Abstract

The problem of surface reconstruction from slope or curvature measurements of rectangular areas has been examined by use of the orthogonality property of Legendre polynomials. A relation is given between the integrals involving slope or curvature data and the coefficients used in the expansion of the surface with such polynomials. It is shown that those coefficients are retrieved independently of each other. The efficiency and stability of the method have been tested with numerical examples.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Assa, J. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
    [Crossref]
  2. C. J. Lin, F. P. Chiang, “A Ligtenberg method for plate bending using laser speckle,” Mech. Res. Commun. 7, 241–246 (1980).
    [Crossref]
  3. F.-P. Chiang, R.-M. Juang, “Laser speckle interferometry for plate bending problems,” Appl. Opt. 15, 2199–2204 (1976).
    [Crossref] [PubMed]
  4. N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).
  5. D. L. Fried, “Least-square fitting of a wave-front distorsion estimation to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [Crossref]
  6. R. J. Noll, “Phase estimate from slope-type wave-front sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [Crossref]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase difference,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [Crossref]
  8. A. Menikoff, “Wave-front reconstruction with a square aperture,” J. Opt. Soc. Am. A 6, 1027–1030 (1989).
    [Crossref]
  9. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase difference using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [Crossref] [PubMed]
  10. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979).
    [Crossref]
  11. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
    [Crossref]
  12. D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
    [Crossref]
  13. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [Crossref]
  14. G. Subramanian, S. M. Nair, “Direct determination of curvatures of bent plates using a double-glass plate shearing interferometer,” Exp. Mech.376–380 (1985).
    [Crossref]
  15. J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
    [Crossref]
  16. C. C. Lin, L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974).
  17. E. Kreysig, Advanced Engineering Mathematics (Wiley, New York, 1972).
  18. C. L. Dym, I. H. Shames, Solid Mechanics, a Variational Approach (McGraw-Hill, New York, 1973).
  19. V. V. Voitsekhovich, “Phase-retrieval problem and orthogonal expansions: curvature sensing,” J. Opt. Soc. Am. A 12, 2194–2202 (1995).
    [Crossref]

1995 (1)

1990 (1)

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[Crossref]

1989 (2)

1986 (2)

1985 (1)

G. Subramanian, S. M. Nair, “Direct determination of curvatures of bent plates using a double-glass plate shearing interferometer,” Exp. Mech.376–380 (1985).
[Crossref]

1980 (2)

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
[Crossref]

C. J. Lin, F. P. Chiang, “A Ligtenberg method for plate bending using laser speckle,” Mech. Res. Commun. 7, 241–246 (1980).
[Crossref]

1979 (3)

1978 (1)

1977 (1)

1976 (1)

Assa, A.

A. Assa, J. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[Crossref]

Betser, A. A.

A. Assa, J. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[Crossref]

Carpio-Valadez, J. M.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[Crossref]

Chiang, F. P.

C. J. Lin, F. P. Chiang, “A Ligtenberg method for plate bending using laser speckle,” Mech. Res. Commun. 7, 241–246 (1980).
[Crossref]

Chiang, F.-P.

Cubalchini, R.

Dym, C. L.

C. L. Dym, I. H. Shames, Solid Mechanics, a Variational Approach (McGraw-Hill, New York, 1973).

Fournier, N.

N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).

Freischlad, K. R.

Fried, D. L.

Ghiglia, D. C.

Grédiac, M.

N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).

Hung, Y. Y.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[Crossref]

Hunt, B. R.

Juang, R.-M.

Koliopoulos, C. L.

Kreysig, E.

E. Kreysig, Advanced Engineering Mathematics (Wiley, New York, 1972).

Lin, C. C.

C. C. Lin, L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974).

Lin, C. J.

C. J. Lin, F. P. Chiang, “A Ligtenberg method for plate bending using laser speckle,” Mech. Res. Commun. 7, 241–246 (1980).
[Crossref]

Malacara, D.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[Crossref]

Menikoff, A.

Nair, S. M.

G. Subramanian, S. M. Nair, “Direct determination of curvatures of bent plates using a double-glass plate shearing interferometer,” Exp. Mech.376–380 (1985).
[Crossref]

Noll, R. J.

Paris, P.-A.

N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).

Politch, J.

A. Assa, J. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[Crossref]

Romero, L. A.

Sanchez-Mondragon, J. J.

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[Crossref]

Segel, L. A.

C. C. Lin, L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974).

Shames, I. H.

C. L. Dym, I. H. Shames, Solid Mechanics, a Variational Approach (McGraw-Hill, New York, 1973).

Southwell, W. H.

Subramanian, G.

G. Subramanian, S. M. Nair, “Direct determination of curvatures of bent plates using a double-glass plate shearing interferometer,” Exp. Mech.376–380 (1985).
[Crossref]

Surrel, Y.

N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).

Takezaki, J.

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[Crossref]

Voitsekhovich, V. V.

