Abstract

Quadrature operators are useful for obtaining the modulating phase ϕ in interferometry and temporal signals in electrical communications. In carrier-frequency interferometry and electrical communications, one uses the Hilbert transform to obtain the quadrature of the signal. In these cases the Hilbert transform gives the desired quadrature because the modulating phase is monotonically increasing. We propose an n-dimensional quadrature operator that transforms cos(ϕ) into -sin(ϕ) regardless of the frequency spectrum of the signal. With the quadrature of the phase-modulated signal, one can easily calculate the value of ϕ over all the domain of interest. Our quadrature operator is composed of two n-dimensional vector fields: One is related to the gradient of the image normalized with respect to local frequency magnitude, and the other is related to the sign of the local frequency of the signal. The inner product of these two vector fields gives us the desired quadrature signal. This quadrature operator is derived in the image space by use of differential vector calculus and in the frequency domain by use of a n-dimensional generalization of the Hilbert transform. A robust numerical algorithm is given to find the modulating phase of two-dimensional single-image closed-fringe interferograms by use of the ideas put forward.

© 2003 Optical Society of America

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References

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  1. T. Kreis, “Digital holgraphic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
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  2. M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-following regularized phase tracker for demodulation of closed-fringe interferogram,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  3. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
    [CrossRef]
  4. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1543 (1998).
    [CrossRef]
  5. 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]
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  7. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000).
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  9. E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).
  10. J. A. Quiroga, M. Servin, F. J. Cuevas, “Modulo 2π fringe orientation angle estimation by phase unwrapping with a regularized phase-tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
    [CrossRef]
  11. D. Richards, Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002).
  12. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  13. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998).
  14. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, “Phase unwrapping through demodulation using the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1940 (1999).
    [CrossRef]
  15. M. Schwartz, Information Transmission Modulation and Noise (McGraw Hill, New York, 1980).

2002 (1)

2001 (2)

1999 (1)

1998 (1)

1997 (1)

1994 (1)

1986 (1)

1982 (1)

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000).

Cuevas, F. J.

Ghiglia, D. C.

Ina, H.

Kobayashi, S.

Kreis, T.

Larkin, K. G.

Malacara, D.

Marroquin, J. L.

Oldfield, M. A.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998).

Quiroga, J. A.

Richards, D.

D. Richards, Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002).

Rodriguez-Vera, R.

Romero, L. A.

Schwartz, M.

M. Schwartz, Information Transmission Modulation and Noise (McGraw Hill, New York, 1980).

Servin, M.

Stein, E.

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

Takeda, M.

Weiss, G.

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Other (6)

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidian Spaces (Princeton U. Press, Princeton, N.J., 1971).

D. Richards, Advanced Mathematical Methods with Maple (Cambridge U. Press, Cambridge, UK., 2002).

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping, Theory Algorithms, and Software (Wiley, New York, 1998).

M. Schwartz, Information Transmission Modulation and Noise (McGraw Hill, New York, 1980).

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Figures (3)

Fig. 1
Fig. 1

Demodulation steps that use the algorithm proposed in Subsection 4.A applied to a noiseless computer-generated interferogram. (a) Fringe pattern, (b) recovered phase, (c) and (d) the two components of I , (e) fringe orientation angle modulo π, (f) fringe orientation angle modulo 2π.

Fig. 2
Fig. 2

Demodulation of an experimentally obtained specklegram. (a) The fringe pattern; (b) the estimated fringe orientation angle modulo 2π; (c) and (d) the two components of I ; (e) the recovered signal n ϕ I ; (f) the recovered phase after the pixelwise adjustment by use of the Halley method.

Fig. 3
Fig. 3

Errors obtained with the vortex operator against the demodulation algorithm proposed by use of a noiseless computer-generated interferogram. (a) Desired quadrature signal, (b) quadrature signal obtained with the numerical algorithm proposed in Subsection 4.A, (c) quadrature signal obtained with the vortex operator, (d) quadrature signal obtained by use of the operator of Eq. (46).

