Abstract

Shearography is an interferometric method that overcomes several limitations of holography by eliminating the reference beam. It greatly simplifies the optical setup and has much higher tolerance to environmental disturbances. Consequently, the technique has received considerable industrial acceptance, particularly for nondestructive testing. Shearography, however, is generally not applicable to the measurement of an obstructed area, as the area to be measured must be accessible to both illumination and imaging. We present an algorithm based on the principle of tomography that permits the reconstruction of the unavailable phase distribution in an obstructed area from the measured boundary phase distribution. In the process, a set of imaginary rays is projected from many different directions across the area. For each ray, integration of the phase directional derivative along the ray is equal to the phase difference between the boundary points intercepted by the ray. Therefore, a set of linear equations can be established by considering the multiple rays. Each equation expresses the unknown phase derivatives in the obstructed area in terms of the measured boundary phase. Solution of the set of simultaneous equations yields the unknown phase distribution in the blind area. While its applications to shearography are demonstrated, the technique is potentially applicable to all full-field optical measurement techniques such as holography, speckle interferometry, classical interferometry, thermography, moiré, photoelasticity, and speckle correlation techniques.

© 2008 Optical Society of America

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References

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  1. Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61-87 (2005).
    [CrossRef]
  2. D. T. Sandwell, “Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data,” Geophys. Res. Lett. 14, 139-142 (1987).
    [CrossRef]
  3. R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
    [CrossRef]
  4. Y. Zou and X. C. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717-2731 (2004).
    [CrossRef] [PubMed]
  5. W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
    [CrossRef] [PubMed]
  6. B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
    [CrossRef] [PubMed]
  7. H. Anton and R. C. Busby, “QR-decomposition; householder transformations,” in Contemporary Linear Algebra (Wiley, 2003).
  8. Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
    [CrossRef]

2005 (1)

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61-87 (2005).
[CrossRef]

2004 (1)

Y. Zou and X. C. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

2003 (3)

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
[CrossRef] [PubMed]

Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
[CrossRef]

1991 (1)

R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
[CrossRef]

1987 (1)

D. T. Sandwell, “Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data,” Geophys. Res. Lett. 14, 139-142 (1987).
[CrossRef]

Anton, H.

H. Anton and R. C. Busby, “QR-decomposition; householder transformations,” in Contemporary Linear Algebra (Wiley, 2003).

Bull, M. J.

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

Busby, R. C.

H. Anton and R. C. Busby, “QR-decomposition; householder transformations,” in Contemporary Linear Algebra (Wiley, 2003).

Chooi, W. K.

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

Fessler, J. A.

B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
[CrossRef] [PubMed]

Gu, R. J.

R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
[CrossRef]

Ho, H. P.

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61-87 (2005).
[CrossRef]

Hovanesian, J. D.

R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61-87 (2005).
[CrossRef]

Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
[CrossRef]

R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
[CrossRef]

Matthews, S.

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

Morcos, S. K.

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

Noll, D. C.

B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
[CrossRef] [PubMed]

Pan, X. C.

Y. Zou and X. C. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

Sandwell, D. T.

D. T. Sandwell, “Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data,” Geophys. Res. Lett. 14, 139-142 (1987).
[CrossRef]

Shang, H. M.

Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
[CrossRef]

Sutton, B. P.

B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
[CrossRef] [PubMed]

Yang, L.

Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
[CrossRef]

Zou, Y.

Y. Zou and X. C. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

Br. J. Radiol. (1)

W. K. Chooi, S. Matthews, M. J. Bull, and S. K. Morcos, “Multislice helical CT: the value of multiplanar image reconstruction in assessment of the bronchi and small airways disease,” Br. J. Radiol. 76, 536-540 (2003).
[CrossRef] [PubMed]

Geophys. Res. Lett. (1)

D. T. Sandwell, “Biharmonic spline interpolation of GEOS-3 and SEASAT altimeter data,” Geophys. Res. Lett. 14, 139-142 (1987).
[CrossRef]

