Abstract

A novel three-dimensional (3D) reconstruction method based on fringe reflection technique for shape measurement of large specular surfaces is presented in this paper, which effectively integrates path integration technique with zonal wavefront reconstruction algorithm. The height information of specular surface obtained from cross-path integration can then be used as the initial value in a zonal wavefront reconstruction algorithm. This method not only has the advantages of global integration, but also enables user-friendly, high-speed operation. A specific iterative algorithm is adopted to improve the antinoise capability of the measuring system, which accelerates the rate of convergence significantly and even improves the accuracy of the reconstructed 3D surface. Moreover, the proper use of boundary contour extraction of the acquired images reduces the computational load of 3D reconstruction dramatically and hence achieves high reconstruction accuracy and enhances the surface integrity at the boundary. An ultraprecision, diamond-turned planar mirror with diameter of 150 mm has been employed to implement the system calibration. The reconstruction results of simulated and actual hyperbolic surfaces and the gauge blocks identify the validity of this new method. It is demonstrated that the measurement error is about 50 μm with reconstruction points of 150×560 pixels of gauge blocks.

© 2012 Optical Society of America

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References

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  1. J. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21, 331–340 (2010).
    [CrossRef]
  2. I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).
  3. S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
    [CrossRef]
  4. M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
    [CrossRef]
  5. G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
    [CrossRef]
  6. Y.-L. Xiao, X. Su, and W. Chen, “Flexible geometrical calibration for fringe-reflection 3D measurement,” Opt. Lett. 37, 620–622 (2012).
    [CrossRef]
  7. K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).
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    [CrossRef]
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2012 (1)

2010 (3)

J. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21, 331–340 (2010).
[CrossRef]

S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

2009 (3)

F. W. Y. Chan, “A novel optical method without phase unwrapping for subsurface flaw detection,” Opt. Lasers Eng. 47, 186–193 (2009).
[CrossRef]

M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
[CrossRef]

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

2008 (2)

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

S. Ettl, J. Kaminski, M. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
[CrossRef]

2007 (2)

O. A. Skydan, M. J. Lalor, and D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Meas. Sci. Technol. 18, 106–114 (2007).
[CrossRef]

H.-Y. Wang, D.-L. Pan, and D.-S. Xia, “A fast algorithm for two-dimensional Otsu adaptive threshold algorithm,” Acta Automat. Sinica 33, 968–971 (2007).

2004 (1)

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

2003 (1)

Y. Y. Hung and H. M. Shang, “Nondestructive testing of specularly reflective objects using reflection three-dimensional computer vision technique,” Opt. Eng. 42, 1343–1347(2003).
[CrossRef]

1995 (1)

1993 (1)

1988 (1)

Z. Wu and L. Li, “A line-integration based method for depth recovery from surface normals,” Comput. Vis. Graph. Image Process 43, 53–66 (1988).
[CrossRef]

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3, 323–344, (1987).
[CrossRef]

1985 (1)

1982 (1)

E. N. Coleman and R. Jain, “Obtaining shape of textured and specular surfaces using four-source photometry,” Comp. Graph. Image Proc. 18, 309–328 (1982).
[CrossRef]

1980 (1)

Bernabeu, E.

Bothe, T.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Breitbarth, M.

M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
[CrossRef]

Burton, D. R.

O. A. Skydan, M. J. Lalor, and D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Meas. Sci. Technol. 18, 106–114 (2007).
[CrossRef]

Chan, F. W. Y.

F. W. Y. Chan, “A novel optical method without phase unwrapping for subsurface flaw detection,” Opt. Lasers Eng. 47, 186–193 (2009).
[CrossRef]

Chen, W.

Y.-L. Xiao, X. Su, and W. Chen, “Flexible geometrical calibration for fringe-reflection 3D measurement,” Opt. Lett. 37, 620–622 (2012).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Cheng, Y.-Y.

Coleman, E. N.

E. N. Coleman and R. Jain, “Obtaining shape of textured and specular surfaces using four-source photometry,” Comp. Graph. Image Proc. 18, 309–328 (1982).
[CrossRef]

Dang, T.

J. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21, 331–340 (2010).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stock, Pattern Classification (Wiley-Interscience, 2000), pp. 174–176.

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing, 3rd ed. (Prentice Hall, 2010).

Edi, P.

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

Ettl, S.

Ghani, A.

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing, 3rd ed. (Prentice Hall, 2010).

Gonzalez-Cano, A.

Gorthi, S.

S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stock, Pattern Classification (Wiley-Interscience, 2000), pp. 174–176.

Häusler, G.

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

S. Ettl, J. Kaminski, M. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47, 2091–2097 (2008).
[CrossRef]

Heidrich, W.

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

Horbach, J.

J. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21, 331–340 (2010).
[CrossRef]

Hung, Y. Y.

Y. Y. Hung and H. M. Shang, “Nondestructive testing of specularly reflective objects using reflection three-dimensional computer vision technique,” Opt. Eng. 42, 1343–1347(2003).
[CrossRef]

Huntley, J. M.

