Abstract

Optical Haar wavelets are generated by zone plates that are designed to realize the bipolar nature of Haar wavelets. We present a circular Haar wavelet in two-dimensional space, which can extract the edge and corner features simultaneously, and characterize its properties with computer simulations. A comparison with rectangular Haar wavelets is also given.

© 1994 Optical Society of America

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References

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  1. Y. Sheng, D. Roberge, H. H. Szu, Opt. Eng. 31, 1840 (1992).
    [CrossRef]
  2. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, Opt. Lett. 17, 210 (1992).
    [CrossRef] [PubMed]
  3. D. Roberge, Y. Sheng, Proc. Soc. Photo-Opt. Instrum. Eng. 1812, 295 (1992).
  4. X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
    [CrossRef]
  5. T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 163.
  7. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 402.
  8. S. B. Vinas, Z. Jaroszewicz, A. Kolodziejczyk, M. Sypek, Appl. Opt. 31, 192 (1992).
    [CrossRef] [PubMed]

1992

D. Roberge, Y. Sheng, Proc. Soc. Photo-Opt. Instrum. Eng. 1812, 295 (1992).

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

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

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, Opt. Lett. 17, 210 (1992).
[CrossRef] [PubMed]

S. B. Vinas, Z. Jaroszewicz, A. Kolodziejczyk, M. Sypek, Appl. Opt. 31, 192 (1992).
[CrossRef] [PubMed]

Burns, T. J.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Caulfield, H. J.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

Caviris, N. P.

Fielding, K. H.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 163.

Jaroszewicz, Z.

Kanterakis, E. G.

Katz, A.

Kolodziejczyk, A.

Li, Y.

Lu, X. J.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 402.

Pinski, S. D.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Roberge, D.

D. Roberge, Y. Sheng, Proc. Soc. Photo-Opt. Instrum. Eng. 1812, 295 (1992).

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

Rogers, S. K.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Ruck, D. W.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Sheng, Y.

D. Roberge, Y. Sheng, Proc. Soc. Photo-Opt. Instrum. Eng. 1812, 295 (1992).

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

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

Sypek, M.

Szu, H. H.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

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

Tolimieri, R.

Vinas, S. B.

Yang, X.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

Zhang, Y.

Appl. Opt.

Opt. Eng.

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

X. Yang, H. H. Szu, Y. Sheng, H. J. Caulfield, Opt. Eng. 31, 1846 (1992).
[CrossRef]

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, Opt. Eng. 31, 1852 (1992).
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

D. Roberge, Y. Sheng, Proc. Soc. Photo-Opt. Instrum. Eng. 1812, 295 (1992).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 163.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 402.

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Figures (4)

Fig. 1
Fig. 1

Zone pattern of the one-dimensional wavelet zone plate.

Fig. 2
Fig. 2

Computer simulation results of RHW transforms with a width of 10 pixels: (a) one-dimensional RHW transforms, (b) two-dimensional RHW transforms.

Fig. 3
Fig. 3

(a) Zone pattern of a CHW zone plate and (b) its Fourier spectrum intensity |H(f)|2.

Fig. 4
Fig. 4

Computer simulation results of CHW transforms for a CHW zone plate diameter of (a) 10 pixels and (b) 20 pixels.

Equations (6)

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h ( x , y ) = rect ( x + 0.5 ) - rect ( x - 0.5 )
h ( x , y ) = rect ( x - 0.5 , y - 0.5 ) + rect ( x + 0.5 , y + 0.5 ) - rect ( x + 0.5 , y - 0.5 ) - rect ( x - 0.5 , y + 0.5 ) .
P ( x , y ) = rect [ ( x + D / 4 ) / ( D / 2 ) ] - rect [ ( x - D / 4 ) / ( D / 2 ) ] .
P ( r ) = circ ( 2 r / D ) - 2 circ ( 2 2 r / D ) ,
H ( f ) = ( D / 2 ) [ J 1 ( π D f ) / f - 2 J 1 ( 2 π D f / 2 ) / f ] .
P ( r ) d x d y = 0 .

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