Abstract

The problem of unwrapping a noisy principal-value phase field or, equivalently, reconstructing an unwrapped phase field from noisy and possibly incomplete phase differences may be considered ill-posed in the sense of Hadamard. We apply the Thikonov regularization theory to find solutions that correspond to minimizers of positive-definite quadratic cost functionals. These methods may be considered generalizations of the classical least-squares solution to the unwrapping problem; the introduction of the regularization term permits the reduction of noise (even if this noise does not generate integration-path inconsistencies) and the interpolation of the solution over regions with missing data in a stable and controlled way, with a minimum increase of computational complexity. Algorithms for finding direct solutions with transform methods and implementations of iterative procedures are discussed as well. Experimental results on synthetic test images are presented to illustrate the performance of these methods.

© 1995 Optical Society of America

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  1. D. L. Fried, “Least-squares fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc Am 67, 370–375 (1977).
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  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  3. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  4. J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13 (1902).
  5. M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
    [CrossRef]
  6. J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. 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]
  9. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.
  10. A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).
  11. D. C. Ghiglia, G. A. Masting, L. A. Romero, “Cellularautomata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  12. W. E. L. Grimson, “A computational theory of visual surface interpolation,” Phil. Trans. R. Soc. London Ser. B 298, 395–427 (1982).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  14. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
    [CrossRef]
  15. G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press,Baltimore, Md., 1990).
  16. D. Hillis, The Connection Machine (MIT Press, Cambridge, Mass., 1985).

1994

1993

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

1988

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

1987

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
[CrossRef]

D. C. Ghiglia, G. A. Masting, L. A. Romero, “Cellularautomata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
[CrossRef]

1982

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Phil. Trans. R. Soc. London Ser. B 298, 395–427 (1982).
[CrossRef]

1979

1978

1977

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
[CrossRef]

D. L. Fried, “Least-squares fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc Am 67, 370–375 (1977).
[CrossRef]

1963

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

1902

J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13 (1902).

Bertero, M.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Fried, D. L.

D. L. Fried, “Least-squares fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc Am 67, 370–375 (1977).
[CrossRef]

Ghiglia, D. C.

Gollub, G. H.

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

Grimson, W. E. L.

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Phil. Trans. R. Soc. London Ser. B 298, 395–427 (1982).
[CrossRef]

Hadamard, J.

J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13 (1902).

Hillis, D.

D. Hillis, The Connection Machine (MIT Press, Cambridge, Mass., 1985).

Hudgin, R. H.

Hunt, B. R.

Marroquin, J. L.

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
[CrossRef]

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.

Masting, G. A.

Mitter, S.

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
[CrossRef]

Noll, R. J.

Poggio, T.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Rodriguez-Vera, R.

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.

Romero, L. A.

Servin, M.

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.

Tapia, M.

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Thikonov, A. N.

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Torre, V.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Van Loan, C. F.

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

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

CVGIP Graph. Models Image Process.

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” CVGIP Graph. Models Image Process. 55, 408–417 (1993).
[CrossRef]

J. Am. Stat. Assoc.

J. L. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82 (397), 76–89 (1987).
[CrossRef]

J. Opt. Soc Am

D. L. Fried, “Least-squares fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc Am 67, 370–375 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phil. Trans. R. Soc. London Ser. B

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Phil. Trans. R. Soc. London Ser. B 298, 395–427 (1982).
[CrossRef]

Princeton University Bulletin

J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13 (1902).

Proc. IEEE

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–889 (1988).
[CrossRef]

Sov. Math. Dokl.

A. N. Thikonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

Other

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random-field models,” submitted to J. Opt. Soc. Am. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

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

D. Hillis, The Connection Machine (MIT Press, Cambridge, Mass., 1985).

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Figures (4)

Fig. 1
Fig. 1

Average roughness (solid curve) and average fit error (with respect to the noisy unwrapped field; dashed curve) as functions of λ for the example shown in Fig. 2.

Fig. 2
Fig. 2

(a) Original (unwrapped) phase. Following panels: reconstructed phase (with automatic scale correction) for (b) λ = 0, (c) λ = 0.8, (d) λ = 5, and (e) λ = 10; (f) shows the solution for λ = 10 without scale correction.

Fig. 3
Fig. 3

Set S (sites with valid information) is represented by the white region; (b) solution obtained with the PCG algorithm for λ = 0; (c) same as (b) with λ = 1; (d) solution obtained with the CG algorithm and λ = 1.

Fig. 4
Fig. 4

(a) Original phase; (b) set D (where the solution is to be computed) is represented by the white region: (c) wrapped phase; (d) solution obtained with the CG algorithm and λ = 1.

