Abstract

We discuss the accuracy limits for the localization of surfaces in three-dimensional (3-D) space. Such a localization is necessary for the registration of different views of an object, taken by 3-D sensors from several directions. A quantitative analysis shows that the lateral localization accuracy of a small surface area is proportional to the local curvature of the surface. This confirms the intuitive conjecture that our visual system performs localization of 3-D objects via sharp features. The longitudinal localization accuracy depends only on the noise of the data and is usually much better than the lateral localization accuracy, suggesting that surfaces are to be registered only along the longitudinal directions.

© 2001 Optical Society of America

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References

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  1. L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in picture,” Prog. Opt. 32, 145–201 (1993).
    [CrossRef]
  2. H. Baher, Analog and Digital Signal Analysis (Wiley, Chichester, UK, 1994).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  4. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1991).
  5. L. P. Yaroslavsky, Digital Picture Processing (Springer-Verlag, Berlin, 1985).
    [CrossRef]
  6. Y. Chen, G. Medioni, “Object modeling by registration of multiple range images,” Image Vision Comput. 10, 145–155 (1992).
    [CrossRef]
  7. J. Feldmar, N. Ayache, “Rigid, local and locally affine registration of free-form surfaces,” Int. J. Comput. Vision 18, 99–119 (1996).
    [CrossRef]
  8. S. Karbacher, G. Häusler, H. Schönfeld, “Reverse engineering using optical range sensors,” in Handbook of Computer Vision and Applications, Vol. 3: Systems and Applications, B. Jähne, H. Haussecker, P. Geissler, eds. (Academic Press, Boston, 1999).
  9. Features should be here understood as both specific points of the object surface and certain properties assigned to these points.
  10. G. Häusler, D. Ritter, “Feature-based object recognition and localization in 3D-space using a single video image,” Comput. Vision Image Understand. 73, 64–81 (1999).
    [CrossRef]
  11. M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  12. C. S. Chua, R. Jarvis, “Point signatures: a new representation for 3D object recognition,” Int. J. Comput. Vision 25, 63–85 (1997).
    [CrossRef]
  13. J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
    [CrossRef] [PubMed]
  14. A. Gueziec, “Large deformable splines, crest lines and matching,” Proceedings of the Fourth International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 650–657.
  15. J. P. Thirion, “New feature points based on geometric invariants for 3D image registration,” Int. J. Comput. Vision 18, 121–137 (1996).
    [CrossRef]
  16. P. J. Besl, N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
    [CrossRef]
  17. Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” Int. J. Comput. Vision 13, 119–152 (1994).
    [CrossRef]
  18. T. Masuda, K. Sakaue, N. Yokoya, “Registration and integration of multiple range images for 3-D model construction,” Proceedings of the 13th International Conference on Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1996), Vol. 1, pp. 879–883.
    [CrossRef]
  19. K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
    [CrossRef] [PubMed]
  20. B. K. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Am. A 4, 629–642 (1987).
    [CrossRef]
  21. B. K. Horn, H. M. Hilden, S. Negahdaripour, “Closed-form solution of absolute orientation using orthonormal matrices,” J. Opt. Soc. Am. A 5, 1127–1135 (1988).
    [CrossRef]
  22. J. B. A. Maintz, M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1–36 (1998).
    [CrossRef]
  23. M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
    [CrossRef]
  24. S. Seeger, X. Laboureux, “Feature extraction and registration,” in Principles of 3D Image Analysis and Synthesis, B. Girod, G. Greiner, H. Niemann, eds. (Kluwer Academic, Boston, 2000).
  25. R. J. Campbell, P. J. Flynn, “A survey of free-form object representation and recognition techniques,” Comput. Vision Image Understand. 81, 166–210 (2001).
    [CrossRef]
  26. The analysis is done in one dimension to simplify the formulation. The extension to two variables would be straightforward.
  27. Another advantage of this approach is that R(ε) does not need to be normalized through the support width.
  28. The proof can be done analytically or verified by simple numerical simulations.
  29. Because we consider the two signals only within their overlapping domain, the Gaussian white-noise assumption over this domain is a justified restriction.
  30. I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).
  31. From now on “localization error” designates the “standard deviation of the localization error” as well, and the localization accuracy is defined as the inverse of the localization error.
  32. Actually 2Mx + 1 (Mx + 12) is the total (half) number of points. Here and in future developments of this paper they are approximated through 2Mx and Mx, respectively.

