Abstract

We present an effective method for the accurate three-dimensional (3D) measurement of small industrial parts under a complicated noisy background, based on stereo vision. To effectively extract the nonlinear features of desired curves of the measured parts in the images, a strategy from coarse to fine extraction is employed, based on a virtual motion control system. By using the multiscale decomposition of gray images and virtual beam chains, the nonlinear features can be accurately extracted. By analyzing the generation of geometric errors, the refined feature points of the desired curves are extracted. Then the 3D structure of the measured parts can be accurately reconstructed and measured with least squares errors. Experimental results show that the presented method can accurately measure industrial parts that are represented by various line segments and curves.

© 2010 Optical Society of America

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    [CrossRef]

2009

2008

B. Li and S. T. Acton, “Automatic active model initialization via Poisson inverse gradient,” IEEE Trans. on Image Process. 17, 1406-1420 (2008).
[CrossRef]

2007

B. Li and S. T. Acton, “Active contour external force using vector field convolution for image segmentation,” IEEE Trans. Image Process. 16, 2096-2106 (2007).
[CrossRef] [PubMed]

2006

R. Anchini, C. Liguori, V. Paciello, and A. Paolillo, “A comparison between stereo-vision techniques for the reconstruction of 3-D coordinates of objects,” IEEE Trans. Instrum. Meas. 55, 1459-1466(2006).
[CrossRef]

Y. J. Zhang, Z. X. Zhang, and J. Q. Zhang, “Automatic measurement of industrial sheetmetal parts with CAD data and non-metric image sequence,” Comput. Vision Image Underst. 102, 52-59 (2006).
[CrossRef]

2005

A. Wozniak and M. Dobosz, “Factors inflliguencing probing accuracy of a coordinate measuring machine,” IEEE Trans. Instrum. Meas. 54, 2540-2548 (2005).
[CrossRef]

2004

2003

Y. S. Chen and B. T. Chen, “Measuring of a three-dimensional surface by use of a spatial distance computation,” Appl. Opt. 42, 1958-1972 (2003).
[CrossRef] [PubMed]

S. Malassiotis and M. G. Strintzis, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

M. Sotiris and G. S. Michael, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

2002

2001

2000

H. H. Joon and S. P. Jong, “Contour matching using epipolar geometry,” IEEE Trans. Pattern Anal. Machine Intell. 22, 358-370(2000).
[CrossRef]

S. H. Park, K. M. Lee, and S. U. Lee, “A line feature matching technique based on an eigenvector approach,” Comput. Vision Image Underst. 77, 263-283 (2000).
[CrossRef]

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

1999

H. Qjidaa and L. Radouane, “Robust line fitting in a noisy image by the method of moments,” IEEE Trans. Pattern Anal. Machine Intell. 21, 1216-1223 (1999).
[CrossRef]

1998

C. Molina, G. P. Prause, P. Radeva, and M. Sonka, “Catheter path reconstruction from biplane angiography using 3D snakes,” Proc. SPIE 3338, 504-512 (1998).
[CrossRef]

1997

C. S. Zhao, “Epipolar parameterization for reconstructing 3d rigid curve,” Patt. Recog. 30, 1817-1827 (1997).
[CrossRef]

1995

P. L. Rosin and G. A. W. West, “Nonparametric segmentation of curves into various representations,” IEEE Trans. Pattern Anal. Machine Intell. 17, 1140-1153(1995).
[CrossRef]

1993

I. D. Coope and L. C. W. Dixon, “Circle fitting by linear and nonlinear least squares,” J. Optim. Theory Appl. 76, 381-388(1993).
[CrossRef]

V. Caselles, “A Geometric model for active contours,” Numer. Math. 66, 1-31 (1993).
[CrossRef]

1992

S. G. Mallat and S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Machine Intell. 14, 710-732 (1992).
[CrossRef]

1990

1989

S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091-2110 (1989).
[CrossRef]

