Abstract

Based on the wave-front filtering concept of wavelet optics, which we proposed, here we consider that the wave fronts of a light wave are filtered by the lens when the wave fronts pass through the lens. After filtering, the weight of the light field is redistributed, and a Gaussian frequency-modulated complex-valued wavelet function is introduced with the weight function. Subsequent analysis indicates that the lens has a wavelet-transforming characteristic and that the introduction of a Gaussian frequency-modulated complex-valued wavelet function conforms to the actual lens.

© 2002 Optical Society of America

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References

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  1. C. K. Chui, An Introduction to Wavelets (Academic, San Diego, Calif., 1992), pp. 16–22.
  2. W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
    [Crossref]
  3. Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
    [Crossref]
  4. P. G. Block, S. K. Rogers, D. W. Ruck, Optical wavelet transform from computer-generated holography,” Appl. Opt. 33, 5275–5278 (1994).
    [Crossref] [PubMed]
  5. L. Tan, J. Ma, Q. Ran, “The elementary theory of wavelet optical diffraction,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS ’99), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 113–114.
  6. L. Tan, J. Ma, Q. Wang, Q. Ran, “Filtering theory and application of wavelet optics at the spatial-frequency domain,” Appl. Opt. 40, 257–260 (2001).
    [Crossref]

2001 (1)

1996 (1)

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

1995 (1)

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

1994 (1)

Block, P. G.

Caufield, H. J.

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

Chui, C. K.

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, Calif., 1992), pp. 16–22.

Jin, G. F.

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

Li, Y.

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

Ma, J.

L. Tan, J. Ma, Q. Wang, Q. Ran, “Filtering theory and application of wavelet optics at the spatial-frequency domain,” Appl. Opt. 40, 257–260 (2001).
[Crossref]

L. Tan, J. Ma, Q. Ran, “The elementary theory of wavelet optical diffraction,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS ’99), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 113–114.

Ran, Q.

L. Tan, J. Ma, Q. Wang, Q. Ran, “Filtering theory and application of wavelet optics at the spatial-frequency domain,” Appl. Opt. 40, 257–260 (2001).
[Crossref]

L. Tan, J. Ma, Q. Ran, “The elementary theory of wavelet optical diffraction,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS ’99), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 113–114.

Rogers, S. K.

Ruck, D. W.

Sheng, Y.

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

Szu, H. H.

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

Tan, L.

L. Tan, J. Ma, Q. Wang, Q. Ran, “Filtering theory and application of wavelet optics at the spatial-frequency domain,” Appl. Opt. 40, 257–260 (2001).
[Crossref]

L. Tan, J. Ma, Q. Ran, “The elementary theory of wavelet optical diffraction,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS ’99), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 113–114.

Wang, Q.

Wang, W. L.

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

Wu, M. X.

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

Yan, Y. B.

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

Appl. Opt. (2)

Opt. Eng. (1)

W. L. Wang, G. F. Jin, Y. B. Yan, M. X. Wu, “Image feature extraction with optical Harr wavelet transform,” Opt. Eng. 34, 1238–1242 (1995).
[Crossref]

Proc. IEEE (1)

Y. Li, H. H. Szu, Y. Sheng, H. J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[Crossref]

Other (2)

L. Tan, J. Ma, Q. Ran, “The elementary theory of wavelet optical diffraction,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference (CLEO/QELS ’99), Vol. 6 of 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 113–114.

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, Calif., 1992), pp. 16–22.

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Equations (28)

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ha,b;c,dx, y=a-1/2c-1/2hb-xa, d-yc.
ha,b;c,dx, y=a-1/2c-1/2hb-xa, d-yc
UTx, y=- UTx, yha,b;c,dx, ydxdy.
UTx, y=UTx, y  ha,b;c,dx, y,
ÛTp, q=ÛTp, qĤp, q.
ha,b;c,dx, y=a-1/2c-1/2hb-xa, d-yc,
ha,0,c,0x, y=exp-12p-xΔx2exp-12q-yΔy2×expj2πfxp+fyq+Cfxp2+fyq2,
exp-12p-xΔx2exp-12q-yΔy2×expj2πfxp+fyq+Cfxp2+fyq2
exp-12p-xΔx2exp-12q-yΔy2
Δx, y=Δ0-x2+y221R1-1R2,
tLx, y=expjknΔ0exp-j k2fx2+y2Px, y,
Px, y=1in aperture0other,
1f=n-11R1-1R2.
UTx0, y0=At0x0, y0,
UTx, y=- UTx, yha,b;c,dx, ydxdy =- UTx, yexp-12p-xΔx2×exp-12q-yΔy2expj2πfxp+fyq+1+dfxp2+fyq2dxdy =expj2πCfxp2+fyq2×- UTx, yexp-12p-xΔx2×exp-12q-yΔy2×expj2πfxp+fyqdxdy.
Ufxf, yf=Aiλfexpj k2fxf2+yf2×- t0x, yexp -j 2πλffxx+fyydxdy.
exp-12p-xΔx2exp-12q-yΔy2
Ifxf, yf=|UTx, y|2=Aλf2|Ft0x, y|2.
C=k2f1-d0f,UTx, y=- UTx, yha,b;c,dx, ydxdy =expj2π k2f1-d0ffxp2+fyq2×-exp-12p-xΔx2×exp-12q-yΔy2expj2π×fxp+fyqdxdy.
Ufxf, yf=Ajλfexpj k2f1-d0fxf2+yf2×- t0x0, y0exp-j 2πλf×x0xf+yyfdx0dy0.
exp-12p-xΔx2exp-12q-yΔy2
expj2πCfxp2+fyq2, C=k2f 1-d0f,
Ifxf, yf=|UTx, y|2=Aλf2|Ft0x, y|2.
UTx, y=- UTx, yha,b;c,dx, ydxdy =- UTx, yexp-12p-xΔx2×exp-12q-yΔy2×expj2πfxp+fyqdxdy.
Ufxf, yf=Ajλf- t0x0, y0×exp -j 2πλfx0xf+y0yfdx0dy0.
fX=xfλf, fY=yfλf,
Ifxf, yf=|UTx, y|2=Aλf2|Ft0x0, y0|2.
C=k2d0+fd1+fd0-d1f2

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