IEEE Trans. Med. Imaging (1)

B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Trans. Med. Imaging 22, 178-188 (2003).
[CrossRef] [PubMed]

J. Appl. Mech. (1)

R. J. Gu, J. D. Hovanesian, and Y. Y. Hung, “Calculations of strains and internal displacement fields using computerized tomography,” J. Appl. Mech. 58, 24-27 (1991).
[CrossRef]

Mater. Sci. Eng. R (1)

Y. Y. Hung and H. P. Ho, “Shearography: an optical measurement technique and applications,” Mater. Sci. Eng. R 49, 61-87 (2005).
[CrossRef]

Opt. Eng. (1)

Y. Y. Hung, H. M. Shang, and L. Yang, “Unified approach for holography and shearography in surface deformation measurement and nondestructive testing,” Opt. Eng. 42, 1197-1207 (2003).
[CrossRef]

Phys. Med. Biol. (1)

Y. Zou and X. C. Pan, “Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT,” Phys. Med. Biol. 49, 2717-2731 (2004).
[CrossRef] [PubMed]

Other (1)

H. Anton and R. C. Busby, “QR-decomposition; householder transformations,” in Contemporary Linear Algebra (Wiley, 2003).

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Figures (11)

Fig. 1
Fig. 1

Typical optical layout for digital shearography.

Fig. 2
Fig. 2

Schematic diagram of the tomographic technique for phase reconstruction.

Fig. 3
Fig. 3

Discretization of Eq. (4) on a rectangular domain.

Fig. 4
Fig. 4

(a) Simulated quadratic surface with data lost. (b) Reconstruction result.

Fig. 5
Fig. 5

(a) Simulated wrinkled saddle surface with some data lost. (b) Reconstruction result.

Fig. 6
Fig. 6

(a) Simulated surface composed of scaled and translated Gaussian distributions. (b) Cracked surface for reconstruction. (c) Reconstructed results using bicubic interpolation. (d) Reconstructed results using nearest interpolation. (e) Reconstructed results using biharmonic interpolation. (f) Reconstructed results using the CTR method.

Fig. 7
Fig. 7

(a) Experimental phase map of a human back. (b) Cracked surface for reconstruction. (c) Reconstructed results using bicubic interpolation. (d)  Reconstructed results using nearest interpolation. (e) Reconstructed results using biharmonic interpolation. (f) Reconstructed results using the CTR method. (g) 3D plot of reconstruction result of (e). (h) 3D plot of reconstruction result of (f).

Fig. 8
Fig. 8

(a) Shearographic phase map of a central loaded plate with lost data. (b)  Reconstruction result by the tomographic method. (c) 3D plot of the reconstructed phase map.

Fig. 9
Fig. 9

(a) Shearographic wrapped phase map with cracked data. (b) Unwrapped phase map with cracked data. (c) Reconstruction result by the proposed tomographic method.

Fig. 10
Fig. 10

(a) Unwrapped phase map with some data lost due to lighting saturation. (b) Reconstruction result.

Fig. 11
Fig. 11

Two surfaces with exactly the same boundary data. (a) Reconstructed from the CTR method. (b) Original simulated data.

Equations (10)

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I = a + b × cos ( ϕ + Δ ) ,
Δ = 2 π λ [ sin α u x + ( 1 + cos α ) w x ] δ x ,
P 1 P 2 f L d L = f ( p 2 ) f ( p 1 ) ,
P 1 P 2 f L d L = P 1 P 2 [ f x ( x , y ) cos θ + f y ( x , y ) sin θ ] d L = f ( p 2 ) f ( p 1 ) ,
i = 1 K [ f x ( x i , y i ) cos θ + f y ( x i , y i ) sin θ ] L ( S i ) = f ( p 2 ) f ( p 1 ) ,
A p q x q = b p ,
A p q = Q p q R q q ,
Q q p T ( Q p q R q q ) x q = Q q p T b p .
R q q x q = Q q p T b p .
x q = R q q 1 Q q p T b p ,

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