I. Weiss, “Line fitting in a noisy image,” IEEE Trans. Pattern Anal. Machine Intell. 11, 325-329 (1989).
[CrossRef]

1988

J. Illingworth and J. Kittler, “A survey of the Hough transform,” Comput. Vision Graphics Image Process. 44, 87-116(1988).
[CrossRef]

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vis. 1, 321-331 (1988).
[CrossRef]

1987

U. M. Landau, “Estimation of a circular arc center and its radius,” Comput. Vision Graph. Image Process. 38, 317-326(1987).
[CrossRef]

1979

Y. Nakagawa and A. Rosenfeld, “A note on polygonal and elliptical approximation of mechanical parts,” Patt. Recog. 11, 133-142 (1979).
[CrossRef]

1974

A. Albano, “Representation of digitized contours in terms of conic arcs and straight-line segments,” Comput. Graph. Image Process . 3, 23-33 (1974).
[CrossRef]

1972

R. O. Duda and P. E. Hart, “Use of the Hough transform to detect lines and curves in pictures,” Commun. ACM 15, 11-15(1972).
[CrossRef]

Acton, S. T.

B. Li and S. T. Acton, “Automatic active model initialization via Poisson inverse gradient,” IEEE Trans. on Image Process. 17, 1406-1420 (2008).
[CrossRef]

B. Li and S. T. Acton, “Active contour external force using vector field convolution for image segmentation,” IEEE Trans. Image Process. 16, 2096-2106 (2007).
[CrossRef] [PubMed]

Albano, A.

A. Albano, “Representation of digitized contours in terms of conic arcs and straight-line segments,” Comput. Graph. Image Process . 3, 23-33 (1974).
[CrossRef]

Anchini, R.

R. Anchini, C. Liguori, V. Paciello, and A. Paolillo, “A comparison between stereo-vision techniques for the reconstruction of 3-D coordinates of objects,” IEEE Trans. Instrum. Meas. 55, 1459-1466(2006).
[CrossRef]

Bajcsy, R.

A. Leonardis and R. Bajcsy, “Finding parametric curves in an image,” Proceedings of the Second European Conference on Computer Vision (Springer, 1992), pp. 653-657.

Cai, L. L.

Cañero, C.

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

Caselles, V.

V. Caselles, “A Geometric model for active contours,” Numer. Math. 66, 1-31 (1993).
[CrossRef]

Chen, B. T.

Chen, Y. S.

Chu, S. C.

Coope, I. D.

I. D. Coope and L. C. W. Dixon, “Circle fitting by linear and nonlinear least squares,” J. Optim. Theory Appl. 76, 381-388(1993).
[CrossRef]

Costen, F.

Dixon, L. C. W.

I. D. Coope and L. C. W. Dixon, “Circle fitting by linear and nonlinear least squares,” J. Optim. Theory Appl. 76, 381-388(1993).
[CrossRef]

Dobosz, M.

A. Wozniak and M. Dobosz, “Factors inflliguencing probing accuracy of a coordinate measuring machine,” IEEE Trans. Instrum. Meas. 54, 2540-2548 (2005).
[CrossRef]

Du, H.

Duda, R. O.

R. O. Duda and P. E. Hart, “Use of the Hough transform to detect lines and curves in pictures,” Commun. ACM 15, 11-15(1972).
[CrossRef]

Graebling, P.

Hart, P. E.

R. O. Duda and P. E. Hart, “Use of the Hough transform to detect lines and curves in pictures,” Commun. ACM 15, 11-15(1972).
[CrossRef]

Hirsch, E.

Hyoungseop, K.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Iizuka, K.

Illingworth, J.

J. Illingworth and J. Kittler, “A survey of the Hough transform,” Comput. Vision Graphics Image Process. 44, 87-116(1988).
[CrossRef]

Ishida, S.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Itai, Y.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Jong, S. P.

H. H. Joon and S. P. Jong, “Contour matching using epipolar geometry,” IEEE Trans. Pattern Anal. Machine Intell. 22, 358-370(2000).
[CrossRef]

Joon, H. H.