Appl. Opt. (1)

Exp. Mech. (2)

A. Assa, J. Politch, A. A. Betser, “Slope and curvature measurement by a double-frequency-grating shearing interferometer,” Exp. Mech. 19, 129–137 (1979).
[Crossref]

G. Subramanian, S. M. Nair, “Direct determination of curvatures of bent plates using a double-glass plate shearing interferometer,” Exp. Mech.376–380 (1985).
[Crossref]

J. Appl. Mech. (1)

J. Takezaki, Y. Y. Hung, “Direct measurement of flexural strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Mech. Res. Commun. (1)

C. J. Lin, F. P. Chiang, “A Ligtenberg method for plate bending using laser speckle,” Mech. Res. Commun. 7, 241–246 (1980).
[Crossref]

Opt. Eng. (1)

D. Malacara, J. M. Carpio-Valadez, J. J. Sanchez-Mondragon, “Wave-front fitting with discrete ortogonal polynomials in a unit radius circle,” Opt. Eng. 29, 672–675 (1990).
[Crossref]

Opt. Lett. (1)

Other (4)

N. Fournier, M. Grédiac, P.-A. Paris, Y. Surrel, “Phase-stepped deflectometry applied to shape measurement of bent plates,” Exp. Mech. (submitted for publication).

C. C. Lin, L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974).

E. Kreysig, Advanced Engineering Mathematics (Wiley, New York, 1972).

C. L. Dym, I. H. Shames, Solid Mechanics, a Variational Approach (McGraw-Hill, New York, 1973).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Reconstruction of the surface where δ = 0%: (a) The original surface w(x, y). (b) The surface [∂w(x, y)]/∂x considered as input in the reconstruction procedure. (c) The surface [∂w(x, y)]/∂y considered as input in the reconstruction procedure. (d) Difference between the actual and reconstructed surfaces.

Fig. 2
Fig. 2

Reconstruction of the surface with noisy data where δ = 20%: (a) The noisy surface [∂w(x, y)/∂x] considered as input in the reconstruction procedure. (b) The noisy surface [∂w(x, y)/∂y] considered as input in the reconstruction procedure. (c) Difference between the actual and reconstructed surfaces. (d) The reconstructed surface.

Tables (2)

Tables Icon

Table 1 Normalized Coefficients Aij Calculated with the Surfaces [∂w(x, y)]/∂x and [∂w(x, y)]/∂y (Upper and Lower Values for Each Row i/j, Respectively)a

Tables Icon

Table 2 Normalized PV and rms Errors versus the Magnitude δ of the Random Error

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

wx, y=i=0j=0AijPixPjy.
-11PixPjxdx=0,  if ij,
-11Pi2xdx=22i+1.
P0x=1,P1x=x,P2x=32x2-12,P3x=52x3-32x,P4x=358x4-154x2+38.
Pi1=1,i0,Pi-1=1,i0, i even,Pi-1=-1,i0, i odd.
i+1Pi+1x=2i+1xPix-iPi-1x.
dPi+1xdx-dPi-1xdx=2i+1Pix.
0xPitdt=12i+1Pi+1x-Pi-1x+a,
wx, y=i=0j=0AijPixPjy.
-1x1,  -1y1
Akl42k+12l+1=-11-11wx, yPkxPlydxdy.
Akl42k+12l+1=--11-11wx, yx0xPktdt×Plydxdy+D0xPktdt×wx, yPlycosn, xds,
Akl42l+1=--11-11wx, yxPk+1x-Pk-1xPlydxdy+DPk+1x-Pk-1xwx, yPlycosn, xds.
DPk+1x-Pk-1xwx, yPlycosn, xds=Pk+11-Pk-11-11w1, yPlydy-Pk+1-1-Pk-1-1-11w-1, yPlydy.
Pk+1-1=Pk-1-1,  k1.
Akl=2l+14-11-11wx, yx×Pk-1x-Pk+1xPlydxdy.
Akl=2k+14-11-11wx, yy×Pl-1y-Pl+1yPkxdxdy.
Ix,k,l=-11-11wx, yxPkxPlydxdy,  Iy,k,l=-11-11wx, yyPkxPlydxdy,  k, l0,
Akl=2l+14Ix,k-1,l-Ix,k+1,l  k1, l0  =2k+14Iy,k,l-1-Iy,k,l+1,  k0, k1.
k0,  l0,
Ixx,k,l=-11-112wx, yx2PkxPlydxdy,Iyy,k,l=-11-11w2x, yy2PkxPlydxdy,Ixy,k,l=-11-11w2x, yxyPkxPlydxdy,Iyx,k,l=-11-11w2x, yyxPkxPlydxdy.
Ix,k,l=12k+1Ixx,k-1,l-Ixx,k+1,l,Ix,k,l=12l+1Ixy,k,l-1-Ixy,k,l+1,Iy,k,l=12l+1Iyy,k,l-1-Iyy,k,l+1,Iy,k,l=12k+1Iyx,k-1,l-Iyx,k+1,l.
Akl=2l+1412k-1Ixx,k-2,l+12k+3Ixx,k+2,l-22k+12k-12k+3Ixx,k,l,  k2, l0  =2k+1412l-1Iyy,k,l-2+12l+3Iyy,k,l+2-22l+12l-12l+3Iyy,k,l,  k0, l2  =14Ixy,k-1,l-1+Ixy,k+1,l+1-Ixy,k-1,l+1-Ixy,k+1,l-1,  k1, l1  =14Iyx,k-1,l-1+Iyx,k+1,l+1-Iyx,k-1,l+1-Iyx,k+1,l-1,  k1, l1.
wx, y=10-2 sinπx+1sin3πy+12, -1x1, -1y1,
wx, yx=π10-2 cosπx+1sin3πy+12,
wx, yy=3π210-2 sinπx+1cos3πy+12.

Metrics