Abstract

Usually a wavelet transform is based on dilated–translated wavelets. We propose a symplectic-transformed–translated wavelet family ψr,s*(zκ) (r,s are the symplectic transform parameters, s2r2=1, κ is a translation parameter) generated from the mother wavelet ψ and the corresponding wavelet transformation Wψf(r,s;κ)=(d2zπ)f(z)ψr,s*(zκ). This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.

© 2006 Optical Society of America

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  1. S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).
  2. C. S. Burrus, R. A. Gopinath, and H.-T. Guo, Introduction to Wavelets and Wavelet Transforms (A Primer) (Prentice-Hall, 1998).
  3. C. K. Chiu, Introduction to Wavelets (Academic, 1992).
  4. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
    [CrossRef]
  5. M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).
  6. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 407 (2006).
    [CrossRef] [PubMed]
  7. D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
    [CrossRef]
  8. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).
  9. R. J. Glauber, Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  10. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
    [CrossRef]
  11. J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985) and references therein.
  12. H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  14. D. F. Walls, Nature 306, 141 (1983).
    [CrossRef]
  15. R. Loudon and P. L. Knight, J. Mol. Spectrosc. 34, 709 (1987).
  16. H.-Y. Fan and H.-L. Lu, Opt. Lett. 28, 680 (2003).
    [CrossRef] [PubMed]
  17. H.-Y. Fan, Opt. Lett. 28, 2177 (2003).
    [CrossRef] [PubMed]
  18. H.-Y. Fan and H.-L. Lu, Opt. Lett. 31, 2622 (2006).
    [CrossRef] [PubMed]

2006 (3)

2003 (2)

1996 (1)

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

1987 (1)

R. Loudon and P. L. Knight, J. Mol. Spectrosc. 34, 709 (1987).

1983 (1)

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

1963 (2)

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Agarwal, G. S.

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Burrus, C. S.

C. S. Burrus, R. A. Gopinath, and H.-T. Guo, Introduction to Wavelets and Wavelet Transforms (A Primer) (Prentice-Hall, 1998).

Chiu, C. K.

C. K. Chiu, Introduction to Wavelets (Academic, 1992).

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
[CrossRef]

Fan, H.-Y.

Glauber, R. J.

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gopinath, R. A.

C. S. Burrus, R. A. Gopinath, and H.-T. Guo, Introduction to Wavelets and Wavelet Transforms (A Primer) (Prentice-Hall, 1998).

Guo, H.-T.

C. S. Burrus, R. A. Gopinath, and H.-T. Guo, Introduction to Wavelets and Wavelet Transforms (A Primer) (Prentice-Hall, 1998).

Jaffard, S.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

James, D. F. V.

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Klauder, J. R.

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985) and references therein.

Knight, P. L.

R. Loudon and P. L. Knight, J. Mol. Spectrosc. 34, 709 (1987).

Loudon, R.

R. Loudon and P. L. Knight, J. Mol. Spectrosc. 34, 709 (1987).

Lu, H.-L.

Meyer, Y.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

Pinsky, M. A.

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).

Ryan, R. D.

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

Skargerstam, B. S.

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985) and references therein.

Walls, D. F.

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

J. Mol. Spectrosc. (1)

R. Loudon and P. L. Knight, J. Mol. Spectrosc. 34, 709 (1987).

Nature (1)

D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

Opt. Commun. (2)

H.-Y. Fan and H.-L. Lu, Opt. Commun. 258, 51 (2006).
[CrossRef]

D. F. V. James and G. S. Agarwal, Opt. Commun. 126, 207 (1996).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (2)

R. J. Glauber, Phys. Rev. 130, 2529 (1963).
[CrossRef]

R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[CrossRef]

Other (8)

J. R. Klauder and B. S. Skargerstam, Coherent States (World Scientific, 1985) and references therein.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

S. Jaffard, Y. Meyer, and R. D. Ryan, Wavelets, Tools for Science & Technology (Society for Industrial and Applied Mathematics, 2001).

C. S. Burrus, R. A. Gopinath, and H.-T. Guo, Introduction to Wavelets and Wavelet Transforms (A Primer) (Prentice-Hall, 1998).

