Abstract

The system impulse response representing the Fresnel diffraction is shown to form a wavelet family of functions. The scale parameter of the wavelet family represents the depth (distance). This observation relates the diffraction-holography-related studies and the wavelet theory. The results may be used in various optical applications such as designing robust volume optical elements for optical signal processing and finding new formulations for optical inverse problems. The results also extend the wavelet concept to the nonbandpass family of functions with the implication of new applications in signal processing. The presented wavelet structure, for example, is a tool for space–depth analysis.

© 1993 Optical Society of America

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References

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  1. I. Daubechies, IEEE Trans. Inf. Theory 36, 961 (1990).
    [CrossRef]
  2. S. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 31, 674 (1989).
    [CrossRef]
  3. Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
    [CrossRef]
  4. Y. Li, Y. Zhang, Opt. Eng. 31, 1865 (1992).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), Chap. 8, pp. 370–387.
  6. G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
    [CrossRef]
  7. L. Onural, P. D. Scott, Opt. Eng. 26, 1124 (1987).
  8. L. Onural, M. T. Özgen, J. Opt. Soc. Am. A 9, 252 (1992).
    [CrossRef]
  9. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Part I,” J. Opt. Soc. Am. A (to be published).

1992 (3)

Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
[CrossRef]

Y. Li, Y. Zhang, Opt. Eng. 31, 1865 (1992).
[CrossRef]

L. Onural, M. T. Özgen, J. Opt. Soc. Am. A 9, 252 (1992).
[CrossRef]

1990 (1)

I. Daubechies, IEEE Trans. Inf. Theory 36, 961 (1990).
[CrossRef]

1989 (1)

S. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 31, 674 (1989).
[CrossRef]

1987 (1)

L. Onural, P. D. Scott, Opt. Eng. 26, 1124 (1987).

1976 (1)

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), Chap. 8, pp. 370–387.

Daubechies, I.

I. Daubechies, IEEE Trans. Inf. Theory 36, 961 (1990).
[CrossRef]

Li, Y.

Y. Li, Y. Zhang, Opt. Eng. 31, 1865 (1992).
[CrossRef]

Mallat, S.

S. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 31, 674 (1989).
[CrossRef]

Mendlovic, D.

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Part I,” J. Opt. Soc. Am. A (to be published).

Onural, L.

L. Onural, M. T. Özgen, J. Opt. Soc. Am. A 9, 252 (1992).
[CrossRef]

L. Onural, P. D. Scott, Opt. Eng. 26, 1124 (1987).

Ozaktas, H. M.

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Part I,” J. Opt. Soc. Am. A (to be published).

Özgen, M. T.

Roberge, D.

Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
[CrossRef]

Scott, P. D.

L. Onural, P. D. Scott, Opt. Eng. 26, 1124 (1987).

Sheng, Y.

Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
[CrossRef]

Szu, H. H.

Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
[CrossRef]

Thompson, B. J.

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), Chap. 8, pp. 370–387.

Zhang, Y.

Y. Li, Y. Zhang, Opt. Eng. 31, 1865 (1992).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. Daubechies, IEEE Trans. Inf. Theory 36, 961 (1990).
[CrossRef]

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

S. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 31, 674 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

G. A. Tyler, B. J. Thompson, Opt. Acta 23, 685 (1976).
[CrossRef]

Opt. Eng. (3)

L. Onural, P. D. Scott, Opt. Eng. 26, 1124 (1987).

Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
[CrossRef]

Y. Li, Y. Zhang, Opt. Eng. 31, 1865 (1992).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, London, 1975), Chap. 8, pp. 370–387.

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. Part I,” J. Opt. Soc. Am. A (to be published).

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Equations (20)

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ψ z ( x , y ) = 1 j λ z exp ( j 2 π λ z ) ξ η ψ ( ξ , η ) × exp { j π λ z [ ( x - ξ ) 2 + ( y - η ) 2 ] } d ξ d η ,
h z ( x , y ) = 1 j λ z exp ( j 2 π λ z ) exp [ j π λ z ( x 2 + y 2 ) ] ,
ψ z ( x , y ) = ψ ( x , y ) * * h z ( x , y ) ,
w ( a , b ) = a - 1 / 2 w ( x - b a ) ,
C w = ω ω - 1 W ( ω ) 2 d ω < .
ϕ = C w - 1 w ( a , b ) , ϕ w ( a , b ) d a a 2 d b ,
f , g = f ( x ) ¯ g ( x ) d x .
w ( a , b , c ) = K a w ( x - b a , y - c a ) .
h ( x , y ) = exp [ j ( x 2 + y 2 ) ]
h ( a , b , c ) = K a h ( x - b a , y - c a ) .
K a = exp ( j 2 π λ z ) j λ z = exp { j [ 2 ( π a λ ) 2 - π 2 ] } 1 π a 2 .
ψ ( x , y ) = ψ z ( x , y ) * * h z * ( x , y ) , ψ ( x , y ) = 1 - j λ z exp ( - j 2 π λ z ) ξ η ψ z ( ξ , η ) × exp { - j π λ z [ ( x - ξ ) 2 + ( y - η ) 2 ] } d ξ d η .
ψ ( x , y ) = 1 C z 1 - j λ z exp ( - j 2 π λ z ) ξ η ψ ( ξ , η , z ) × exp { - j π λ z [ ( x - ξ ) 2 + ( y - η ) 2 ] } d ξ d η d z ,
I z ( x , y ) = ψ ( x , y ) * * h z ( x , y ) 2 ,
I z ( x , y ) = ( 1 - s ) * * h z ) 2 1 - s * * h z - s * * * h z * .
I z ( x , y ) = 1 - s ( x , y ) * * 2 Re { h z ( x , y ) } .
I z ( x , y ) = 1 - s * * g z .
g z ( x , y ) = 2 λ z sin [ π λ z ( x 2 + y 2 ) ] .
g ( a , b , c ) ( x , y ) = K a g ( x - b a , y - c a ) .
K a = 2 λ z = 2 π a 2

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