H. H. Joon and S. P. Jong, “Contour matching using epipolar geometry,” IEEE Trans. Pattern Anal. Machine Intell. 22, 358-370(2000).
[CrossRef]

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vis. 1, 321-331 (1988).
[CrossRef]

Katsuragawa, S.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Kittler, J.

J. Illingworth and J. Kittler, “A survey of the Hough transform,” Comput. Vision Graphics Image Process. 44, 87-116(1988).
[CrossRef]

Komatsu, S.

Lallement, A.

Landau, U. M.

U. M. Landau, “Estimation of a circular arc center and its radius,” Comput. Vision Graph. Image Process. 38, 317-326(1987).
[CrossRef]

Lee, K. M.

S. H. Park, K. M. Lee, and S. U. Lee, “A line feature matching technique based on an eigenvector approach,” Comput. Vision Image Underst. 77, 263-283 (2000).
[CrossRef]

Lee, S. U.

S. H. Park, K. M. Lee, and S. U. Lee, “A line feature matching technique based on an eigenvector approach,” Comput. Vision Image Underst. 77, 263-283 (2000).
[CrossRef]

Leonardis, A.

A. Leonardis and R. Bajcsy, “Finding parametric curves in an image,” Proceedings of the Second European Conference on Computer Vision (Springer, 1992), pp. 653-657.

Li, B.

B. Li and S. T. Acton, “Automatic active model initialization via Poisson inverse gradient,” IEEE Trans. on Image Process. 17, 1406-1420 (2008).
[CrossRef]

B. Li and S. T. Acton, “Active contour external force using vector field convolution for image segmentation,” IEEE Trans. Image Process. 16, 2096-2106 (2007).
[CrossRef] [PubMed]

Liguori, C.

R. Anchini, C. Liguori, V. Paciello, and A. Paolillo, “A comparison between stereo-vision techniques for the reconstruction of 3-D coordinates of objects,” IEEE Trans. Instrum. Meas. 55, 1459-1466(2006).
[CrossRef]

Lin, C. Y.

C. Y. Lin, “A new approach to automatic reconstruction of a 3-D world using active stereo vision,” Comput. Vision Image Underst. 85, 117-143 (2002).
[CrossRef]

Malassiotis, S.

S. Malassiotis and M. G. Strintzis, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

Mallat, S. G.

S. G. Mallat and S. Zhong, “Characterization of signals from multiscale edges,” IEEE Trans. Pattern Anal. Machine Intell. 14, 710-732 (1992).
[CrossRef]

S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091-2110 (1989).
[CrossRef]

Mauri, J.

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

Mauris, G.

Meenavathi, M. B.

J. Prakash, M. B. Meenavathi, and K. Rajesh, “Linear feature extraction using combined approach of Hough transform, eigen values and raster scan algorithms,” in Proceedings of the Fourth International Conference on Intelligent Sensing and Information Processing (IEEE, 2006), pp. 65-70.
[CrossRef]

Michael, G. S.

M. Sotiris and G. S. Michael, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

Mohr, R.

C. S. Zhao and R. Mohr, “Epipolar parameterization for reconstructing a 3D rigid curve,” in Proceedings of the IEEE Symposium on Computer Vision (IEEE, 1995), pp. 67-72.
[CrossRef] [PubMed]

Molina, C.

C. Molina, G. P. Prause, P. Radeva, and M. Sonka, “Catheter path reconstruction from biplane angiography using 3D snakes,” Proc. SPIE 3338, 504-512 (1998).
[CrossRef]

Nakagawa, Y.

Y. Nakagawa and A. Rosenfeld, “A note on polygonal and elliptical approximation of mechanical parts,” Patt. Recog. 11, 133-142 (1979).
[CrossRef]

Nakamura, T.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Ohzu, H.

Onana, V. P.

Paciello, V.

R. Anchini, C. Liguori, V. Paciello, and A. Paolillo, “A comparison between stereo-vision techniques for the reconstruction of 3-D coordinates of objects,” IEEE Trans. Instrum. Meas. 55, 1459-1466(2006).
[CrossRef]

Paolillo, A.

