Abstract

We present a modification of the multigrid method for the minimization of quadratic cost functions that result from the formulation of several problems in the digital processing of fringe-pattern images by using the Bayesian estimation-theory framework. With this modification the method can be applied to irregular domains and results in substantial savings in computational cost. We compare it with other state-of-the-art numerical techniques and present examples of its application to image smoothing without edge effects, robust quadrature filtering for the phase demodulation of spatial-carrier fringe patterns, and robust phase unwrapping.

© 1998 Optical Society of America

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References

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  1. J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).
  2. J. L. Marroquin, M. Servin, J. E. Figueroa, “Robust quadrature filters,” J. Opt. Soc. Am. A 14,779–791 (1997).
  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).
  4. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intel. 6,721–741 (1984).
  5. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
  6. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72,156–160 (1982).
  7. Th. Kreis, “Digital holographic interference phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3,847–855 (1986).
  8. D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).
  9. M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “A fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36,8381–8390 (1997).
  10. M. Rivera, R. Rodriguez-Vera, J. L. Marroquin, “Robust procedure for fringe analysis,” Appl. Opt. 36,8391–8396 (1997).
  11. J. M. Huntly, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28,3268–3270 (1989).
  12. K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36,4830–4838 (1997).
  13. 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).
  14. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12,2393–2400 (1995).
  15. M. Rivera, “Integración de gradientes inconsistentes,” Ph.D. dissertation (Centro de Investigaciones en Optica, Universidad de Guanajuato, Mexico, 1997).
  16. G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).
  17. W. L. Briggs, Multigrid Tutorial (Lancaster Press, Lancaster, Pa., 1987).
  18. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15,1536–1544 (1998).

1998 (1)

1997 (5)

1995 (1)

1994 (1)

1989 (1)

1987 (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).

1986 (1)

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intel. 6,721–741 (1984).

1982 (1)

Briggs, W. L.

W. L. Briggs, Multigrid Tutorial (Lancaster Press, Lancaster, Pa., 1987).

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

Figueroa, J. E.

Gauthier, P.

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intel. 6,721–741 (1984).

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intel. 6,721–741 (1984).

Ghiglia, D. C.

Gollub, G. H.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Huntly, J. M.

Ina, H.

Kobayashi, S.

Kreis, Th.

Marroquin, J.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).

Marroquin, J. L.

Mitter, S.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).

Poggio, T.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).

Rivera, M.

Rodriguez-Vera, R.

Romero, L. A.

Servin, M.

Stetson, K. A.

Takeda, M.

Van Loan, C. F.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Wahid, J.

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Mach. Intel. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intel. 6,721–741 (1984).

J. Am. Statis. Assoc. (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Statis. Assoc. 82,76–89 (1987).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Other (5)

M. Rivera, “Integración de gradientes inconsistentes,” Ph.D. dissertation (Centro de Investigaciones en Optica, Universidad de Guanajuato, Mexico, 1997).

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

W. L. Briggs, Multigrid Tutorial (Lancaster Press, Lancaster, Pa., 1987).

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).

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Figures (9)

Fig. 1
Fig. 1

Grids used in the multigrid scheme. Note that some pixels disappear in coarser grids but the modeled surface is the same.

Fig. 2
Fig. 2

Propagation of the link variables through multiple grids. Note how discontinuities in the finer grids are preserved in the coarser ones.

Fig. 3
Fig. 3

Example used for the numerical tests: (a) noisy synthetic image, (b) mask (indicator function of the set Q), and (c) image recovered with the parameter λ = 1000.

Fig. 4
Fig. 4

Comparison of the decreased rates of the mean-squared error (MSE) for different schemes. The multigrid (MG) scheme had eight grids. PCG, preconditioned conjugate gradient; GS, Gauss–Seidel; CG, conjugate gradient.

Fig. 5
Fig. 5

Multigrid phase demodulation: (a) electronic speckle-pattern interferometry of an aluminum plate under thermal strain and (b) wrapped phase recovered with the RQF’s multigrid algorithm.