2001

R. J. Campbell, P. J. Flynn, “A survey of free-form object representation and recognition techniques,” Comput. Vision Image Understand. 81, 166–210 (2001).
[CrossRef]

2000

M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
[CrossRef]

1999

G. Häusler, D. Ritter, “Feature-based object recognition and localization in 3D-space using a single video image,” Comput. Vision Image Understand. 73, 64–81 (1999).
[CrossRef]

1998

J. B. A. Maintz, M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1–36 (1998).
[CrossRef]

1997

C. S. Chua, R. Jarvis, “Point signatures: a new representation for 3D object recognition,” Int. J. Comput. Vision 25, 63–85 (1997).
[CrossRef]

1996

J. P. Thirion, “New feature points based on geometric invariants for 3D image registration,” Int. J. Comput. Vision 18, 121–137 (1996).
[CrossRef]

J. Feldmar, N. Ayache, “Rigid, local and locally affine registration of free-form surfaces,” Int. J. Comput. Vision 18, 99–119 (1996).
[CrossRef]

1994

Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” Int. J. Comput. Vision 13, 119–152 (1994).
[CrossRef]

1993

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in picture,” Prog. Opt. 32, 145–201 (1993).
[CrossRef]

1992

Y. Chen, G. Medioni, “Object modeling by registration of multiple range images,” Image Vision Comput. 10, 145–155 (1992).
[CrossRef]

P. J. Besl, N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

1988

1987

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

B. K. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Am. A 4, 629–642 (1987).
[CrossRef]

1986

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
[CrossRef] [PubMed]

Arun, K. S.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Audette, M. A.

M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
[CrossRef]

Ayache, N.

J. Feldmar, N. Ayache, “Rigid, local and locally affine registration of free-form surfaces,” Int. J. Comput. Vision 18, 99–119 (1996).
[CrossRef]

Baher, H.

H. Baher, Analog and Digital Signal Analysis (Wiley, Chichester, UK, 1994).

Besl, P. J.

P. J. Besl, N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Blostein, S. D.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).

Campbell, R. J.

R. J. Campbell, P. J. Flynn, “A survey of free-form object representation and recognition techniques,” Comput. Vision Image Understand. 81, 166–210 (2001).
[CrossRef]

Canny, J.

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
[CrossRef] [PubMed]

Chen, Y.

Y. Chen, G. Medioni, “Object modeling by registration of multiple range images,” Image Vision Comput. 10, 145–155 (1992).
[CrossRef]

Chua, C. S.

C. S. Chua, R. Jarvis, “Point signatures: a new representation for 3D object recognition,” Int. J. Comput. Vision 25, 63–85 (1997).
[CrossRef]

do Carmo, M. P.

M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Feldmar, J.

J. Feldmar, N. Ayache, “Rigid, local and locally affine registration of free-form surfaces,” Int. J. Comput. Vision 18, 99–119 (1996).
[CrossRef]

Ferrie, F. P.

M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
[CrossRef]

Flynn, P. J.

R. J. Campbell, P. J. Flynn, “A survey of free-form object representation and recognition techniques,” Comput. Vision Image Understand. 81, 166–210 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Gueziec, A.

A. Gueziec, “Large deformable splines, crest lines and matching,” Proceedings of the Fourth International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 650–657.

Häusler, G.

G. Häusler, D. Ritter, “Feature-based object recognition and localization in 3D-space using a single video image,” Comput. Vision Image Understand. 73, 64–81 (1999).
[CrossRef]

S. Karbacher, G. Häusler, H. Schönfeld, “Reverse engineering using optical range sensors,” in Handbook of Computer Vision and Applications, Vol. 3: Systems and Applications, B. Jähne, H. Haussecker, P. Geissler, eds. (Academic Press, Boston, 1999).

Hilden, H. M.

Horn, B. K.

Huang, T. S.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

Jarvis, R.