R. Anchini, C. Liguori, V. Paciello, and A. Paolillo, “A comparison between stereo-vision techniques for the reconstruction of 3-D coordinates of objects,” IEEE Trans. Instrum. Meas. 55, 1459-1466(2006).
[CrossRef]

Park, S.

Park, S. H.

S. H. Park, K. M. Lee, and S. U. Lee, “A line feature matching technique based on an eigenvector approach,” Comput. Vision Image Underst. 77, 263-283 (2000).
[CrossRef]

Prakash, J.

J. Prakash, M. B. Meenavathi, and K. Rajesh, “Linear feature extraction using combined approach of Hough transform, eigen values and raster scan algorithms,” in Proceedings of the Fourth International Conference on Intelligent Sensing and Information Processing (IEEE, 2006), pp. 65-70.
[CrossRef]

Prause, G. P.

C. Molina, G. P. Prause, P. Radeva, and M. Sonka, “Catheter path reconstruction from biplane angiography using 3D snakes,” Proc. SPIE 3338, 504-512 (1998).
[CrossRef]

Qjidaa, H.

H. Qjidaa and L. Radouane, “Robust line fitting in a noisy image by the method of moments,” IEEE Trans. Pattern Anal. Machine Intell. 21, 1216-1223 (1999).
[CrossRef]

Radeva, P.

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

C. Molina, G. P. Prause, P. Radeva, and M. Sonka, “Catheter path reconstruction from biplane angiography using 3D snakes,” Proc. SPIE 3338, 504-512 (1998).
[CrossRef]

Radouane, L.

H. Qjidaa and L. Radouane, “Robust line fitting in a noisy image by the method of moments,” IEEE Trans. Pattern Anal. Machine Intell. 21, 1216-1223 (1999).
[CrossRef]

Rajesh, K.

J. Prakash, M. B. Meenavathi, and K. Rajesh, “Linear feature extraction using combined approach of Hough transform, eigen values and raster scan algorithms,” in Proceedings of the Fourth International Conference on Intelligent Sensing and Information Processing (IEEE, 2006), pp. 65-70.
[CrossRef]

Ren, Z. G.

Rosenfeld, A.

Y. Nakagawa and A. Rosenfeld, “A note on polygonal and elliptical approximation of mechanical parts,” Patt. Recog. 11, 133-142 (1979).
[CrossRef]

Rosin, P. L.

P. L. Rosin and G. A. W. West, “Nonparametric segmentation of curves into various representations,” IEEE Trans. Pattern Anal. Machine Intell. 17, 1140-1153(1995).
[CrossRef]

Rudant, J. P.

Shimotahira, H.

Sonka, M.

C. Molina, G. P. Prause, P. Radeva, and M. Sonka, “Catheter path reconstruction from biplane angiography using 3D snakes,” Proc. SPIE 3338, 504-512 (1998).
[CrossRef]

Sotiris, M.

M. Sotiris and G. S. Michael, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

Strintzis, M. G.

S. Malassiotis and M. G. Strintzis, “Stereo vision system for precision dimensional inspection of 3D holes,” Machine Vis. Apps. 15, 101-113 (2003).
[CrossRef]

Suhara, H.

Terzopoulos, D.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vis. 1, 321-331 (1988).
[CrossRef]

Toledo, R.

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

Tonyé, E.

Trouvé, E.

Villanueva, J.

C. Cañero, P. Radeva, R. Toledo, J. Villanueva, and J. Mauri, “3-D curve reconstrucion by biplane snakes,” Proc. ICPR 4, 563-566 (2000).
[CrossRef]

Wah, C.

Wang, Z. Y.

Weiss, I.

I. Weiss, “Line fitting in a noisy image,” IEEE Trans. Pattern Anal. Machine Intell. 11, 325-329 (1989).
[CrossRef]

West, G. A. W.

P. L. Rosin and G. A. W. West, “Nonparametric segmentation of curves into various representations,” IEEE Trans. Pattern Anal. Machine Intell. 17, 1140-1153(1995).
[CrossRef]

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Int. J. Comput. Vis. 1, 321-331 (1988).
[CrossRef]

Wozniak, A.