Ihrke, I.

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

Jain, R.

E. N. Coleman and R. Jain, “Obtaining shape of textured and specular surfaces using four-source photometry,” Comp. Graph. Image Proc. 18, 309–328 (1982).
[CrossRef]

Jüptner, W. P. O.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Kaminski, J.

Klette, R.

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in Third Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP, 2002), pp. 204–210.

Knauer, M.

Knauer, M. C.

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

Kühmstedt, P.

M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
[CrossRef]

Kutulakos, K. N.

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

Lalor, M. J.

O. A. Skydan, M. J. Lalor, and D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Meas. Sci. Technol. 18, 106–114 (2007).
[CrossRef]

Leitz, K.-H.

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

Lensch, H. P. A.

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

Li, L.

Z. Wu and L. Li, “A line-integration based method for depth recovery from surface normals,” Comput. Vis. Graph. Image Process 43, 53–66 (1988).
[CrossRef]

Li, W.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Magnor, M.

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

Notni, G.

M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
[CrossRef]

Pan, D.-L.

H.-Y. Wang, D.-L. Pan, and D.-S. Xia, “A fast algorithm for two-dimensional Otsu adaptive threshold algorithm,” Acta Automat. Sinica 33, 968–971 (2007).

Quan, C.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Quiroga, J. A.

Radzi, A.

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

Rastogi, P.

S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Richter, C.

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

Saldner, H. O.

Shang, H. M.

Y. Y. Hung and H. M. Shang, “Nondestructive testing of specularly reflective objects using reflection three-dimensional computer vision technique,” Opt. Eng. 42, 1343–1347(2003).
[CrossRef]

Skydan, O. A.

O. A. Skydan, M. J. Lalor, and D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Meas. Sci. Technol. 18, 106–114 (2007).
[CrossRef]

Southwell, W. H.

Stock, D. G.

R. O. Duda, P. E. Hart, and D. G. Stock, Pattern Classification (Wiley-Interscience, 2000), pp. 174–176.

Su, X.

Tay, C. J.

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3, 323–344, (1987).
[CrossRef]

von Kopylow, C.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Wang, H.-Y.

H.-Y. Wang, D.-L. Pan, and D.-S. Xia, “A fast algorithm for two-dimensional Otsu adaptive threshold algorithm,” Acta Automat. Sinica 33, 968–971 (2007).

Wei, T.

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in Third Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP, 2002), pp. 204–210.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing, 3rd ed. (Prentice Hall, 2010).

Wu, Z.

Z. Wu and L. Li, “A line-integration based method for depth recovery from surface normals,” Comput. Vis. Graph. Image Process 43, 53–66 (1988).
[CrossRef]

Wyant, J. C.

Xia, D.-S.

H.-Y. Wang, D.-L. Pan, and D.-S. Xia, “A fast algorithm for two-dimensional Otsu adaptive threshold algorithm,” Acta Automat. Sinica 33, 968–971 (2007).

Xiao, Y.-L.

Yusuf, K.

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

Zhou, R.

R. Zhou, Adaptive Optics (National Defence Industry, 1996).

Acta Automat. Sinica (1)

H.-Y. Wang, D.-L. Pan, and D.-S. Xia, “A fast algorithm for two-dimensional Otsu adaptive threshold algorithm,” Acta Automat. Sinica 33, 968–971 (2007).

Appl. Opt. (4)

Comp. Graph. Image Proc. (1)

E. N. Coleman and R. Jain, “Obtaining shape of textured and specular surfaces using four-source photometry,” Comp. Graph. Image Proc. 18, 309–328 (1982).
[CrossRef]

Comput. Vis. Graph. Image Process (1)

Z. Wu and L. Li, “A line-integration based method for depth recovery from surface normals,” Comput. Vis. Graph. Image Process 43, 53–66 (1988).
[CrossRef]

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3, 323–344, (1987).
[CrossRef]

J. Opt. Lett. (1)

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry—a novel tool to acquire 3D microtopography with nanometer height resolution,” J. Opt. Lett. 33, 396–398 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

Mach. Vis. Appl. (1)

J. Horbach and T. Dang, “3D reconstruction of specular surfaces using a calibrated projector-camera setup,” Mach. Vis. Appl. 21, 331–340 (2010).
[CrossRef]

Meas. Sci. Technol. (1)

O. A. Skydan, M. J. Lalor, and D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Meas. Sci. Technol. 18, 106–114 (2007).
[CrossRef]

Opt. Eng. (1)

Y. Y. Hung and H. M. Shang, “Nondestructive testing of specularly reflective objects using reflection three-dimensional computer vision technique,” Opt. Eng. 42, 1343–1347(2003).
[CrossRef]

Opt. Lasers Eng. (3)