C. S. Chua, R. Jarvis, “Point signatures: a new representation for 3D object recognition,” Int. J. Comput. Vision 25, 63–85 (1997).
[CrossRef]

Karbacher, S.

S. Karbacher, G. Häusler, H. Schönfeld, “Reverse engineering using optical range sensors,” in Handbook of Computer Vision and Applications, Vol. 3: Systems and Applications, B. Jähne, H. Haussecker, P. Geissler, eds. (Academic Press, Boston, 1999).

Laboureux, X.

S. Seeger, X. Laboureux, “Feature extraction and registration,” in Principles of 3D Image Analysis and Synthesis, B. Girod, G. Greiner, H. Niemann, eds. (Kluwer Academic, Boston, 2000).

Maintz, J. B. A.

J. B. A. Maintz, M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1–36 (1998).
[CrossRef]

Masuda, T.

T. Masuda, K. Sakaue, N. Yokoya, “Registration and integration of multiple range images for 3-D model construction,” Proceedings of the 13th International Conference on Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1996), Vol. 1, pp. 879–883.
[CrossRef]

McKay, N. D.

P. J. Besl, N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Medioni, G.

Y. Chen, G. Medioni, “Object modeling by registration of multiple range images,” Image Vision Comput. 10, 145–155 (1992).
[CrossRef]

Mühlig, H.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).

Musiol, G.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).

Negahdaripour, S.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1991).

Peters, T. M.

M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
[CrossRef]

Ritter, D.

G. Häusler, D. Ritter, “Feature-based object recognition and localization in 3D-space using a single video image,” Comput. Vision Image Understand. 73, 64–81 (1999).
[CrossRef]

Sakaue, K.

T. Masuda, K. Sakaue, N. Yokoya, “Registration and integration of multiple range images for 3-D model construction,” Proceedings of the 13th International Conference on Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1996), Vol. 1, pp. 879–883.
[CrossRef]

Schönfeld, H.

S. Karbacher, G. Häusler, H. Schönfeld, “Reverse engineering using optical range sensors,” in Handbook of Computer Vision and Applications, Vol. 3: Systems and Applications, B. Jähne, H. Haussecker, P. Geissler, eds. (Academic Press, Boston, 1999).

Seeger, S.

S. Seeger, X. Laboureux, “Feature extraction and registration,” in Principles of 3D Image Analysis and Synthesis, B. Girod, G. Greiner, H. Niemann, eds. (Kluwer Academic, Boston, 2000).

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).

Thirion, J. P.

J. P. Thirion, “New feature points based on geometric invariants for 3D image registration,” Int. J. Comput. Vision 18, 121–137 (1996).
[CrossRef]

Viergever, M. A.

J. B. A. Maintz, M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1–36 (1998).
[CrossRef]

Yaroslavsky, L. P.

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in picture,” Prog. Opt. 32, 145–201 (1993).
[CrossRef]

L. P. Yaroslavsky, Digital Picture Processing (Springer-Verlag, Berlin, 1985).
[CrossRef]

Yokoya, N.

T. Masuda, K. Sakaue, N. Yokoya, “Registration and integration of multiple range images for 3-D model construction,” Proceedings of the 13th International Conference on Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1996), Vol. 1, pp. 879–883.
[CrossRef]

Zhang, Z.

Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” Int. J. Comput. Vision 13, 119–152 (1994).
[CrossRef]

Comput. Vision Image Understand.

G. Häusler, D. Ritter, “Feature-based object recognition and localization in 3D-space using a single video image,” Comput. Vision Image Understand. 73, 64–81 (1999).
[CrossRef]

R. J. Campbell, P. J. Flynn, “A survey of free-form object representation and recognition techniques,” Comput. Vision Image Understand. 81, 166–210 (2001).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

K. S. Arun, T. S. Huang, S. D. Blostein, “Least-squares fitting of two 3D point sets,” IEEE Trans. Pattern Anal. Mach. Intell. 9, 698–700 (1987).
[CrossRef] [PubMed]

J. Canny, “A computational approach to edge detection,” IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986).
[CrossRef] [PubMed]

P. J. Besl, N. D. McKay, “A method for registration of 3D shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Image Vision Comput.