Equations (38)

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g ( x , y ) = ϕ ( x , y ) + n ( x , y ) + 2 π k ( x , y )
g x ( x , y ) = ϕ ( x , y ) x + n x ( x , y ) + 2 π k x ( x , y ) ,
g y ( x , y ) = ϕ ( x , y ) y + n y ( x , y ) + 2 π k y ( x , y ) ,
ϕ = g + 2 π f ,
u = h ,
lim λ 0 R λ h = u + ,
* u = * h ,
U λ ( u ) = u h 2 + λ P u 2 ,
R λ = ( * + λ P * P ) 1 * .
h = u + n ,
P h | u ( u ) = P n ( u h ) = 1 K exp ( u h 2 / 2 σ 2 ) ,
P u ( u ) = 1 Z exp ( P ̂ u 2 ) ,
P u | h ( u ) = 1 Z exp ( 1 2 σ 2 u h 2 P ̂ u 2 )
g x ( x , y ) = g ( x , y ) g ( x 1 , y ) + 2 π k x , g y ( x , y ) = g ( x , y ) g ( x , y 1 ) + 2 π k y ,
ϕ h 2 = ( x , y ) S x D x [ ϕ ( x , y ) ϕ ( x 1 , y ) g x ( x , y ) ] 2 + ( x , y ) S y D y [ ϕ ( x , y ) ϕ ( x , y 1 ) g y ( x , y ) ] 2
P 1 ϕ 2 = ( x , y ) D x [ ϕ ( x , y ) ϕ ( x 1 , y ) ] 2 + ( x , y ) D y [ ϕ ( x , y ) ϕ ( x , y 1 ) ] 2 .
P 2 ϕ 2 = α ( x , y ) D x x [ ϕ ( x 1 , y ) 2 ϕ ( x , y ) + ϕ ( x + 1 , y ) ] 2 + b ( x , y ) D y y [ ϕ ( x , y 1 ) 2 ϕ ( x , y ) + ϕ ( x , y + 1 ) ] 2 + c ( x , y ) D x y [ ϕ ( x , y ) ϕ ( x , y + 1 ) ϕ ( x + 1 , y ) + ϕ ( x + 1 , y + 1 ) ] 2
( 1 + λ ) T ϕ = ρ ,
ρ ( x , y ) = g x ( x , y ) g x ( x 1 , y ) + g y ( x , y ) g y ( x , y 1 ) .
P 2 T P 2 ϕ 4 ϕ x 4 + 4 ϕ y 4 + 2 4 ϕ x 2 y 2 ,
U ( ϕ ) = ϕ h 2 + λ P 2 ϕ 2 ,
( Q ϕ ) ( x , y ) = k = 0 1 { A x ( x , y , k ) [ ϕ ( x + k , y ) ϕ ( x + k 1 , y ) ] + A y ( x , y , k ) [ ϕ ( x , y + k ) ϕ ( x , y + k 1 ) ] } + λ ( k = 0 2 { B x ( x , y , k ) [ ϕ ( x k , y ) 2 ϕ ( x k + 1 , y ) + ϕ ( x k + 2 , y ) ] + B y ( x , y + k ) [ ϕ ( x , y k ) 2 ϕ ( x , y k + 1 ) + ϕ ( x , y k + 2 ) ] } + n = 0 1 k = 0 1 C ( x , y , k , n ) [ ϕ ( x n , y k ) ϕ ( x n , y k + 1 ) ϕ ( x n + 1 , y k ) + ϕ ( x n + 1 , y k + 1 ) ] , )
ρ ( x , y ) = k = 0 1 { A x ( x , y , k ) [ g x ( x + k , y ) g x ( x + k 1 , y ) ] + A y ( x , y , k ) [ g y ( x , y + k ) g y ( x , y + k 1 ) ] } ,
ϕ x = 0 for x { 0 , M 1 } ,
ϕ y = 0 for y { 0 , N 1 } ,
f ̂ ( u , υ ) = x = 0 M 1 y = 0 N 1 4 f ( x , y ) cos [ π u 2 M ( 2 x + 1 ) ] × cos [ π υ 2 N ( 2 y + 1 ) ]
f ( x , y ) = 1 M N u = 0 M 1 υ = 0 N 1 w ( u ) w ( υ ) f ̂ ( u , υ ) cos [ π u 2 M ( 2 x + 1 ) ] × cos [ π υ 2 N ( 2 y + 1 ) ] ,
ϕ ̂ ( u , υ ) = ρ ̂ ( u , υ ) F ( u , υ ) ,
F ( u , υ ) = ( 2 + 16 λ ) [ cos ( π u M ) + cos ( π υ N ) ] 2 λ [ cos ( 2 π u M ) + cos ( 2 π υ N ) ] 8 λ cos ( π u M ) cos ( π υ N ) 20 λ 4 ,
g x ( M , y ) = g x ( 0 , y ) = 0 for 0 y N 1 , g y ( x , N ) = g y ( x , 0 ) = 0 0 x M 1 .
g x ( x , y ) = 0 f o r ( x , y ) S D x , g y ( x , y ) = 0 f o r ( x , y ) S D y
F ( λ ) = ϕ λ h 2 = σ 2 ,
R ( λ ) = P 2 ϕ λ 2 = r
( x , y ) S [ α ϕ λ ( x , y ) = b ϕ 0 ( x , y ) ] 2 ,
ϕ ( x , y ) = α ϕ λ ( x , y ) ,
α = | S | ( x , y ) S ϕ λ ( x , y ) ϕ 0 ( x , y ) [ ( x , y ) S ϕ λ ( x , y ) ] [ ( x , y ) S ϕ 0 ( x , y ) ] | S | ( x , y ) S ϕ λ 2 ( x , y ) [ ( x , y ) S ϕ λ ( x , y ) ] 2 ,
ϕ ( x , y ) = 4 r 4 ( x , y ) 3 r 2 ( x , y ) ,
r ( x , y ) = α [ ( x x 0 ) 2 + ( y y 0 ) 2 ] 1 / 2 ,

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