S. Gorthi and P. Rastogi, “Fringe projection techniques: Wither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

C. Quan, W. Chen, and C. J. Tay, “Phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48, 235–243 (2010).
[CrossRef]

F. W. Y. Chan, “A novel optical method without phase unwrapping for subsurface flaw detection,” Opt. Lasers Eng. 47, 186–193 (2009).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. Breitbarth, P. Kühmstedt, and G. Notni, “Calibration of a combined system with phase measuring deflectometry and fringe projection,” Proc. SPIE 7389, 738909 (2009).
[CrossRef]

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation method for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

WSEAS Trans. Appl. Theor. Mech. (1)

K. Yusuf, P. Edi, A. Radzi, and A. Ghani, “3D shape of specular surface measurement using five degrees of freedom camera system,” WSEAS Trans. Appl. Theor. Mech. 4, 74–84(2009).

Other (5)

I. Ihrke, K. N. Kutulakos, H. P. A. Lensch, M. Magnor, and W. Heidrich, “State of the art in transparent and specular object reconstruction,” in Eurographics 2008 (European Association for Computer Graphics, 2008).

R. Zhou, Adaptive Optics (National Defence Industry, 1996).

T. Wei and R. Klette, “Height from gradient using surface curvature and area constraints,” in Third Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP, 2002), pp. 204–210.

R. O. Duda, P. E. Hart, and D. G. Stock, Pattern Classification (Wiley-Interscience, 2000), pp. 174–176.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing, 3rd ed. (Prentice Hall, 2010).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic diagram of fringe reflectometry.

Fig. 2.
Fig. 2.

Iterative process for 3D shape reconstruction of specular surfaces.

Fig. 3.
Fig. 3.

Captured image of checkerboard pattern with marking point.

Fig. 4.
Fig. 4.

Reference phase distribution based upon virtual reference plane technique; (a) horizonal direction and (b) vertical direction.

Fig. 5.
Fig. 5.

Relative position between LCD screen (top) and super flat planar mirror (bottom).

Fig. 6.
Fig. 6.

Southwell model based on zonal wavefront reconstruction.

Fig. 7.
Fig. 7.

Boundary contour extraction. (a) Phase unwrapping, (b) cosine values of retrieved true phases, (c) threshold segmentation [25], (d) addition operation, and (e) contour extraction [24].

Fig. 8.
Fig. 8.

(a) Simulated hyperboloid surface. (b) Gradient distribution of x direction. (c) Gradient distribution of y direction.

Fig. 9.
Fig. 9.

Photo of measurement setup.

Fig. 10.
Fig. 10.

3D shape measurement of gauge blocks. (a) Gauge block group. (b) Contour of measurement region. (c) Reconstruction result of four gauge blocks. (d) Repeatability error for measurement of the gauge block with height of 0.5 mm.

Fig. 11.
Fig. 11.

3D reconstruction results of two gauge blocks with heights of 6.45 and 6.50 mm.

Fig. 12.
Fig. 12.

3D shape measurement of the optical hyperboloid surface. (a) Horizontal projection image. (b) Phase-unwrapping result in horizontal direction. (c) Vertical projection image. (d) Phase-unwrapping result in vertical direction. (e) Contour of measurement region. (f) 3D reconstruction result.

Tables (2)

Tables Icon

Table 1. Analysis of 3D Reconstruction Results of Hyperbolic Surface With 1% Gaussian Noise

Tables Icon

Table 2. Reconstruction Errors of the Gauge Blocks

Equations (16)

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

I(x,y)=A(x,y)+B(x,y)cos[(2π/p)x+ϕ(x,y)+Δϕ(x,y)],
dtanθ+ztanθΔϕ(x,y)·p/2πdz=tan(θ+2α),
gx(x,y)=z(x,y)xgy(x,y)=z(x,y)y,
z(x,y)=z(x0,y0)+γgx(x,y)dx+gy(x,y)dy,
12(gi+1,jx+gi,jx)=1h(zi+1,jzi,j),i=1N1,j=1N12(gi,j+1y+gi,jy)=1h(zi,j+1zi,j),i=1N,j=1N1,
ki,jzi,j[zi+1,j+zi1,j+zi,j+1+zi,j1]=[gi1,jxgi+1,jx+gi,j1ygi,j+1y]h/2,
ki,j={2i=1orNi=1orN3otheri,j4i,j=2N1.
AZ=b.
W=Ω[(ΔZG)2]dxdymin,
zi,j=zi,j+bi,j,
zi,j=[zi+1,j+zi1,j+zi,j+1+zi,j1]/ki,j,
bi,j=[gi1,jxgi+1,jx+gi,j1ygi,j+1y]h/2ki,j.
{g0,jx=g1,jx,gN+1,jx=gN,jxgi,0y=gi,1y,gi,N+1y=gi,Nyz0,j=zN+1,j=zi,0=zi,N+1=0.
zi,j(m+1)=zi,j(m)+bi,j.
max|zi,j(m+1)zi,j(m)|106,
x2a2+y2b2(z45)2c2=1,

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