Y. Chen, G. Medioni, “Object modeling by registration of multiple range images,” Image Vision Comput. 10, 145–155 (1992).
[CrossRef]

Int. J. Comput. Vision

J. Feldmar, N. Ayache, “Rigid, local and locally affine registration of free-form surfaces,” Int. J. Comput. Vision 18, 99–119 (1996).
[CrossRef]

Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” Int. J. Comput. Vision 13, 119–152 (1994).
[CrossRef]

C. S. Chua, R. Jarvis, “Point signatures: a new representation for 3D object recognition,” Int. J. Comput. Vision 25, 63–85 (1997).
[CrossRef]

J. P. Thirion, “New feature points based on geometric invariants for 3D image registration,” Int. J. Comput. Vision 18, 121–137 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Med. Image Anal.

J. B. A. Maintz, M. A. Viergever, “A survey of medical image registration,” Med. Image Anal. 2, 1–36 (1998).
[CrossRef]

M. A. Audette, F. P. Ferrie, T. M. Peters, “An algorithmic overview of surface registration techniques for medical imaging,” Med. Image Anal. 4, 201–217 (2000).
[CrossRef]

Prog. Opt.

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in picture,” Prog. Opt. 32, 145–201 (1993).
[CrossRef]

Other

H. Baher, Analog and Digital Signal Analysis (Wiley, Chichester, UK, 1994).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1991).

L. P. Yaroslavsky, Digital Picture Processing (Springer-Verlag, Berlin, 1985).
[CrossRef]

S. Karbacher, G. Häusler, H. Schönfeld, “Reverse engineering using optical range sensors,” in Handbook of Computer Vision and Applications, Vol. 3: Systems and Applications, B. Jähne, H. Haussecker, P. Geissler, eds. (Academic Press, Boston, 1999).

Features should be here understood as both specific points of the object surface and certain properties assigned to these points.

T. Masuda, K. Sakaue, N. Yokoya, “Registration and integration of multiple range images for 3-D model construction,” Proceedings of the 13th International Conference on Pattern Recognition (IEEE Computer Society, Los Alamitos, Calif., 1996), Vol. 1, pp. 879–883.
[CrossRef]

A. Gueziec, “Large deformable splines, crest lines and matching,” Proceedings of the Fourth International Conference on Computer Vision (IEEE Computer Society, Los Alamitos, Calif., 1993), pp. 650–657.

M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976).

S. Seeger, X. Laboureux, “Feature extraction and registration,” in Principles of 3D Image Analysis and Synthesis, B. Girod, G. Greiner, H. Niemann, eds. (Kluwer Academic, Boston, 2000).

The analysis is done in one dimension to simplify the formulation. The extension to two variables would be straightforward.

Another advantage of this approach is that R(ε) does not need to be normalized through the support width.

The proof can be done analytically or verified by simple numerical simulations.

Because we consider the two signals only within their overlapping domain, the Gaussian white-noise assumption over this domain is a justified restriction.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Mühlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 2000).

From now on “localization error” designates the “standard deviation of the localization error” as well, and the localization accuracy is defined as the inverse of the localization error.

Actually 2Mx + 1 (Mx + 12) is the total (half) number of points. Here and in future developments of this paper they are approximated through 2Mx and Mx, respectively.

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Figures (8)

Fig. 1
Fig. 1

Reverse engineering: data acquisition, registration, and modeling. (a) Original, (b) single views, (c) registered views, and (d) virtual 3-D model.

Fig. 2
Fig. 2

“Trombone Angel” from the Bamberg Dome, Germany. (left) Photo of the original, (right) 3-D virtual model.

Fig. 3
Fig. 3

Relative position between a(x) and b(x).

Fig. 4
Fig. 4

Constraints applied to the curve to simplify the analysis.

Fig. 5
Fig. 5

Digitized curve.

Fig. 6
Fig. 6

Constraints applied to the surface to simplify the analysis.

Fig. 7
Fig. 7

Object part for which the localization error is calculated.

Fig. 8
Fig. 8

Points having the same local Frenet coordinate systems.

