Abstract

According to the wavefront filtering idea of wavelet optics, the transfer function of an optical system is described with a wavelet scale function. In the transfer function described with a wavelet scale function, different scale parameters a,c and shift parameters b,d correspond to different subtransfer functions, which correspond to different situations of the optical system. According to the request of the optical system, by adjusting all these scale parameters, not only can we obtain the optical images under different conditions, but we can also obtain the singular points under this scale parameter; hence a more ideal output can be obtained by such processing. The transfer function described with a wavelet scale function can be adjusted according to the request of the optical system, which makes the described transfer function self-adjustable. According to all types of disturbing effects to the system, by adjusting the scale and shift parameters, the practical form of the transfer function of an optical system can be confirmed, which satisfies the request of the self-adjustability of the optical imaging system. The result of our analysis shows that describing the transfer function of an optical system with a wavelet scale function is not only feasible but also satisfies the request of the self-adjustability of the optical imaging system, and different optical systems can be described by different wavelet scale parameters. This work breaks from the formal additional describing mode of the transfer function of an optical system and makes description of the transfer function of an optical system convenient.

© 2006 Optical Society of America

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References

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  1. H. Szu, Y. Shen, and J. Chen, "Wavelet transform as a bank of the matched filters," Appl. Opt. 31, 3267-3277 (1992).
    [CrossRef] [PubMed]
  2. H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
    [CrossRef]
  3. W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
    [CrossRef]
  4. C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
    [CrossRef]
  5. S. Qiu, "Gabor-type matrix algebra and fast computations of dual and tight Gabor wavelets," Opt. Eng. 36, 276-282 (1997).
    [CrossRef]
  6. B. Y. Soon, M. S. Alam, and M. A. Karim, "Improved feature extraction by use of a joint wavelet transform correlator," Appl. Opt. 37, 821-827 (1998).
    [CrossRef]
  7. Y. Shen, S. Deschenes, and H. J. Caulfield, "Monochromatic electromagnetic wavelets and the Huygens principle," Appl. Opt. 37, 828-833 (1998).
    [CrossRef]
  8. T. Liying, M. Jing, Q. Ran, and Q. Wang, "Filtering theory and application of wavelet optics," Appl. Opt. 40, 257-260 (2001).
    [CrossRef]
  9. T. Liying, M. Jing, and Q. Ran, "The elementary theory of wavelet optical diffraction," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, 1999), pp. 113-114.

2001 (1)

1998 (2)

1997 (2)

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

S. Qiu, "Gabor-type matrix algebra and fast computations of dual and tight Gabor wavelets," Opt. Eng. 36, 276-282 (1997).
[CrossRef]

1995 (1)

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

1992 (2)

H. Szu, Y. Shen, and J. Chen, "Wavelet transform as a bank of the matched filters," Appl. Opt. 31, 3267-3277 (1992).
[CrossRef] [PubMed]

H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
[CrossRef]

Alam, M. S.

Caulfield, H. J.

Chen, J.

Cho, C. S.

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

Deschenes, S.

Ha, S.-W.

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

Jin, G.

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

Jing, M.

T. Liying, M. Jing, Q. Ran, and Q. Wang, "Filtering theory and application of wavelet optics," Appl. Opt. 40, 257-260 (2001).
[CrossRef]

T. Liying, M. Jing, and Q. Ran, "The elementary theory of wavelet optical diffraction," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, 1999), pp. 113-114.

Karim, M. A.

Kim, J.-C.

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

Liying, T.

T. Liying, M. Jing, Q. Ran, and Q. Wang, "Filtering theory and application of wavelet optics," Appl. Opt. 40, 257-260 (2001).
[CrossRef]

T. Liying, M. Jing, and Q. Ran, "The elementary theory of wavelet optical diffraction," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, 1999), pp. 113-114.

Lohmann, A.

H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
[CrossRef]

Nam, K. G.

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

Qiu, S.

