Abstract

It has been demonstrated that the vectorial fringe-direction field is very important to demodulate fringe patterns without a dominant (or carrier) frequency. Unfortunately, the computation of this direction-filed is by far the most difficult task in the full interferogram phase-demodulation process. In this paper we present an algorithm to estimate this fringe-direction vector-field of a single n-dimensional fringe pattern. Despite that our theoretical results are valid at any dimension in the Euclidean space, we present some computer-simulated results in three dimensions because it is the most useful case in practical applications. As herein demonstrated, our method is based on linear matrix and vector analysis, this translates into a low computational cost.

©2010 Optical Society of America

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References

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  1. K. G. Larkin, D. J. Bone, and 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]
  2. M. Servín, J. A. Quiroga, and J. L. Marroquín, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
    [Crossref]
  3. X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
    [Crossref]
  4. J. A. Quiroga, M. Servín, and F. 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]
  5. Jesús Villa, Ismael De la Rosa, Gerardo Miramontes, and Juan Antonio Quiroga, “Phase recovery from a single fringe pattern using an orientational vector field regularized estimator,” J. Opt. Soc. Am. A 22, 2766–2773 (2005).
    [Crossref]
  6. J. A. Quiroga, Manuel Servín, J. Luis Marroquín, and Daniel Crespo “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” J. Opt. Soc. Am. A 22, 439–444 (2005).
    [Crossref]
  7. D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
    [Crossref]
  8. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [Crossref] [PubMed]
  9. Ng TW., “Photoelastic stress analysis using an object steep-loading method,” Exp. Mech. 37, 137–141 (1997).
    [Crossref]
  10. J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
    [Crossref]

2005 (2)

2004 (2)

D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
[Crossref]

J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
[Crossref]

2003 (1)

2002 (1)

2001 (1)

1999 (1)

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
[Crossref]

1997 (1)

Ng TW., “Photoelastic stress analysis using an object steep-loading method,” Exp. Mech. 37, 137–141 (1997).
[Crossref]

1996 (1)

Arnold, J. F.

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
[Crossref]

Baird, J. P.

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
[Crossref]

Bone, D. J.

Crespo, D.

D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
[Crossref]

Crespo, Daniel

Cuevas, F.

De la Rosa, Ismael

Gomez-Pedrero, J. A.

D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
[Crossref]

Gómez-Pedrero, J. A.

J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
[Crossref]

Larkin, K. G.

Marroquín, J. L.

Marroquín, J. Luis

Miramontes, Gerardo

Oldfield, M. A.

Quiroga, J. A.

J. A. Quiroga, Manuel Servín, J. Luis Marroquín, and Daniel Crespo “Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern,” J. Opt. Soc. Am. A 22, 439–444 (2005).
[Crossref]

J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
[Crossref]

D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
[Crossref]

M. Servín, J. A. Quiroga, and J. L. Marroquín, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[Crossref]

J. A. Quiroga, M. Servín, and F. 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]

Quiroga, Juan Antonio

Servín, M.

Servín, Manuel

Ströbel, B.

TW., Ng

Ng TW., “Photoelastic stress analysis using an object steep-loading method,” Exp. Mech. 37, 137–141 (1997).
[Crossref]

Villa, J.

J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
[Crossref]

Villa, Jesús

Zhou, X.

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
[Crossref]

Appl. Opt. (3)

X. Zhou, J. P. Baird, and J. F. Arnold, “Fringe orientation estimation by use of a gaussian gradient-filter and neighboring-direction averaging,” Appl. Opt. 38,, 795–804 (1999).
[Crossref]

D. Crespo, J. A. Quiroga, and J. A. Gomez-Pedrero, “Fast algorithm for estimation of the orientation term of a general quadrature transform with application to demodulation of an n-dimensional fringe pattern,” Appl. Opt. 43,, 6139–6146 (2004).
[Crossref]

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
[Crossref] [PubMed]

Exp. Mech. (1)

Ng TW., “Photoelastic stress analysis using an object steep-loading method,” Exp. Mech. 37, 137–141 (1997).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Las. Eng. (1)

J. Villa, J. A. Quiroga, and J. A. Gómez-Pedrero, “Measurement of retardation in digital photoelasticity by load stepping using a sinusoidal least-squares fitting,” Opt. Las. Eng. 41, 127–137 (2004).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. Relation between vectors n ϕ and n f in a 3D fringe pattern. (A) They point out to the same direction, (B) or they have opposite sign, but they are parallel at every site in the fringe pattern. Note that d k and n f form an orthonormal set.