Fig. 6
Fig. 6

Multigrid phase unwrapping: (a) synthetic noisy wrapped phase, (b) in black, sites where the gradient g x , g y is inconsistent, (c) unwrapped phase, and (d) unwrapped phase of (c) rewrapped to show the preservation of the dynamic range.

Fig. 7
Fig. 7

Cardboard test object.

Fig. 8
Fig. 8

(a) Fringes projected across the test object shown in Fig. 7. (b) Image of (a) filtered with a multigrid membrane smoother. (c) Mask (indicator function of the set Q). (d) Wrapped phase recovered with the multigrid RQF algorithm. (e) Unwrapped phase obtained with the multigrid phase-unwrapping algorithm.

Fig. 9
Fig. 9

Same as Fig. 8(e), but the wrapped phase was obtained by use of the method of Takeda et al.6

Equations (29)

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U ( f ) = r L q ( r ) [ f ( r ) g ( r ) ] 2 + λ ( r , s ) [ f ( r ) f ( s ) ] 2 q ( r ) q ( s ) .
U ( f ) = r L q ( r ) | f ( r ) 2 g ( r ) | 2 + λ r , s | f ( r ) f ( s ) exp [ i ω 0 · ( r s ) ] | 2 q ( r ) q ( s ) ,
ϕ ( r ) = arctan   [ Im f ( r ) Re f ( r ) ] ω 0 · r .
g x ( r ) g x ( x , y ) = ϕ ( x , y ) ϕ ( x 1 , y ) + 2 π k ( x , y ) ,
U ( f x , f y ) = r L q ( r ) { [ f x ( r ) g x ( r ) p x ( r ) ] 2 + [ f y ( r ) g y ( r ) p y ( r ) ] 2 } + λ r L [ Δ y f x ( r ) Δ x f y ( r ) ] 2 m ( r ) ,
p x ( r ) = { 1 if | Δ y F [ Δ y g x ( r ) Δ x g y ( r ) ] | 2 π 0 otherwise , p y ( r ) = { 1 if | Δ x F [ Δ y g x ( r ) Δ x g y ( r ) ] | 2 π 0 otherwise ,
f ( t + 1 ) ( r ) = q ( r ) [ g ( r ) + λ s N ( r ) f ( t ) ( s ) q ( s ) 1 + λ s N ( r ) q ( s ) ] ,
U [ f ( r ) ] f ( r ) = 0
g ( r ) = q ( r ) { f ( r ) + λ s N ( r ) [ f ( r ) f ( s ) ] q ( s ) } .
h 1 ( x , y ) = q 1 ( x , y ) q 1 ( x 1 , y ) , υ 1 ( x , y ) = q 1 ( x , y ) q 1 ( x , y 1 ) ,
q n + 1 ( x , y ) = q n ( 2 x , 2 y ) , h n + 1 ( x , y ) = h n ( 2 x , 2 y ) h n ( 2 x 1 , 2 y ) , υ n + 1 ( x , y ) = υ n ( 2 x , 2 y ) υ n ( 2 x , 2 y 1 ) .
ρ n + 1 ( x , y ) = q n + 1 ( x , y ) 4 r n ( 2 x , 2 y ) + F n ( r n , 2 x , 2 y ) 4 + G n ( 2 x , 2 y ) ,
F n ( r n , x , y ) = h n ( x , y ) r n ( x 1 , y ) + h n ( x + 1 , y ) × r n ( x + 1 , y ) + υ n ( x , y ) r n ( x , y 1 ) + υ n ( x , y + 1 ) r n ( x , y + 1 ) ,
G n ( x , y ) = h n ( x , y ) + h n ( x + 1 , y ) + υ n ( x , y ) + υ n ( x , y + 1 ) .