Equations (65)

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H 1 [ cos ( ω 0 x ) ] = - sin ( ω 0 x ) for ω 0 > 0 ;
H 1 [ sin ( ω 0 x ) ] = cos ( ω 0 x )
F { H 1 [ g ( x ) ] } = - i   sign ( u ) g ˆ ( u ) = - i   u | u |   g ˆ ( u ) ,
g ˆ ( u ) = F { g ( x ) } .
H 1 { cos [ ω 0 x + ψ ( x ) ] } = - sin [ ω 0 x + ψ ( x ) ] ,
ω 0 + d ψ ( x ) d x > 0 , x .
Q 1 { cos ( ϕ ) } = d ϕ / d x | d ϕ / d x |   H 1 { cos ( ϕ ) } = - sin ( ϕ ) ,
I a ( x ,   y ) = a ( x ,   y ) + b ( x ,   y ) cos [ ψ ( x ,   y ) + u 0 x + v 0 y ] .
u 0 > ψ ( x ,   y ) x , v 0 > ψ ( x ,   y ) y , ( x ,   y ) .
I ( x ,   y ) = b ( x ,   y ) cos [ ψ ( x ,   y ) + u 0 x + v 0 y ] ,
I ( x ,   y ) = b ( x ,   y ) 2 ( exp { i [ - ψ ( x ,   y ) - u 0 x - v 0 y ] } + exp { i [ ψ ( x ,   y ) + u 0 x + v 0 y ] } ) .
I ˆ ( u ,   v ) = 1 2 [ f - ( u ,   v ) + f + ( u ,   v ) ] ,
H 2 { I ( x ,   y ) } = - b ( x ,   y ) sin [ ψ ( x ,   y ) + u 0 x + v 0 y ] = F - 1 - i ω 0 q | ω 0 q |   [ f + ( u ,   v ) + f - ( u ,   v ) ] ,
ω 0 = u 0 e 1 + v 0 e 2 , q = u e 1 + v e 2 ,
S { I ˆ ( u ,   v ) } = 1 2 [ 1 + H 2 { I ˆ ( u ,   v ) } ] = U ( u ,   v ) + iV ( u ,   v ) ,
ψ ( x ,   y ) + u 0 x + v 0 y = tan - 1 Im ( A ) Re ( A ) ,
A ( x ,   y ) = F - 1 { U ( u ,   v ) + iV ( u ,   v ) } .
Q n { b ( r ) cos [ ϕ ( r ) ] } = - b ( r ) sin [ ϕ ( r ) ] ,
I ( r ) = cos [ ϕ ( r ) ] b ( r ) + b ( r ) { cos [ ϕ ( r ) ] } .
I ( r ) b ( r ) { cos [ ϕ ( r ) ] } ;
I ( r ) = - b ( r ) sin [ ϕ ( r ) ] ϕ ( r ) .
I ( r ) ϕ ( r ) = - b ( r ) sin [ ϕ ( r ) ] | ϕ ( r ) | 2 ,
Q n { b ( r ) cos [ ϕ ( r ) ] } = ϕ ( r ) | ϕ ( r ) | 2 I ( r ) = - b ( r ) sin [ ϕ ( r ) ] .
Q n { b ( r ) cos [ ϕ ( r ) ] } = ϕ ( r ) | ϕ ( r ) | I ( r ) | ϕ ( r ) | = n ϕ I ( r ) | ϕ ( r ) | ,
tan [ θ 2 π ( x ,   y ) ] = ω y ( x ,   y ) ω x ( x ,   y ) ,
ω x = ϕ ( x ,   y ) x , ω y = ϕ ( x ,   y ) y .
cos [ θ 2 π ( x ,   y ) ] i + sin [ θ 2 π ( x ,   y ) ] j
= ω x i ( ω x 2 + ω y 2 ) 1 / 2 + ω y j ( ω x 2 + ω y 2 ) 1 / 2
= ϕ ( x ,   y ) | ϕ ( x ,   y ) | .
n ϕ = ϕ ( x ,   y ,   z ) | ϕ ( x ,   y ,   z ) | = cos   α 2 π ( x ,   y ,   z ) i + cos   β 2 π ( x ,   y ,   z ) j + cos   γ 2 π ( x ,   y ,   z ) k ,
cos 2 [ α 2 π ( x ,   y ,   z ) ] + cos 2 [ β 2 π ( x ,   y ,   z ) ]
+ cos 2 [ γ 2 π ( x ,   y ,   z ) ] = 1 ,