A. Wozniak and M. Dobosz, “Factors inflliguencing probing accuracy of a coordinate measuring machine,” IEEE Trans. Instrum. Meas. 54, 2540-2548 (2005).
[CrossRef]

Xie, H. M.

Yamamoto, K. A.

Y. Itai, K. Hyoungseop, S. Katsuragawa, S. Ishida, T. Nakamura, and K. A. Yamamoto, “Automatic segmentation of lung areas based on SNAKES and extraction of abnormal areas,” in Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (IEEE, 2005).
[PubMed]

Yoshikuni, Y.

Zhang, J. Q.

Y. J. Zhang, Z. X. Zhang, and J. Q. Zhang, “Automatic measurement of industrial sheetmetal parts with CAD data and non-metric image sequence,” Comput. Vision Image Underst. 102, 52-59 (2006).
[CrossRef]

Zhang, Y. J.

Y. J. Zhang, Z. X. Zhang, and J. Q. Zhang, “Automatic measurement of industrial sheetmetal parts with CAD data and non-metric image sequence,” Comput. Vision Image Underst. 102, 52-59 (2006).
[CrossRef]

Zhang, Z. X.

Y. J. Zhang, Z. X. Zhang, and J. Q. Zhang, “Automatic measurement of industrial sheetmetal parts with CAD data and non-metric image sequence,” Comput. Vision Image Underst. 102, 52-59 (2006).
[CrossRef]

Zhao, C. S.

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Figures (21)

Fig. 1
Fig. 1

Representing a curve using a chain of line segments: (a) at large scale and (b) at small scale.

Fig. 2
Fig. 2

Schematic attraction distribution on a virtual beam.

Fig. 3
Fig. 3

Schematic force status on joint ( P i ) .

Fig. 4
Fig. 4

Flow chart of the virtual control system.

Fig. 5
Fig. 5

Principle of coarse-to-fine nonlinear feature extraction.

Fig. 6
Fig. 6

Using a line segment to fit a curve section.

Fig. 7
Fig. 7

Schematic principle of the epipolar line method.

Fig. 8
Fig. 8

Projection of a 3D curve on the X Y plane.

Fig. 9
Fig. 9

(a) Initial position of the virtual beam chain in the gray image with large Gaussian white noise and salt-pepper noise and (b) the initial position of the active contour in the gray image with small Gaussian white noise and salt-pepper noise.

Fig. 10
Fig. 10

Gradient modulus distributions: (a) of the gray image in Fig. 9a and (b) of the gray image in Fig. 9b.

Fig. 11
Fig. 11

Results of nonlinear feature extraction: (a) by using the proposed method and (b) by using the active contour method.

Fig. 12
Fig. 12

Developed image acquisition system.

Fig. 13
Fig. 13

Gray images of a cylindrical part with a round hole: (a) acquired by the left CCD and b) acquired by the right CCD.

Fig. 14
Fig. 14

Initial positions of the virtual beam chains and the virtual resultant forces on the virtual joints [(a) large initial position error of 70 pixels is added].

Fig. 15
Fig. 15

Nonlinear feature extraction results by using the beam chain at scale 1 and the wavelet decomposition at scale 2 3 .

Fig. 16
Fig. 16

Initial positions and force states of the beam chain at scale 1 / 2 .

Fig. 17
Fig. 17

Nonlinear feature extraction results by using the beam chain at scale 1 / 2 and the wavelet decomposition at scale 2 2 .

Fig. 18
Fig. 18

Nonlinear feature extraction results by using the beam chain at scale 1 / 2 2 and the wavelet decomposition at scale 2 1 .

Fig. 19
Fig. 19

Nonlinear feature extraction results by using the beam chain at scale 1 / 2 3 and the wavelet transform modulus at scale 2 0 . We replace the wavelet decomposition at the scale 2 0 by the original gradient.

Fig. 20
Fig. 20

Final position of the virtual beam chain at the scale 1 / 2 3 .

Fig. 21
Fig. 21

Reconstructed points of the 3D curve.