e ˆ n ( 2 x + 1 , 2 y ) = e n + 1 ( x , y ) h n ( 2 x + 1 , 2 y ) + e n + 1 ( x + 1 , y ) h n ( 2 x + 2 , 2 y ) h n ( 2 x + 1 , 2 y ) + h n ( 2 x + 2 , 2 y ) ,
e ˆ n ( 2 x , 2 y + 1 ) = e n + 1 ( x , y ) υ n ( 2 x , 2 y + 1 ) + e n + 1 ( x , y + 1 ) υ n ( 2 x , 2 y + 2 ) υ n ( 2 x , 2 y + 1 ) + υ n ( 2 x , 2 y + 2 ) ,
e ˆ n ( 2 x + 1 , 2 y + 1 ) = F n ( e n , 2 x + 1 , 2 y + 1 ) G n ( 2 x + 1 , 2 y + 1 ) ,
e n ( t + 1 ) ( x , y ) = q n ( x , y ) { ρ n ( x , y ) + λ F n [ e n ( t ) , x , y ] 1 + λ G n ( x , y ) } ,
r ( x , y ) = { y 64 if y > 64 and x < 64 1 otherwise .
U n ( f n ) = r L q n ( r ) | f n ( r ) 2 g n ( r ) | 2 + λ r , s | f n ( r ) f n ( s ) exp [ i 2 n 1 ω 0 · ( r s ) ] | 2 q n ( r ) q n ( s ) .
f n ( t + 1 ) ( r ) = q n ( r ) { g n ( r ) + λ s N ( r ) f n ( t ) ( s ) exp [ i 2 n 1 ω 0 · ( r s ) ] q n ( s ) 1 + λ s N ( r ) q n ( s ) } ,
g n ( r ) = q n ( r ) 2 f n ( r ) + λ s N ( r ) | f n ( r ) f n ( s ) × exp [ i 2 n 1 ω 0 · ( r s ) ] | q n ( s ) .
  2.3 : Set n = n + 1 , ω n + 1 = 2 * ω n . 3 : Set n = M 1 , ω n = ( ω n + 1 ) / 2 .   4.3 : Set n = n 1 , ω n = ( ω n + 1 ) / 2 .
U ( f x , n , f y , n ) = ( r ) L q n ( r ) { [ f x , n ( r ) g x , n ( r ) ] 2 + [ f y ( r ) g y , n ( r ) ] 2 } + λ r L [ Δ y f x , n ( r ) Δ x f y , n ( r ) ] 2 m n ( r ) ,
g x , 1 ( r ) = g x ( r ) p x ( r ) ,   g x , 1 ( r ) = g x ( r ) p x ( r ) ,
f n ( t + 1 ) ( r ) [ f x , n ( t + 1 ) ( r ) f y , n ( t + 1 ) ( r ) ] = [ q n ( r ) { g x , n ( r ) + λ s N x ( r ) R x , n ( s ) m n ( s ) 1 + λ s N x ( r ) m n ( s ) } q n ( r ) { g y , n ( r ) + λ s N y ( r ) R y , n ( s ) m n ( s ) 1 + λ s N x ( r ) m n ( s ) } ] ,
R x , n ( r ) f x , n ( x , y 1 ) f y , n ( x , y ) + f y , n ( x 1 , y ) , R x , n ( r + 1 ) f x , n ( x , y + 1 ) + f y , n ( x , y + 1 ) + f y , n ( x 1 , y + 1 ) , R y , n ( r ) f y , n ( x , y 1 ) f x , n ( x , y ) + f x , n ( x , y 1 ) , R y , n ( r + 1 ) f y , n ( x + 1 , y ) + f x , n ( x + 1 , y ) + f x , n ( x + 1 , y 1 ) .
g n ( r ) = [ g ( r ) g ( r ) ] = q n ( r ) [ [ f x , n ( r ) f y , n ( r ) ] + λ [ s N x ( r ) { f x , n ( r ) R x , n ( s ) } m n ( s ) s N y ( r ) { f y , n ( r ) R y , n ( s ) } m n ( s ) ] ] .
g ( x , y ) = 60 exp { σ 1 2 [ ( x / 128 1 ) 2 + ( y / 128 1 ) 2 ] } + 60 exp { σ 2 2 [ ( x / 128 + 1 ) 2 + ( y / 128 + 1 ) 2 ] } 6 sin [ 3 π ( x / 128 1 ) ] + 1.65 η ( x , y ) ,

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