, Optical Wave in Crystals (Wiley, New York, 1994), pp. 241–243.

Zakhor, A.

R. Rinaldo, A. Zakhor, “Inverse and approximation problem for two-dimensional fractal sets,” IEEE Trans. Image Process. 3, 802–820 (1994).
[CrossRef] [PubMed]

Appl. Opt. (1)

BYTE (1)

M. F. Barnsley, A. Sloan, “A better way to compress image,” BYTE 13, 215–223 (Jan.1988).

IEEE Trans. Image Process. (3)

S. Pei, C. Tseng, C. Lin, “Wavelet transform and scale space filtering of fractal images,” IEEE Trans. Image Process. 4, 682–687 (1995).
[CrossRef] [PubMed]

R. Rinaldo, A. Zakhor, “Inverse and approximation problem for two-dimensional fractal sets,” IEEE Trans. Image Process. 3, 802–820 (1994).
[CrossRef] [PubMed]

S. Pei, C. Tseng, C. Lin, “A parallel decoding algorithm for IFS codes without transient behavior,” IEEE Trans. Image Process. 5, 411–415 (1996).
[CrossRef] [PubMed]

Opt. Rev. (1)

H. T. Chang, C. J. Kuo, “An optical decoding architecture for the random iteration algorithm of iterated function system codes,” Opt. Rev. 1, 146–149 (1994).
[CrossRef]

Other (8)

H. T. Chang, C. J. Kuo, “A fully parallel algorithm for fractal image decoding using IFS codes,” IEEE Trans. Image Process. (to be published).

A. Yariv, P. Yeh, Optical Wave in Crystals (Wiley, New York, 1994), pp. 241–243.

1995/96 OptoSigma Catalog: Optics Opto-Mechanics (OptoSigma Corporation, Santa Ana, Calif., 1995).

The ranges of the mirror loss and the beam-splitter unbalance are chosen according to the data given in Ref. 13.

H. A. Cohen, “Deterministic scanning and hybrid algorithms for fast decoding of IFS encoded image sets,” in Proceedings of the IEEE 1992 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1992), Vol. 3, pp. 509–512.
[CrossRef]

M. Barnsley, L. Hurd, Fractal Image Compression (Peters, Wellesley, Mass., 1993).

M. Kawamata, H. Kanbada, T. Higuchi, “Determination of IFS codes using scale–space correlation functions,” Proceedings of the IEEE Workshop on Intelligent Signal Processing Communication Systems (Institution of Electrical and Electronics Engineers, New York, 1992), pp. 219–233.

M. Barnsley, Fractals Everywhere (Academic, Boston, Mass., 1988), Chap. 3.

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Figures (10)

Fig. 1
Fig. 1

Block diagram of the proposed parallel-decoding algorithm.

Fig. 2
Fig. 2

Single optical CAT module in the proposed parallel-decoding architecture.

Fig. 3
Fig. 3

Block diagram of the proposed parallel-decoding architecture.

Fig. 4
Fig. 4

Decoded fractal images based on the random-iteration algorithm: (a) Sierpinski triangle. (b) Fractal tree.

Fig. 5
Fig. 5

(a)–(c) Three subimages of the Sierpinski triangle generated by the three CAT’s. (d) Their union result.

Fig. 6
Fig. 6

(a)–(d) Four subimages of the fractal tree generated by the four CAT’s. (e) Their union result.

Fig. 7
Fig. 7

Decoded fractal tree image with a different uniformity on each subimage.

Fig. 8
Fig. 8

Absolute value of the determinant of the matrix A affected by (a) a mirror loss of α = 0–1, (b) a beam-splitter unbalance of β = 0–0.5, and (c) a device nonlinearity of σ = 0.1–50. BS, beam splitter.

Fig. 9
Fig. 9

Normalized error ∊% between the ideal image and the images affected by (a) a mirror loss of α = 0–0.01, (b) a beam-splitter unbalance of β = 0–0.01, and (c) a device nonlinearity of σ = 1–50.

Fig. 10
Fig. 10

Examples of the degraded images under (a) a mirror loss of α = 1%, (b) a beam-splitter unbalance of β = 0.5%, (c) a device nonlinearity of σ = 2, and (d) the compounding effects of (a)–(c).

Tables (2)

Tables Icon

Table 1 IFS Codes for a Sierpinski Triangle

Tables Icon

Table 2 IFS Codes for a Fractal Tree

Equations (20)

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W x y = A x y + b = a c b d   x y + e f ,
| det A | = | ad - bc | < 1 ,
F = i = 1 N   W i F ,
W 1 x y = 0.5 0.0 0.0 0.5   x y + 0 0 , W 2 x y = 0.5 0.0 0.0 0.5   x y + 1 0 , W 3 x y = 0.5 0.0 0.0 0.5   x y + 0.5 0.5 .
N = i = 1 N p i p max ,   N N .
N ˆ = i = 1 N p i p min ,   N ˆ N .
W x y = 2 T a T c T b T d   x y + T e T f - 1 1 1 1   x y - 1 1 ,
V ζ = V π π sin - 1 ζ 2 ,   ζ = a f ,
W x y = 2 T a T c T b T d   x 1 - α y 1 - α 2 + T e T f 1 - α - 1 1 1 1   x y - 1 1 .
W x y = A m x y + b m = k a + 1 - 1 k c + 1 - 1 k 2 b + 1 - 1 k 2 d + 1 - 1   x y + e k f + 1 - 1 ,
det ( A m ) = k 3 ( ad - bc ) + k 2 ( 1 - k ) ( b - d ) + k ( 1 - k ) ( 1 + k ) ( c - a ) .
T ξ = ξ + 1 0.5 + β , ξ = a ,   c ,   e , T ξ = ξ + 1 0.5 - β , ξ = b ,   d ,   f ,
W x y = A bs x y + b bs = a + 2 β 1 + a c + 2 β 1 + c b - 2 β 1 + b d - 2 β 1 + d   x y + e + 2 β 1 + e f - 2 β 1 + f ,
det ( A bs ) = ad - bc - 2 β ( a - b + c - d ) - 4 β 2 × [ ( 1 + a ) ( 1 + d ) - ( 1 + b ) ( 1 + c ) ] .
g ( u ) = L   erf u 2 σ = 2 L π 0 u / 2 σ exp ( - v 2 ) d v ,
T ξ = g ¯ ξ + 1 2 ,   ξ = a f .
W x y = A n x y + b n = g ¯ a g ¯ c g ¯ b g ¯ d   x y + g ¯ e g ¯ f .
det ( A n ) = g ¯ ( a ) g ¯ ( d ) - g ¯ ( b ) g ¯ ( c ) ,
H = 0 i , j < 512   | F 1 i ,   j - F 2 i ,   j | ,
ε % = H 512 2 × 100 % .

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