Abstract

To provide a theoretical background for the superiority of the signal-domain phase-only correlation (SDPOC) technique proposed here for microdisplacement measurement, we study the first- and the second-order statistical properties of the complex amplitude of an analytic signal of a white-light speckle pattern, under the assumption of a Gaussian random process, and give a formula for the autocorrelation function of the pseudophase associated with the complex analytic signal. Based on these results, we show mathematically that SDPOC has a performance advantage over conventional intensity-based correlation techniques.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Gabor, “Theory of communications,” J. IEE 93, 429–457 (1946).
  2. K. G. Larkin, D. J. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  3. Y. Watanabe, I. Yamaguchi, “Digital Hilbert transformation for separation measurement of thickness and refractive indices of layered objects by use of a wavelength-scanning heterodyne interference confocal microscope,” Appl. Opt. 41, 4497–4502 (2002).
    [CrossRef] [PubMed]
  4. W. Wang, N. Ishii, Y. Miyamoto, M. Takeda, “Pseudophase information about the complex analytic signal of speckle fields and its applications. Part I: Microdisplacement measurement based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
    [CrossRef] [PubMed]
  5. S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1997), pp. 38, 64, 375.
  6. J. W. Goodman, Statistical Optics (Wiley, 1985), p. 105.
  7. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer, 1984).
  8. H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
    [CrossRef]
  9. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 399, 404, 408.
  10. I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
    [CrossRef]
  11. D. D. Duncan, S. J. Kirkpatrick, “Performance analysis of a maximum likelihood speckle motion estimator,” Opt. Express 10, 929–941 (2002).
    [CrossRef]
  12. M. Sjodahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
    [CrossRef] [PubMed]

2005 (1)

2004 (1)

H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
[CrossRef]

2002 (2)

2001 (1)

1996 (1)

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

1993 (1)

1946 (1)

D. Gabor, “Theory of communications,” J. IEE 93, 429–457 (1946).

Benckert, L. R.

Bone, D. J.

Duncan, D. D.

D. D. Duncan, S. J. Kirkpatrick, “Performance analysis of a maximum likelihood speckle motion estimator,” Opt. Express 10, 929–941 (2002).
[CrossRef]

Freund, I.

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communications,” J. IEE 93, 429–457 (1946).

Garcia, D.

H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), p. 105.

Hahn, S. L.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1997), pp. 38, 64, 375.

Ishii, N.

Kessler, D. A.

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

Kirkpatrick, S. J.

D. D. Duncan, S. J. Kirkpatrick, “Performance analysis of a maximum likelihood speckle motion estimator,” Opt. Express 10, 929–941 (2002).
[CrossRef]

Larkin, K. G.

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 399, 404, 408.

Miyamoto, Y.

Oldfield, M. A.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer, 1984).

Schreier, H. W.

H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
[CrossRef]

Sjodahl, M.

Suttan, M. A.

H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
[CrossRef]

Takeda, M.

Wang, W.

Watanabe, Y.

Yamaguchi, I.

Appl. Opt. (3)

Exp. Mech. (1)

H. W. Schreier, D. Garcia, M. A. Suttan, “Advances in light microscope stereo vision,” Exp. Mech. 44, 278–288 (2004).
[CrossRef]

J. IEE (1)

D. Gabor, “Theory of communications,” J. IEE 93, 429–457 (1946).

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

I. Freund, D. A. Kessler, “Phase autocorrelation of random wave fields,” Opt. Commun. 124, 321–332 (1996).
[CrossRef]

Opt. Express (1)

D. D. Duncan, S. J. Kirkpatrick, “Performance analysis of a maximum likelihood speckle motion estimator,” Opt. Express 10, 929–941 (2002).
[CrossRef]

Other (4)

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, 1960), pp. 399, 404, 408.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1997), pp. 38, 64, 375.

J. W. Goodman, Statistical Optics (Wiley, 1985), p. 105.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena,J. C. Dainty, ed. (Springer, 1984).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Theoretical autocorrelation functions calculated for intensity and pseudophase autocorrelation.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