Q 2 { I ( x ,   y ) } = ϕ ( x ,   y ) | ϕ ( x ,   y ) | I ( x ,   y ) | ϕ ( x ,   y ) | ;
n ϕ ( x ,   y ) I ( x ,   y ) = | ϕ ( x ,   y ) | sin [ ϕ ( x ,   y ) ] ,
ϕ 0 ( x ,   y ) = tan - 1 n ϕ ( x ,   y ) I ( x ,   y ) I ( x ,   y ) = tan - 1 | ϕ ( x ,   y ) | sin [ ϕ ( x ,   y ) ] cos [ ϕ ( x ,   y ) ]
f ( x ,   y ) = cos [ ϕ ( x ,   y ) ] - I ( x ,   y ) = 0 ,
ϕ k + 1 ( x ,   y ) = ϕ k ( x ,   y ) - 2 f ( x ,   y ) f ϕ ( x ,   y ) 2 f ϕ 2 ( x ,   y ) + f ϕ ϕ ( x ,   y ) f ( x ,   y ) ,
k = 0 ,   1 ,   2 , ,
ϕ k + 1 ( x ,   y )
= ϕ k ( x ,   y ) + 2 f ( x ,   y ) sin [ ϕ k ( x ,   y ) ] 2   sin 2 [ ϕ k ( x ,   y ) ] - cos [ ϕ k ( x ,   y ) ] f ( x ,   y ) ,
k = 0 ,   1 ,   2 , .
ϕ ( x ,   y ) = cos - 1 [ I ( x ,   y ) ] .
H 2 { b   cos ( ϕ ) } = F - 1 - iu ( u 2 + v 2 ) 1 / 2 e 1 + - iv ( u 2 + v 2 ) 1 / 2 e 2 F { b   cos ( ϕ ) } ,
H 2 { I ( x ,   y ) } = F - 1 F [ I ( x ,   y ) ] ( u 2 + v 2 ) 1 / 2 .
H 2 { I ( x ,   y ) } F - 1 F I ( x ,   y ) [ ω x 2 ( x ,   y ) + ω y 2 ( x ,   y ) ] 1 / 2 ,
H 2 { I ( x ,   y ) } I ( x ,   y ) | ϕ ( x ,   y ) | ;
H n { I ( r ) } I ( r ) | ϕ ( r ) | ,
H n { I ( r ) } F - 1 - q | q |   I ˆ ( u ) ,
r = ( x 1 , , x n ) , q = ( u 1 , , u n ) .
tan [ θ π ( x ,   y ) ] = I ( x ,   y ) / y I ( x ,   y ) / x .
θ π = θ 2 π + k π ,
2 θ π = 2 θ 2 π + 2 k π = W ( 2 θ 2 π ) .
Q 2 { cos ( ϕ ) } = ω 0 n ϕ | ω 0 n ϕ |   H 2 { cos ( ϕ ) } = - sin ( ϕ ) ,
sin ( ϕ )   ϕ x + I x = 0 ,
sin ( ϕ )   ϕ y + I y = 0 ,
H [ ω x ( x ,   y ) ,   ω y ( x ,   y ) ] I ( x ,   y ) F - 1 { H ( u ,   v ) I ˆ ( u ,   v ) } ,
I ( x ,   y ) = b ( x ,   y ) cos [ ϕ ( x ,   y ) ]
ω x ( x ,   y ) = ϕ ( x ,   y ) x , ω y ( x ,   y ) = ϕ ( x ,   y ) y .
b ( x ,   y ) cos [ ϕ ( x ,   y ) ]
n , m ( x - n ,   y - m ) b ( n ,   m ) cos [ ϕ ( n ,   m )
+ ω x ( n ,   m ) ( x - n ) + ω y ( n ,   m ) ( y - m ) ] ,
F - 1 [ H ( u ,   v ) I ( u ,   v ) ]
n , m ( x - n ,   y - m ) b ( n ,   m ) H [ ω x ( n ,   m ) , ω y ( n ,   m ) ] × cos [ p ( x ,   y ,   n ,   m ) ] ,
p ( x ,   y ,   n ,   m ) = ϕ ( n ,   m ) + ω x ( n ,   m ) ( x - n ) + ω y ( n ,   m ) ( y - m ) .
F - 1 { H ( u ,   v ) F [ I ( x ,   y ) ] } H [ ω x ( x ,   y ) , ω y ( x ,   y ) ] I ( x ,   y ) .

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