Equations (46)

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

Rε=-ax-bx-ε2dx.
Rε=- a2xdx-2 - axbx-εdx+- b2x-εdx,
Rε=-2 - axbx-εdx+-a2x+b2xdx.
Rε=-2Cε+K,
Rε=xinfxsupax-bx-ε2dx,
CNε=xinfxsup axbx-εdxxinfxsup a2xdx1/2xinfxsup b2x-εdx1/2.
axinf=±axsup,axinfaxinf=axsupaxsup.
Rε, η=xinfxsupax+ε+η-bx2dx,
Rεε0, η0=0,  Rηε0, η0=0.
Rεε0, η0=Rε0, 0+ε02Rε20, 0+η02Rεη0, 0=0,Rηε0, η0=Rη0, 0+η02Rη20, 0+ε02Rεη0, 0=0,
ε0= dx axnxdx- axdx nxdx a2x-axnxdx dx- axdx2,η0=a2x-axnxdx nxdx- axdx axnxdx a2x-axnxdx dx- axdx2.
ε0=xp-ΔTxp+ΔT axnxdxxp-ΔTxp+ΔT a2xdx,  η0=xp-ΔTxp+ΔT nxdxxp-ΔTxp+ΔTdx,
Eε0=Eη0=0,  Eε02=N023a2xpΔT3,Eη02=N02ΔT,  Eε0η0=0,
κxp=axp1+a2xp3/2.
lx¯=0,  lz¯=0,
σlx2=N023κxp2ΔT3,
σlz2=N02ΔT,
σlxlz=0,
σlz2=σn2Tox2MxTox=σn22Mx,
σ=σnM,
Rε, μ, η=xinfxsupyinfysupax+ε, y+μ+η-bx, y2dxdy,
Rεε0, μ0, η0=0,  Rμε0, μ0, η0=0,Rηε0, μ0, η0=0.
axxp, yp=0,  ayxp, yp=0,2axyxp, yp=0.
Eε0=Eμ0=Eη0=0,Eε02=N023axx2xp, ypΔTx32ΔTy,Eμ02=N023ayy2xp, ypΔTy32ΔTx,Eη02=N02ΔTx2ΔTy,Eε0μ0=Eε0η0=Eμ0η0=0,
axxxp, yp=κxxp, yp=κ1,ayyxp, yp=κyxp, yp=κ2,
lx¯=0,  ly¯=0,  lz¯=0,
σlx2=N043κ12ΔTx3ΔTy,
σly2=N043κ22ΔTy3ΔTx,
σlz2=N04ΔTxΔTy,
σlxlz=0,  σlxly=0,  σlylz=0,
σlx1D2=N01D23κx2ΔTox3=σn2Tox23κx2Mx3Tox3=σn223κx2Mx3Tox2,σlx2D2=N02D23κx2ΔTx32ΔTy=σn2ToxToy23κx2Mx3Tox32MyToy=σn223κx2Mx3Tox212My,
σlx2D2=σlx1D22My,
σlx=32σnToM2κx,σlz=σn2M,
σlx65 μm,  σlz25 μm.
σlzσlx=ToxMxκx3.
σlx2=11-λ2N04π2  νx2|Aνx, νy|2dνxdνy,σly2=11-λ2N04π2  νy2|Aνx, νy|2dνxdνy,σlxly=λ1-λ2N04π2 νx2|Aνx, νy|2dνxdνy  νy2|Aνx, νy|2dνxdνy1/2,
λ= νxνy|Aνx, νy|2dνxdνy νx2|Aνx, νy|2dνxdνy  νy2|Aνx, νy|2dνxdνy1/2.
4π2  νx2|Aνx, νy|2dνxdνy= 2ax2x, y2dxdy.
σlx2=N043axx2ΔTx3ΔTy,σlx2=N043ayy2ΔTy3ΔTx,σlxly=0.
σl=constant,  σl1κ,
σl21iK κi2,
1σlK points2=iK1σli2,
1σlK+1 points2=1σlK points2+1σlK+12,
σlK+1 points<σlK points.
1σlK+L points2=1σlK points2+Lσla2.
σlK points=σloK,

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