Abstract

The image formation and the point-spread function of an optical system are analyzed by use of the wavelet basis function. The image described by a wavelet is no longer an indivisible whole image. It is, rather, a complex image consisting of many wavelet subimages, which come from the changes of different parameters (scale)a andc, and parametersb andd show the positions of wavelet subimages under different scales. A Gaussian frequency-modulated complex-valued wavelet function is introduced to express the point-spread function of an optical system and used to describe the image formation. The analysis, in allusion to the situation of illumination with a monochromatic plain light wave, shows that using the theory of wavelet optics to describe the image formation of an optical system is feasible.

© 2005 Optical Society of America

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References

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  1. H. Szu, Y. Shen, J. Chen, “Wavelet transform as a bank of the matched filters,”Appl. Opt. 31,3267–3277 (1992).
    [CrossRef] [PubMed]
  2. H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
    [CrossRef]
  3. D. Mendlovic, I. Ouzieli, I. Kiryuschev, E. Marom, “Two-dimensional wavelet transform achieved by computer-generated multireference matched filter and Dammann grating,”Appl. Opt. 34,8213–8219 (1995).
    [CrossRef] [PubMed]
  4. B. Y. Soon, M. S. Alam, M. A. Karim, “Improved feature extraction by use of a joint wavelet transform correlator,”Appl. Opt. 37,821–827 (1998).
    [CrossRef]
  5. J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
    [CrossRef]
  6. T. Gharbi, D. Barchiesi, “Local signal processing to evaluate resolution in SNOM images using 1D wavelets,”Opt. Commun. 177,85–93 (2000).
    [CrossRef]
  7. M. Pohit, K. Singh, “Performance of a wavelet matched filter with optimized dilation designed using simulated annealing algorithm,”Opt. Commun. 187,337–346 (2001).
    [CrossRef]
  8. Z. Chen, M. A. Karim, “Frequency-refined mutiresolution decomposition using wavelet splitting,”Opt. Commun. 173,81–93 (2000).
    [CrossRef]
  9. T. Liying, M. Jing, W. Qi, R. Qiwen, “Filtering theory and application of wavelet optics,”Appl. Opt. 40,257–260 (2001).
    [CrossRef]

2001 (2)

M. Pohit, K. Singh, “Performance of a wavelet matched filter with optimized dilation designed using simulated annealing algorithm,”Opt. Commun. 187,337–346 (2001).
[CrossRef]

T. Liying, M. Jing, W. Qi, R. Qiwen, “Filtering theory and application of wavelet optics,”Appl. Opt. 40,257–260 (2001).
[CrossRef]

2000 (3)

Z. Chen, M. A. Karim, “Frequency-refined mutiresolution decomposition using wavelet splitting,”Opt. Commun. 173,81–93 (2000).
[CrossRef]

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

T. Gharbi, D. Barchiesi, “Local signal processing to evaluate resolution in SNOM images using 1D wavelets,”Opt. Commun. 177,85–93 (2000).
[CrossRef]

1998 (1)

1995 (1)

1992 (2)

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

H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
[CrossRef]

Alam, M. S.

Barchiesi, D.

T. Gharbi, D. Barchiesi, “Local signal processing to evaluate resolution in SNOM images using 1D wavelets,”Opt. Commun. 177,85–93 (2000).
[CrossRef]

Chen, J.

Chen, Z.

Z. Chen, M. A. Karim, “Frequency-refined mutiresolution decomposition using wavelet splitting,”Opt. Commun. 173,81–93 (2000).
[CrossRef]

Cywiak, M.

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

Gharbi, T.

T. Gharbi, D. Barchiesi, “Local signal processing to evaluate resolution in SNOM images using 1D wavelets,”Opt. Commun. 177,85–93 (2000).
[CrossRef]

Jing, M.

Karim, M. A.

Z. Chen, M. A. Karim, “Frequency-refined mutiresolution decomposition using wavelet splitting,”Opt. Commun. 173,81–93 (2000).
[CrossRef]

B. Y. Soon, M. S. Alam, M. A. Karim, “Improved feature extraction by use of a joint wavelet transform correlator,”Appl. Opt. 37,821–827 (1998).
[CrossRef]

Kiryuschev, I.

Liying, T.

Lohmann, A.

H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
[CrossRef]

Marom, E.

Mendlovic, D.

Muarkowski, J.

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

Ouzieli, I.

Pohit, M.

M. Pohit, K. Singh, “Performance of a wavelet matched filter with optimized dilation designed using simulated annealing algorithm,”Opt. Commun. 187,337–346 (2001).
[CrossRef]

Qi, W.

Qiwen, R.

Rosner, B.

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

Shen, Y.

Singh, K.