C. K. Chiu, Introduction to Wavelets (Academic, 1992).

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics (Society for Industrial and Applied Mathematics, 1992).
[CrossRef]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets (Book/Cole, 2002).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Equations (27)

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ψ ( x ) d x = 0 .
ψ ( a , b ) ( x ) = 1 a ψ ( x b a ) ,
W ψ f ( a , b ) = 1 a f ( x ) ψ * ( x b a ) d x .
ψ r , s ; κ ( z ) = s ψ [ s ( z κ ) r ( z * κ * ) ] .
W ψ f ( r , s ; κ ) = d 2 z π f ( z ) ψ r , s ; κ * ( z ) , d 2 z = d x d y , z = x + i y .
( z κ z * κ * ) M ( z κ z * κ * ) , M [ s r r * s * ] ,
J = [ 0 I I 0 ] .
W ψ f = s d 2 z π exp ( z β * z * β ) ψ * [ s ( z κ ) r ( z * κ * ) ] = s d 2 z π exp [ ( z + κ ) β * ( z * + κ * ) β ] ψ * ( s z r z * ) = exp ( κ β * κ * β ) s d 2 z π exp [ z ( s * β * r * β ) z * ( s β r β * ) ] ψ * ( z ) = s exp [ κ β * κ * β ] Φ ( s * β * r * β ) ,
W ψ * ( W ψ f ) ( z ) = s * d 2 κ π ( W ψ f ) ( r , s ; κ ) ψ [ s ( z κ ) r ( z * κ * ) ] = s Φ ( s * β * r * β ) d 2 κ π exp [ ( κ + z ) β * ( κ * + z * ) β ] ψ ( s κ + r κ * ) = s exp ( z β * z * β ) Φ ( s * β * r * β ) × d 2 κ π exp [ κ ( s * β * r * β ) + κ * ( s β r β * ) ] ψ ( κ ) = s exp ( z β * z * β ) Φ ( s * β * r * β ) 2 .
W ψ * ( W ψ f ) ( z ) d 2 s s 2 = exp ( z β * z * β ) d 2 s Φ ( s * β * r * β ) 2 s ,
f ( z ) = exp ( z β * z * β ) = d 2 s W ψ * ( W ψ f ) ( z ) s 2 d 2 s Φ ( s * β * r * β ) 2 s .
d 2 s Φ ( s * β * r * β ) 2 s = 1 ,
f ( z ) = d 2 s W ψ * ( W ψ f ) ( z ) s 2 .
W ψ f ( r , s ; κ ) W ψ f * ( r , s ; κ ) d 2 κ d 2 s s 2 = d 2 z f ( z ) f * ( z ) .
F ( β ) = d 2 z π f ( z ) exp ( z β * z * β ) ,
d 2 z f ( α z , α * z * ) f ( z ) = d 2 z d 2 β π F ( β ) e ( α * z * ) β ( α z ) β * d 2 β π F ( β ) e z * β z β * = d 2 β d 2 β F ( β ) F ( β ) e α * β α β * δ ( β β ) δ ( β * β * ) = d 2 β F ( β ) F ( β ) e α * β α β * ,
d 2 z f ( z ) ψ * [ s ( z κ ) r ( z * κ * ) ] = d 2 β F ( β ) Φ * ( s β r β * ) exp ( κ β * κ * β ) .
W ψ f ( r , s ; κ ) W ψ f * ( r , s ; κ ) d 2 κ = s d 2 β d 2 β F ( β ) Φ * ( s β r β * ) F * ( β ) Φ ( s β r β * ) δ ( β β ) δ ( β * β * ) = s d 2 β F ( β ) F * ( β ) Φ ( s β r β * ) 2 ,
d 2 s s 2 W ψ f ( r , s ; κ ) W ψ f * ( r , s ; κ ) d 2 κ = d 2 β F ( β ) F * ( β ) d 2 s s Φ ( s β r β * ) 2 = d 2 β F ( β ) F * ( β ) = d 2 z f ( z ) f * ( z ) ,
W ψ f ( r , s ; κ ) ψ r , s ; κ ( z ) d 2 κ d 2 s π s 2 = f ( z ) ,
W ψ f ( r , s ; κ ) = d 2 z π f ( z ) ψ r , s ; κ * ( z ) = 1 π ψ r , s ; κ * ( z ) .
z = exp ( z a z * a ) 0 ( z z * ) ,
W ψ f ( r , s ; κ ) = s d 2 z π ψ * [ s ( z κ ) r ( z * κ * ) ] f ( z ) = s d 2 z π ψ [ s r r * s * ] ( z κ z * κ * ) z f = ψ F ( r , s , κ ) f ,
F ( r , s , κ ) = s d 2 z π s z r z * z + κ ,
s z r z * [ s r r * s * ] ( z z * ) .
x 2 F ( r , s , κ = 0 ) x 1 = 1 2 π i B exp [ i 2 B ( A x 1 2 2 x 2 x 1 + D x 2 2 ) ] ,
[ A B C D ] = 1 2 [ k + k * + t + t * i ( k k * t + t * ) i ( k k * + t t * ) k + k * t t * ] ,

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