Equations (27)

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

{ W x 2 j I ( x , y ) = I ( x , y ) * ψ x 2 j ( x , y ) W y 2 j I ( x , y ) = I ( x , y ) * ψ y 2 j ( x , y ) ,
{ ψ 1 ( x , y ) = θ ( x , y ) x ψ 2 ( x , y ) = θ ( x , y ) y .
M 2 j I ( x , y ) = | W x 2 j I ( x , y ) | 2 + | W y 2 j I ( x , y ) | 2 ,
A 2 j I ( x , y ) = arctan ( W y 2 j I ( x , y ) W x 2 j I ( x , y ) ) .
A ( X , Y ) = | I ( X , Y ) sin [ γ ( X , Y ) β ] | ,
W = 1 + cos ( k 1 N C 1 π ) 2 W v + W 0 ,
f ( x ) = W W A ( x , y ) D ( x , y ) d y ,
F i 2 = 1 L 0 L W W A ( x , y ) y ( L x ) d y d x + ( Δ X i sin β i j Δ Y i cos β i j ) L 2 0 L W W A ( x , y ) ( L x ) 2 d y d x ,
F i = F i 1 + F i 2 ,
{ F X i 1 ( Δ X i , Δ Y i ) + F X i 2 ( Δ X i , Δ Y i ) = 0 F Y i 1 ( Δ X i , Δ Y i ) + F Y i 2 ( Δ X i , Δ Y i ) = 0 ,
f ( θ ) = 0 θ R ( cos θ cos θ 0 ) R d θ .
sin θ θ cos θ 0 = 0.
{ Δ h 1 = R ( 1 cos θ 0 ) or Δ h 2 = R ( cos θ cos θ 0 ) .
{ Δ h 2 0 | Δ h 2 | | Δ h 1 | .
Δ e max = R ( cos θ sin θ θ ) .
{ sin θ θ θ 3 6 cos θ 1 θ 2 2 .
Δ e max = θ 2 3 R .
l < 2 3 K ,
l a b = 3 3 l .
[ x i y i z i ] = Rot ( X , φ ) Rot ( Y , θ ) ( [ x i y i z i ] [ x c y c z c ] ) ,
[ x i y i z i ] = Rot ( X , Δ φ ) Rot ( Y , Δ θ ) ( [ x i y i z i ] [ Δ x c Δ y c Δ z c ] ) ,
[ x i y i z i ] = [ x i Δ x c + Δ θ z i y i Δ y c Δ φ z i z i Δ z c Δ θ x i + Δ φ y i ] .
A = 1 n i = 1 n ( D i 2 D i 2 ¯ ) 2 ,
{ A Δ x c = 0 , A Δ y c = 0 A Δ θ = 0 , A Δ φ = 0 ,
{ i = 1 n a i c i + i = 1 n b i c i + 2 Δ x i = 1 n c i 2 2 Δ y c i = 1 n d i c i + 2 Δ θ i = 1 n e i c i 2 Δ φ i = 1 n f i c i = 0 i = 1 n a i d i + i = 1 n b i d i + 2 Δ x i = 1 n c i d i 2 Δ y c i = 1 n d i 2 + 2 Δ θ i = 1 n e i d i 2 Δ φ i = 1 n f i d i = 0 i = 1 n a i e i + i = 1 n b i e i + 2 Δ x i = 1 n c i e i 2 Δ y c i = 1 n d i e i + 2 Δ θ i = 1 n e i 2 2 Δ φ i = 1 n f i e i = 0 i = 1 n a i f i + i = 1 n b i f i + 2 Δ x i = 1 n c i f i 2 Δ y c i = 1 n d i f i + 2 Δ θ i = 1 n e i f i 2 Δ φ i = 1 n f i 2 = 0 ,
{ a i = x i 2 x i 2 ¯ , b i = y i 2 y i 2 ¯ c i = x i x i ¯ , d i = y i y i ¯ e i = x i z i x i z i ¯ , f i = y i z i y i z i ¯ .
R = 1 n i = 1 n x i 2 + y i 2 ,

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