M. Pohit, K. Singh, “Performance of a wavelet matched filter with optimized dilation designed using simulated annealing algorithm,”Opt. Commun. 187,337–346 (2001).
[CrossRef]

Soon, B. Y.

Szu, H.

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

H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
[CrossRef]

Telfer, B.

H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
[CrossRef]

Weide, D.

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

Appl. Opt. (4)

Opt. Commun. (4)

J. Muarkowski, M. Cywiak, B. Rosner, D. Weide, “Far field optical imaging with subwavelength resolution,”Opt. Commun. 185,295–303 (2000).
[CrossRef]

T. Gharbi, D. Barchiesi, “Local signal processing to evaluate resolution in SNOM images using 1D wavelets,”Opt. Commun. 177,85–93 (2000).
[CrossRef]

M. Pohit, K. Singh, “Performance of a wavelet matched filter with optimized dilation designed using simulated annealing algorithm,”Opt. Commun. 187,337–346 (2001).
[CrossRef]

Z. Chen, M. A. Karim, “Frequency-refined mutiresolution decomposition using wavelet splitting,”Opt. Commun. 173,81–93 (2000).
[CrossRef]

Opt. Eng. (Bellingham) (1)

H. Szu, B. Telfer, A. Lohmann, “Causal analytical wavelet transform,”Opt. Eng. (Bellingham) 31,1825–1829 (1992).
[CrossRef]

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Equations (26)

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Ui(xi, yi)=-ϕ(xi, yi; x0, y0)U0(x0, y0)dx0dy0,
ϕ(x, y)dxdy=1.
ϕ(x, y)=2kZhkϕ(2x-k, 2y-k).
hk=2Rϕ(x, y)ϕ¯(2x-k, 2y-k)dxdy.
ϕˆ(fx, fy)=k=-  h(k)2ϕˆfx2, fy2exp(-jωk/2)=ϕˆfx2, fy2k=-  h(k)2 exp(-jωk/2)=Hfx2, fy2ϕˆfx2, fy2.
Hfx2, fy2=k=-  h(k)2 exp(-jωk/2)
H(fx, fy)=k=-  h(k)2 exp(-jωk).
ϕa,b;c,d(x, y)=a-1/2c-1/2ϕb-xa, d-yc,
ϕa,0,c,0(x, y)=exp-12 p-xΔx2exp-12 q-yΔy2×exp{j2π[fxp+fyq+C(fxp2+fyq2)]},
exp-12 p-xΔx2exp-12 q-yΔy2×exp{j2π[fxp+fyq+C(fxp2+fyq2)]}
exp-12 p-xΔx2exp-12 q-yΔy2
ϕa,b;c,d(x, y)=a-1/2c-1/2ϕb-xa, d-yc.
ϕa,b;c,d(x, y)=a-1/2c-1/2ϕb-xa, d-yc
g(α, β)=Sˆ(x, y)f(x, y)ϕ(x, y),
dI(α, β)=g(α, β)g*(α, β)ds.
I(α, β)=|g(α, β)|2ds=S(x, y)S*(x, y)f(x, y)f*(x, y)×ϕα-xa, β-ybϕ*α-xa, β-ybdxdy.
I(α, β)=1ab -+ϕα-xa, β-yb2|f(x, y))|2dxdy.
ϕa,b;c,d(x, y)=a-1/2c-1/2ϕb-xa, d-yc,
ϕa,0,c,0(x, y)=exp-12 p-xΔx2exp-12 q-yΔy2×exp{j2π[fxp+fyq+C(fxp2+fyq2)]}.
I(α, β)=-|f(x, y)|2|ϕa,b;c,d(x, y)|2dxdy=-|f(x, y)|2exp-12 p-xΔx2×exp-12 q-yΔy2×exp{j2π[fxp+fyq+(1+d)(fxp2+fyq2)]}2dxdy=exp[j4πC(fxp2+fyq2)]-|f(x, y)|2  exp-12 p-xΔx2×exp-12 q-yΔy2exp[j4π(fxp+fyq)]dxdy.
I(α, β)=-|f(x, y)|2|h(α-x, β-y)|2dxdy.
exp-12 p-xΔx2exp-12 q-yΔy2,
U(xi, yi)=K-P(x, y)ϕa,b;c,d(x, y)dxdy,
p(x, y)=1inputpupil0outputpupil.
U(xi, yi)=-|ϕa,b;c,d(x, y)|2dxdy=-exp-12 p-xΔxexp-12 q-yΔy2×|exp{j2π[fxp+fyq+C(fxp2+fyq2)]}|2=exp[-j4πC(fxp2+fyq2)]-  exp-p-xΔx×exp-q-yΔyexp[-j4πC(fxp+fyq)]dxdy.
exp-12 p-xΔx2exp-12 q-yΔy2,

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