S. Qiu, "Gabor-type matrix algebra and fast computations of dual and tight Gabor wavelets," Opt. Eng. 36, 276-282 (1997).
[CrossRef]

Ran, Q.

T. Liying, M. Jing, Q. Ran, and Q. Wang, "Filtering theory and application of wavelet optics," Appl. Opt. 40, 257-260 (2001).
[CrossRef]

T. Liying, M. Jing, and Q. Ran, "The elementary theory of wavelet optical diffraction," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, 1999), pp. 113-114.

Shen, Y.

Soon, B. Y.

Szu, H.

H. Szu, Y. Shen, and J. Chen, "Wavelet transform as a bank of the matched filters," Appl. Opt. 31, 3267-3277 (1992).
[CrossRef] [PubMed]

H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
[CrossRef]

Telfer, B.

H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
[CrossRef]

Wang, Q.

Wang, W.

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

Wu, M.

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

Yan, Y.

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

Yoon, T.-H.

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

Appl. Opt. (4)

Opt. Eng. (4)

H. Szu, B. Telfer, and A. Lohmann, "Causal analytical wavelet transform," Opt. Eng. 31, 1825-1829 (1992).
[CrossRef]

W. Wang, G. Jin, Y. Yan, and M. Wu, "Image feature extraction with optical Harr wavelet transform," Opt. Eng. 34, 1238-1242 (1995).
[CrossRef]

C. S. Cho, S.-W. Ha, J.-C. Kim, T.-H. Yoon, and K. G. Nam, "Optoelectronic difference-of-Gaussian wavelet transform system," Opt. Eng. 36, 3471-3475 (1997).
[CrossRef]

S. Qiu, "Gabor-type matrix algebra and fast computations of dual and tight Gabor wavelets," Opt. Eng. 36, 276-282 (1997).
[CrossRef]

Other (1)

T. Liying, M. Jing, and Q. Ran, "The elementary theory of wavelet optical diffraction," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, 1999), pp. 113-114.

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Figures (3)

Fig. 1
Fig. 1

(A) Input image. (B) Transfer function when scale parameters a and c are selected to be infinitude. (C) Output image of the system under a Fourier description. (D) Output image of the system under the wavelet description (a = c = ∞).

Fig. 2
Fig. 2

(A) Transfer function when shift parameters b = d = 0. (B) Output image of the system under the wavelet description (b = d = 0).

Fig. 3
Fig. 3

(A) Transfer function when the parameters are a = c = b = d = 2.3. (B) Output image of the system with the wavelet description a = c = b = d = 2.3. (C) Transfer function when the shift parameters are a = c = b = d = 2.4. (D) Output image of the system with the wavelet description a = c = b = d = 2.4.

Equations (25)