u ˜ ( x , y ) = u ˜ ( x , y ) exp [ j θ ( x , y ) ] = u ( r ) ( x , y ) + j π P - + u ( r ) ( ξ , y ) x - ξ d ξ ,
u ˜ ( x ) = u ˜ ( x ) exp [ j θ ( x ) ] = u ( r ) ( x ) + j π P - + u ( r ) ( ξ ) x - ξ d ξ = [ δ ( x ) + j π x ] * u ( r ) ( x ) ,
u ˜ ( x ) = u ( r ) ( x ) + j H { u ( r ) ( x ) } ,
r = Re { u ˜ ( x ) } = u ( r ) ( x ) i = Im { u ˜ ( x ) } = H { u ( r ) ( x ) } .
i = 1 π x * u ( r ) ( x ) = 1 π P - + ( r ) u ( r ) ( x - ξ ) ξ d ξ = 1 π P - + u ( r ) ( x - ξ ) ξ d ξ = 0 ,
Γ ( r , r ) ( Δ x ) = Γ ( i , i ) ( Δ x ) , Γ ( r , i ) ( Δ x ) = - Γ ( i , r ) ( Δ x ) = H { Γ ( r , r ) ( Δ x ) } ,
Γ ( f , g ) f ( x + Δ x ) g ( x ) .
r r = Γ ( r , r ) ( Δ x ) Δ x = 0 σ 2 .
i i = Γ ( i , i ) ( Δ x ) Δ x = 0 σ 2 .
r i = Γ ( r , i ) ( Δ x ) Δ x = 0 = - Γ ( i , r ) ( Δ x ) Δ x = 0 = - i r ,
r i = i r = 0.
M = [ σ 2 0 0 σ 2 ] .
P ( r , i ) = 1 2 π σ 2 exp ( - r 2 + i 2 2 σ 2 ) .
θ = tan - 1 Im { u ˜ } Re { u ˜ } = tan - 1 H { u ( r ) } u ( r ) .
P ( θ ) = { 1 / 2 π - π θ < π 0 otherwise .
P ( r 1 , i 1 , r 2 , i 2 ) = exp ( - ½ X M - 1 X T ) ( 2 π ) 2 det M ,
M = [ r 1 r 1 r 1 i 1 r 1 r 2 r 1 i 2 i 1 r 1 i 1 i 1 i 1 r 2 i 1 i 2 r 2 r 1 r 2 i 1 r 2 r 2 r 2 i 2 i 2 r 1 i 2 i 1 i 2 r 2 i 2 i 2 ] .
r k r k = i k i k σ 2 r k i k = i k r k = 0
r 1 r 2 = Γ ( r , r ) ( Δ x ) = Γ ( i , i ) ( Δ x ) = i 1 i 2 σ 2 ρ 0 ( Δ x ) , r 1 i 2 = Γ ( r , i ) ( Δ x ) = - Γ ( i , r ) ( Δ x ) = - i 1 r 2 = - H { Γ ( r , r ) ( Δ x ) } σ 2 λ 0 ( Δ x ) .
M = σ 2 [ 1 0 ρ 0 λ 0 0 1 - λ 0 ρ 0 ρ 0 - λ 0 1 0 λ 0 ρ 0 0 1 ] .
P ( r 1 , i 1 , r 2 , i 2 ) = exp { - r 1 2 + i 1 2 + r 2 2 + i 2 2 - 2 ρ 0 ( r 1 r 2 + i 1 i 2 ) - 2 λ 0 ( r 1 i 2 - r 2 i 1 ) 2 σ 2 [ 1 - ( ρ 0 2 + λ 0 2 ) ] } 4 π 2 σ 4 [ 1 - ( ρ 0 2 + λ 0 2 ) ] .
P ( θ 1 , θ 2 ) = 1 - ρ 0 2 - λ 0 2 4 π 2 ( 1 - β 2 ) - 3 / 2 × [ β arcsin β + π β 2 + ( 1 - β 2 ) 1 / 2 ] ,
Γ c ( θ , θ ) ( Δ x ) = π arcsin [ μ ( Δ x ) ] - arcsin 2 [ μ ( Δ x ) ] + 1 2 n = 1 μ ( Δ x ) 2 n n 2 ,
exp { j [ θ ( x + Δ x ) - θ ( x ) ] } = exp ( - θ 2 ) exp [ θ ( x + Δ x ) θ ( x ) ] .
C P SDPOC ( Δ x ) exp [ Γ c ( θ , θ ) ( Δ x ) ] .
C I ( Δ x ) = I ( x ) I ( x + Δ x ) I ( x ) I ( x ) = r 1 r 2 r r = σ 2 ρ 0 ( Δ x ) σ 2 = ρ 0 ( Δ x ) .
μ ( Δ x ) = u ˜ ( x ) u ˜ * ( x + Δ x ) u ˜ ( x ) u ˜ * ( x ) = ( r 1 r 2 + i 1 i 2 ) + j ( i 1 r 2 - r 1 i 2 ) ( r r - i i ) = 2 σ 2 [ ρ 0 ( Δ x ) - j λ 0 ( Δ x ) ] 2 σ 2 = ρ 0 ( Δ x ) + j H { ρ 0 ( Δ x ) } .
ρ 0 ( Δ x ) = sinc ( Δ x / p ) ,
μ ( Δ x ) = sinc ( Δ x / p ) + j H { sinc ( Δ x / p ) } = sin ( π Δ x / p ) + j [ 1 - cos ( π Δ x / p ) ] π Δ x / p .

Metrics