Abstract

In one-dimensional (1D) signal analysis, the complex analytic signal built from a real-valued signal and its Hilbert transform is an important tool providing a mathematical foundation for 1D statistical analysis. For a natural extension beyond 1D signal, Riesz transform has been applied to high-dimensional signal processing as a generalized Hilbert transform to construct a vector signal representation and therefore, to enlarge the traditional analytic signal concept. In this paper, we introduce the vector correlations as new mathematical tools for vector calculus for statistical speckle analysis. Based on vector correlations of a real-valued speckle pattern, we present the associated correlation properties, which can be regarded as mathematical foundation for the vector analysis in speckle metrology.

© 2014 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).
  2. S. L. Hahn, Hilbert Transforms in Signal Processing (Arteche, 1996).
  3. J. W. Goodman, Statisitcal Optics (Wiley, 2000).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).
  5. M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
    [CrossRef]
  6. S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
    [CrossRef]
  7. M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in Proc. 22nd DAGM Symp. Mustererkennung (Heidelberg, Germany, 2000).
  8. K. G. Larkin, J. D. Bone, and A. M. 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]
  9. M. N. Nabighian, “Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
    [CrossRef]
  10. M. Riesz, “Sur les fonctions conjuguées,” Mathematische Zeitschrift 27, 218–244 (1928).
    [CrossRef]
  11. W. Wang, T. Yomiaki, R. Ishijima, A. Wada, Y. Miyamoto, and T. Mitruo, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
    [CrossRef]
  12. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” J. Mod. Opt. 28, 1359–1376 (1981).
  13. M. Sjödahl, “Calculation of speckle displacement, decorrelation, and object-point location in imaging systems,” Appl. Opt. 34, 7998–8010 (1995).
    [CrossRef]
  14. P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
    [CrossRef]
  15. W. Wang, R. Ishijima, A. Matsuda, S. G. Hanson, and M. Takeda, “Pseudo-Stokes vector correlation from complex signal representation of a speckle pattern and its applications to micro‐displacement measurement,” Strain 46, 12–18 (2010).
  16. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw, 1999).
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, 1974).
  18. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part I: microdisplacement observation based on phase-only correlation in the signal domain,” Appl. Opt. 44, 4909–4915 (2005).
    [CrossRef]
  19. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda,. “Pseudophase information from the complex analytic signal of speckle fields and its applications. Part II: statistical properties of the analytic signal of a white-light speckle pattern applied to the microdisplacement measurment,” Appl. Opt. 44, 4916–4921 (2005).
    [CrossRef]
  20. E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton, 1970).
  21. E. M. Stein and L. G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton, 1971).

2010 (1)

W. Wang, R. Ishijima, A. Matsuda, S. G. Hanson, and M. Takeda, “Pseudo-Stokes vector correlation from complex signal representation of a speckle pattern and its applications to micro‐displacement measurement,” Strain 46, 12–18 (2010).

2007 (1)

P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
[CrossRef]

2006 (1)

2005 (2)

2001 (2)

1995 (1)

1992 (1)

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

1984 (1)

M. N. Nabighian, “Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

1981 (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” J. Mod. Opt. 28, 1359–1376 (1981).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

1928 (1)

M. Riesz, “Sur les fonctions conjuguées,” Mathematische Zeitschrift 27, 218–244 (1928).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, 1974).

Bone, J. D.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw, 1999).

Felsberg, M.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in Proc. 22nd DAGM Symp. Mustererkennung (Heidelberg, Germany, 2000).

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

Goodman, J. W.

J. W. Goodman, Statisitcal Optics (Wiley, 2000).

Hahn, S. L.

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

S. L. Hahn, Hilbert Transforms in Signal Processing (Arteche, 1996).

Hanson, S. G.

Horváth, P.

P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
[CrossRef]

Hrabovský, M.

P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
[CrossRef]

Ishii, N.

Ishijima, R.

