Abstract

We propose a method for direct control of position, rotation, and scaling of fractal patterns generated on an optical fractal synthesizer. In this method we introduce an iterated-function-system mother function to produce control parameters for arbitrary fractal patterns. We implemented the method experimentally and verified the effectiveness of the method.

© 2000 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry Of Nature (Freeman, New York, 1977).
  2. H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer-Verlag, New York, 1992).
    [CrossRef]
  3. M. Barnsley, Fractals Everywhere (Academic, San Diego, Calif., 1988).
  4. A. Jacquin, “Image coding based on a fractal theory of iterated contractive image transformation,” IEEE Trans. Image Process. 1, 18–30 (1992).
    [CrossRef]
  5. N. Lu, Fractal Imaging (Academic, San Diego, 1997).
  6. G. Held, T. R. Marshall, Data and Image Compression (Wiley, Chichester, UK, 1996).
  7. J. Tanida, A. Uemoto, Y. Ichioka, “Optical fractal synthesizer: concept and experimental verification,” Appl. Opt. 32, 653–658 (1993).
    [CrossRef] [PubMed]
  8. T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
    [CrossRef]

1999 (1)

T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
[CrossRef]

1993 (1)

1992 (1)

A. Jacquin, “Image coding based on a fractal theory of iterated contractive image transformation,” IEEE Trans. Image Process. 1, 18–30 (1992).
[CrossRef]

Barnsley, M.

M. Barnsley, Fractals Everywhere (Academic, San Diego, Calif., 1988).

Held, G.

G. Held, T. R. Marshall, Data and Image Compression (Wiley, Chichester, UK, 1996).

Ichioka, Y.

T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
[CrossRef]

J. Tanida, A. Uemoto, Y. Ichioka, “Optical fractal synthesizer: concept and experimental verification,” Appl. Opt. 32, 653–658 (1993).
[CrossRef] [PubMed]

Jacquin, A.

A. Jacquin, “Image coding based on a fractal theory of iterated contractive image transformation,” IEEE Trans. Image Process. 1, 18–30 (1992).
[CrossRef]

Jürgens, H.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer-Verlag, New York, 1992).
[CrossRef]

Lu, N.

N. Lu, Fractal Imaging (Academic, San Diego, 1997).

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry Of Nature (Freeman, New York, 1977).

Marshall, T. R.

G. Held, T. R. Marshall, Data and Image Compression (Wiley, Chichester, UK, 1996).

Peitgen, H.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer-Verlag, New York, 1992).
[CrossRef]

Sasaki, T.

T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
[CrossRef]

Saupe, D.

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer-Verlag, New York, 1992).
[CrossRef]

Tanida, J.

T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
[CrossRef]

J. Tanida, A. Uemoto, Y. Ichioka, “Optical fractal synthesizer: concept and experimental verification,” Appl. Opt. 32, 653–658 (1993).
[CrossRef] [PubMed]

Uemoto, A.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

A. Jacquin, “Image coding based on a fractal theory of iterated contractive image transformation,” IEEE Trans. Image Process. 1, 18–30 (1992).
[CrossRef]

Opt. Eng. (1)

T. Sasaki, J. Tanida, Y. Ichioka, “Optical implementation of high-accuracy computing based on interval arithmetic and fixed point theorem,” Opt. Eng. 38, 508–513 (1999).
[CrossRef]

Other (5)

N. Lu, Fractal Imaging (Academic, San Diego, 1997).

G. Held, T. R. Marshall, Data and Image Compression (Wiley, Chichester, UK, 1996).

B. B. Mandelbrot, The Fractal Geometry Of Nature (Freeman, New York, 1977).

H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer-Verlag, New York, 1992).
[CrossRef]

M. Barnsley, Fractals Everywhere (Academic, San Diego, Calif., 1988).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of experimental OFS.

Fig. 2
Fig. 2

Time evolution of generated pattern on experimental OFS.

Fig. 3
Fig. 3

Attractors generated by experimental OFS.

Fig. 4
Fig. 4

(a) Standard attractor. (b)–(i) Several variations in position, rotation, reflection, and scaling. Tables 1 and 2 summarize parameters to control the patterns.

Fig. 5
Fig. 5

Relationship between translation vector of affine transform and captured area.

Tables (2)

Tables Icon

Table 1 Parameter Sets for Standard Attractor Used in Experimental Verification

Tables Icon

Table 2 Manipulating Parameters Used in Experimental Verification

Equations (22)

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WA=i=1N wiA,  AR2,
wiA=x; x=aibicidix+eifi, x A,
Bn+1=WBn,  BnR2,
B=WB,  BR2.
wiA=x; x=SsiRθiMjix+ti, xA  i=1, 2,
Ss=s00s,
Rθ=cos θ-sin θsin θcos θ,
Mj=j001,  j1, -1.
si1/k,
wiA=x; x=Ai-1x-b+ai+b, xA,
uiA=x; x=Aix+ai, xA.
Ai=SsiRθiMji,
=SsaRθaMja,
wiA=x; x=RθaMjaSsiRθiMjiMja×R-θax-ta+SsaRθaMjati+ta, xA,
ξ>ξsi+|ki|,
UA=i=1NuiA,  AR2,
α=Uα,  αR2.
β=vα,
vA=x; x=x+b, xA,
β=vUvinvβ=i=1Nvuivinvβ,
vinvA=x; x=-1x-b, xA.
wiA=vuivinvβ=x; x=Ai-1x-b+ai+b, xA.

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