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U i ( x i , y i ) = ϕ ( x i , y i ; x 0 , y 0 ) U 0 ( x 0 , y 0 ) d x 0 d y 0 ,
ϕ ( x , y ) d x d y = 1 .
ϕ ( x , y ) = 2 n Z h n ϕ ( 2 x n , 2 y n ) ,
h n = 2 R ϕ ( x , y ) ϕ ¯ ( 2 x n , 2 y n ) d x d y ,
ϕ ̂ ( f x , f y ) = k = - h ( k ) 2 ϕ ̂ ( f x 2 , f y 2 ) exp ( - jωk / 2 ) = ϕ ̂ ( f x 2 , f y 2 ) k = - h ( k ) 2 exp ( - jωk / 2 ) = H ( f x 2 , f y 2 ) ϕ ̂ ( f x 2 , f y 2 ) ,
H ( f x 2 , f y 2 ) = k = h ( k ) 2 exp ( j ω k / 2 ) or
H ( f x , f y ) = k = h ( k ) 2 exp ( j ω k ) .
U ( x , y ) = U 1 ( x 1 , y 1 ) ϕ ( x x 1 , y y 1 ) d x 1 d y 1 .
U ( x , y ) = U 1 ( x , y ) ϕ ( x , y ) .
U ^ ( f x , f y ) = U ^ 1 ( f x , f y ) ϕ ^ ( f x , f y ) = U ^ 1 ( f x , f y ) ϕ ^ ( f x 2 , f y 2 ) H ( f x 2 , f y 2 ) .
ϕ a , b ; c , d ( x , y ) = exp [ - 1 2 ( b - x a ) 2 ] exp [ - 1 2 ( d - y a ) 2 ] × exp { j 2 π [ f x b + f y d + C ( f x b 2 + f y d 2 ) ] } .
U 2 ( x 2 , x 2 ) = U 1 ( x 1 , y 1 ) ϕ ( x 1 , y 1 ; x 2 , y 2 ) d x 1 d y 1 = U 1 ( x 1 , y 1 ) exp [ 1 2 ( x 2 x 1 Δ x ) 2 ] × exp [ 1 2 ( y 2 y 1 Δ y ) 2 ] exp { j 2 π [ f x x 2 + f y y 2 + C ( f x x 2     2 + f y y 2     2 ) ] } d x 1 d y 1 .
Φ ( f x , f y ) = exp [ 1 2 ( b x a ) 2 ] exp [ - 1 2 ( d y c ) 2 ] × exp { j π [ 4 ( f x b + f y d ) + 2 C ( f x b 2 + f y d 2 ) ] } d x d y .
Φ ( f x , f y ) = exp { j π [ 4 ( f x b + f y d ) + 2 C ( f x b 2 + f y d 2 ) ] } d x d y .
Φ ( f x , f y ) = exp [ 1 2 ( x 2 a 2 + y 2 c 2 ) ] d x d y .
exp [ - 1 2 ( b - x a ) 2 ] exp [ - 1 2 ( d - y c ) 2 ]
U i ( x i , y i , t ) = - U g ( x ¯ 0 , y ¯ 0 , t ) ϕ ( x i - x ¯ 0 , y i - y ¯ 0 ) d x ¯ 0 d y ¯ 0 .
I i ( x i , y i ) = U i ( x i , y i ; t ) U i * ( x i , y i ; t ) .
F ^ ( ϕ ) = Φ ( p , q ) ,
I ( x 2 , y 2 ) = | U 1 ( x 1 , y 1 ) | 2 × | ϕ ( x 1 , y ; x 2 , y 2 ) | 2 d x 1 d y 1 = | U 1 ( x 1 , y 1 ) | 2 { exp [ 1 2 ( x 2 x 1 Δx ) 2 ] × exp [ 1 2 ( y 2 y 1 Δy ) 2 ] exp [ j 2 π ( f x x 2 + f y y 2 ) ] × exp [ j 2 πC ( f x x 2 2 + f y y 2 2 ) ] } 2 d x 1 d y 1
= exp [ j 4 π C ( f x x 2     2 + f y y 2     2 ) ] × | U 1 ( x 1 , y 1 ) | 2 × exp [ ( x 2 x 1 Δ x ) 2 ] × exp [ ( y 2 y 1 Δ y ) 2 ] × exp [ j 4 π ( f x x 2 + f y y 2 ) ] d x 1 d y 1 .
I ( x 2 , y 2 ) = | U 1 ( x 1 , y 1 ) | 2 | h ( x 2 x 1 , y 2 y 1 ) | 2 d x 1 d y 1 = | U 1 ( x 1 , y 1 ) | 2 exp [ j 4 π ( f x x 2 + f y y 2 ) ] d x 1 d y 1 .
exp [ - ( x 2 - x 1 Δ x ) 2 ] exp [ - ( y 2 - y 1 Δ y ) 2 ]
Φ ( f x , f y ) = exp { j π [ 4 ( f x b + f y d ) + 2 C ( f x b 2 + f y d 2 ) ] } d x d y .
Φ ( f x , f y ) = exp [ 1 2 ( x 2 a 2 + y 2 c 2 ) ] d x d y .

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