W. Wang, R. Ishijima, A. Matsuda, S. G. Hanson, and M. Takeda, “Pseudo-Stokes vector correlation from complex signal representation of a speckle pattern and its applications to micro‐displacement measurement,” Strain 46, 12–18 (2010).

W. Wang, T. Yomiaki, R. Ishijima, A. Wada, Y. Miyamoto, and T. Mitruo, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14, 120–127 (2006).
[CrossRef]

Larkin, K. G.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Matsuda, A.

W. Wang, R. Ishijima, A. Matsuda, S. G. Hanson, and M. Takeda, “Pseudo-Stokes vector correlation from complex signal representation of a speckle pattern and its applications to micro‐displacement measurement,” Strain 46, 12–18 (2010).

Mitruo, T.

Miyamoto, Y.

Nabighian, M. N.

M. N. Nabighian, “Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

Oldfield, A. M.

Riesz, M.

M. Riesz, “Sur les fonctions conjuguées,” Mathematische Zeitschrift 27, 218–244 (1928).
[CrossRef]

Sjödahl, M.

Šmíd, P.

P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
[CrossRef]

Sommer, G.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in Proc. 22nd DAGM Symp. Mustererkennung (Heidelberg, Germany, 2000).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, 1974).

Stein, E. M.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton, 1970).

E. M. Stein and L. G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton, 1971).

Takeda, M.

Wada, A.

Wang, W.

Weiss, L. G.

E. M. Stein and L. G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton, 1971).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” J. Mod. Opt. 28, 1359–1376 (1981).

Yomiaki, T.

Appl. Opt (1)

P. Šmíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to measure object’s in-plane velocity,” Appl. Opt 46, 3709–3715 (2007).
[CrossRef]

Appl. Opt. (3)

Geophysics (1)

M. N. Nabighian, “Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

IEEE Trans. Signal Process. (1)

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

J. Inst. Elect. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. 93, 429–457 (1946).

J. Mod. Opt. (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” J. Mod. Opt. 28, 1359–1376 (1981).

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

Mathematische Zeitschrift (1)

M. Riesz, “Sur les fonctions conjuguées,” Mathematische Zeitschrift 27, 218–244 (1928).
[CrossRef]

Opt. Express (1)

Proc. IEEE (1)

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

Strain (1)

W. Wang, R. Ishijima, A. Matsuda, S. G. Hanson, and M. Takeda, “Pseudo-Stokes vector correlation from complex signal representation of a speckle pattern and its applications to micro‐displacement measurement,” Strain 46, 12–18 (2010).

Other (8)

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw, 1999).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, 1974).

E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton, 1970).

E. M. Stein and L. G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton, 1971).

M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in Proc. 22nd DAGM Symp. Mustererkennung (Heidelberg, Germany, 2000).

S. L. Hahn, Hilbert Transforms in Signal Processing (Arteche, 1996).

J. W. Goodman, Statisitcal Optics (Wiley, 2000).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

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Figures (5)

Fig. 1.
Fig. 1.

Construction of a vector representation from a real-valued 2D signal.

Fig. 2.
Fig. 2.

Vector signal representation in the Cartesian and spherical coordinates.

Fig. 3.
Fig. 3.

Sample image of a vector representation for 2D speckle pattern.

Fig. 4.
Fig. 4.

Modulus of U⃗fV⃗f.

Fig. 5.
Fig. 5.

Angle of U⃗fV⃗f.