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Fig. 2.
Fig. 2. (A) Sequence of phase-shifted interferograms where z-axis indicates the phase-shift. (B) Gray-level-codified direction-maps (Black represents −π rad and white π rad).

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Fig. 3.
Fig. 3. (A) Simulated photoelastic fringe patterns by load-steeping. (B) Three-dimensional phase-map obtained using the n-dimensional fringe direction-estimator and the quadrature transform [2].

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Equations (20)

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f ( x ) = a ( x ) + b ( x ) cos ϕ ( x ) , x = ( x 1 , x 2 , . . . , x n ) ∈ L ,
Q n { f ( x ) } = n ϕ · ∇ f ∥ ∇ ϕ ∥ ,
n ϕ = ∇ ϕ ∥ ∇ ϕ ∥ = Σ k = 1 n ( ∂ ϕ ∂ x k ) e k ∥ ∇ ϕ ∥ = Σ k = 1 n cos ( α k ) e k .
n f = ∇ f ∥ ∇ f ∥ = − sin ϕ Σ k = 1 n ( ∂ ϕ ∂ x k ) e k ∣ sin ϕ ∣ ∥ ∇ ϕ ∥ = − sgn ( sin ϕ ) n ϕ .
θ = tan − 1 ( ∂ f ∂ x 2 ∂ f ∂ x 1 ) , θ ∈ [ − π 2 , π 2 ) ,
n ϕ = cos ( α ) e 1 + sin ( α ) e 2 .
Q S { f } = sgn [ cos ( β k ) ] sgn [ cos ( φ k ) ] ,
β k = tan − 1 ( ∂ f ∂ x k + 1 ∂ f ∂ x k )
d 1 = ( d 11 , d 12 , . . . , d 1 n ) T ,
d 2 = ( d 21 , d 22 , . . . , d 2 n ) T ,
⋮
d n − 1 = ( d ( n − 1 ) 1 , d ( n − 1 ) 2 , . . . , d ( n − 1 ) n ) T ,
d k ( x ) · p ( x ) = 0 , x ∀ L .
U x ( p ) = Σ x ˜ ∈ Γ { Σ k = 1 n − 1 [ p ( x ) · d k ( x ˜ ) ] 2 + μ ∥ p ( x ) − p ( x ˜ ) ∥ 2 s ( x ˜ ) } .
A p = b ,
A = ( Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 1 ( x ˜ ) 2 + μ s ( x ˜ ) } Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 2 ( x ˜ ) d k 1 ( x ˜ ) } ⋯ Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k n ( x ˜ ) d k 1 ( x ˜ ) } Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 1 ( x ˜ ) d k 2 ( x ˜ ) } Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 2 ( x ˜ ) 2 + μ s ( x ˜ ) } ⋯ Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k n ( x ˜ ) d k 2 ( x ˜ ) } ⋮ ⋮ ⋱ ⋮ Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 1 ( x ˜ ) d k n ( x ˜ ) } Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k 2 ( x ˜ ) d k n ( x ˜ ) } ⋯ Σ x ˜ ∈ Γ { Σ k = 1 n − 1 d k n ( x ˜ ) 2 + μ s ( x ˜ ) } ) ,
b = ( μ Σ x ˜ ∈ Γ { p 1 ( x ˜ ) s ( x ˜ ) } μ Σ x ˜ ∈ Γ { p 2 ( x ˜ ) s ( x ˜ ) } ⋮ μ Σ x ˜ ∈ Γ { p n ( x ˜ ) s ( x ˜ ) } ) .
ϕ ( x , y ) = 80 2 − ( x − 50 ) 2 − ( y − 50 ) 2 .
θ = tan − 1 ( p 2 p 1 ) .
ε = Σ ∥ n ϕ − p ∥ 2 Σ ∥ n ϕ ∥ 2 ,

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