Equations (33)

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A⃗f(x,y)=[k(2)(x,y)e^1k(1)(x,y)e^2+δ(x,y)e^3]*f(x,y),
k(x,y)={k(1)(x,y)=x2π(x2+y2)3/2k(2)(x,y)=y2π(x2+y2)3/2,
I{k(x,y)}={jωxωx2+ωy2jωyωx2+ωy2.
sgn(ω)={+1ω>00ω=01ω<0.
H{f(x)}=jsgn(ω)I{f(x)}
A⃗f(x,y)=U⃗f(x,y)+V⃗f(x,y),
{V⃗f(x,y)=f(x,y)e^3U⃗f(x,y)=[k(2)*f]e^1[k(1)*f]e^2=Uf(1)e^1+Uf(2)e^2.
f⃗(x,y)g⃗(x,y)=m=13[f(m)*(x,y)*g(m)(x,y)]=m=13Γf,g(m,m)(x,y),
f⃗(x,y)g⃗(x,y)=Γf,g(2,3)(x,y)e^1+Γf,g(3,1)(x,y)e^2+Γf,g(1,2)(x,y)e^3Γf,g(3,2)(x,y)e^1Γf,g(1,3)(x,y)e^2Γf,g(2,3)(x,y)e^3.
U⃗f(x,y)U⃗f(x,y)=[k(2)(x,y)*k(2)(x,y)+k(1)(x,y)*k(1)(x,y)]*Γf,f(x,y).
k(2)*k(2)+k(1)*k(1)=I1{(I{k(1)})2+(I{k(2)})2}=I1{1}=δ(x,y),
U⃗f(x,y)U⃗f(x,y)=Γf,f(x,y)=V⃗f(x,y)V⃗f(x,y).
U⃗f(x,y)V⃗f(x,y)=V⃗f(x,y)U⃗f(x,y)=[k(1)(x,y)e^1+k(2)(x,y)e^2]*Γf,f(x,y).
U⃗fV⃗f|x,y=0=V⃗fU⃗f|x,y=0=0.
A⃗f(x,y)A⃗f(x,y)=U⃗f(x,y)U⃗f(x,y)+V⃗f(x,y)V⃗f(x,y)=2Γf,f(x,y).
A⃗f(x,y)A⃗f(x,y)=2U⃗f(x,y)V⃗f(x,y)=2(k(1)e^1+k(2)e^2)*Γf,f(x,y).
A⃗g(x,y)=U⃗g(x,y)+V⃗g(x,y)·
A⃗f(x,y)A⃗g(x,y)=2Γf,g(x,y).
U⃗f(x,y)U⃗g(x,y)=0,
V⃗f(x,y)U⃗g(x,y)=[k(1)e^1+k(2)e^2]*Γf,g,
U⃗f(x,y)V⃗g(x,y)=[k(1)e^1+k(2)e^2]*Γf,g,
V⃗f(x,y)V⃗g(x,y)=0.
A⃗f(x,y)A⃗g(x,y)=2(k(1)e^1+k(2)e^2)*Γf,g.
Γf,f(x,y)=exp(x2+y22σ2).
U⃗f(x,y)V⃗f(x,y)=2π4σM(32,2,x2+y22σ2)(xe^1+ye^2),
{|U⃗fV⃗f|=2π4σx2+y2M(32,2,x2+y22σ2)Angle{U⃗fV⃗f}=atan2(x,y),
k(1)(x,y)*Γf,f(x,y)=I1{k(1)(ωx,ωy)Γf,f(ωx,ωy)}=jσ22π++ωxωx2+ωy2exp(σ2(ωx2+ωy2)2)exp(jωxx+jωyy)dωxdωy.
k(1)(r,θ)*Γf,f(r)=jσ22π0+exp(σ2ρ22)ρdρ02πexp(jφ)+exp(jφ)2exp[jρrcos(θφ)]dφ.
Jn(x)=12πππexp[j(nφxsinφ)]dφ,
k(1)(r,θ)*Γf,f(r)=σ2cosθ0+exp(σ2ρ22)ρJ1(ρr)dρ.
k(2)(r,θ)*Γf,f(r)=σ2sinθ0+exp(σ2ρ22)ρJ1(ρr)dρ.
0+ea2ρ2ρμ1Jν(bρ)dρ=Γ(μ+ν2)(b2a)ν2aμΓ(ν+1)M(μ+ν2,ν+1,b24a2),
0+exp(σ2ρ22)ρJ1(ρr)dρ=2π4σrM(32,2,r22σ2).

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