Abstract

A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.

© 2016 Optical Society of America

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References

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  1. W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
    [Crossref]
  2. S. Flöry, “Fitting curves and surfaces to point clouds in the presence of obstacles,” Comput. Aided Geom. Des. 26(2), 192–202 (2009).
    [Crossref]
  3. F. Remondino, “From point cloud to surface: the modelling and visualization problem,” in International Workshop on Visualization and Animation of Reality-Based 3D Models, http://dx.doi.org/10.3929/ethz-a-004655782 (2003).
  4. J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974).
    [Crossref]
  5. P. Dierkx, Curve and Surface Fitting with Splines (Oxford University, 1993).
  6. P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
    [Crossref]
  7. M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. 45(27), 6954–6964 (2006).
    [Crossref] [PubMed]
  8. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
    [Crossref] [PubMed]
  9. M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003).
    [Crossref] [PubMed]
  10. V. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981).
    [Crossref]
  11. J. A. Díaz and R. Navarro, “Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation,” Appl. Opt. 53(10), 2051–2057 (2014).
    [Crossref] [PubMed]
  12. G. M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier Transform,” Open Opt. J. 1(1), 1–3 (2007).
    [Crossref]
  13. K. P. Murphy, Machine Learning a Probabilistic Perspective (Massachusetts Institute of Technology, 2012).
  14. A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970).
    [Crossref]
  15. J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE, 2004).
  16. M. M. Lipschutz, Differential Geometry (McGraw Hill, 1970).
  17. D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
    [Crossref]
  18. J. Yao, J. Huang, P. Meemon, M. Ponting, and J. P. Rolland, “Simultaneous estimation of thickness and refractive index of layered gradient refractive index optics using a hybrid confocal-scan swept-source optical coherence tomography system,” Opt. Express 23(23), 30149–30164 (2015).
    [Crossref] [PubMed]
  19. J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
    [Crossref]
  20. J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
    [Crossref]

2015 (3)

J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
[Crossref]

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

J. Yao, J. Huang, P. Meemon, M. Ponting, and J. P. Rolland, “Simultaneous estimation of thickness and refractive index of layered gradient refractive index optics using a hybrid confocal-scan swept-source optical coherence tomography system,” Opt. Express 23(23), 30149–30164 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (1)

D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
[Crossref]

2012 (1)

P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
[Crossref]

2009 (1)

S. Flöry, “Fitting curves and surfaces to point clouds in the presence of obstacles,” Comput. Aided Geom. Des. 26(2), 192–202 (2009).
[Crossref]

2007 (1)

G. M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier Transform,” Open Opt. J. 1(1), 1–3 (2007).
[Crossref]

2006 (2)

W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
[Crossref]

M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. 45(27), 6954–6964 (2006).
[Crossref] [PubMed]

2003 (1)

M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003).
[Crossref] [PubMed]

2001 (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

1981 (1)

1974 (1)

J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974).
[Crossref]

1970 (1)

A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970).
[Crossref]

Alonso, J.

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
[Crossref]

Ares, M.

Bo, P.

P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
[Crossref]

Collins, M. J.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Dai, G. M.

G. M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier Transform,” Open Opt. J. 1(1), 1–3 (2007).
[Crossref]

Davis, B.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Díaz, J. A.

Flöry, S.

S. Flöry, “Fitting curves and surfaces to point clouds in the presence of obstacles,” Comput. Aided Geom. Des. 26(2), 192–202 (2009).
[Crossref]

Gómez-Pedrero, J. A.

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

Halliday, J.

J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974).
[Crossref]

Hayes, J. G.

J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974).
[Crossref]

Hoerl, A. E.

A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970).
[Crossref]

Huang, J.

Iskander, D. R.

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

Jiang, X.

J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
[Crossref]

Kennard, R. W.

A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970).
[Crossref]

Klyce, S. D.

M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003).
[Crossref] [PubMed]

Leach, R. K.

J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
[Crossref]

Ling, R.

P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
[Crossref]

Liu, Y.

W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
[Crossref]

Mahajan, V.

Meemon, P.

Navarro, R.

Ponting, M.

Pottmann, H.

W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
[Crossref]

Quirgoa, J. A.

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

Quiroga, J. A.

D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
[Crossref]

Rodriguez-Ibañez, D.

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

Rodríguez-Ibáñez, D.

D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
[Crossref]

Rolland, J. P.

Royo, S.

Smolek, M. K.

M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003).
[Crossref] [PubMed]

Wang, J.

J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
[Crossref]

Wang, W.

P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
[Crossref]

W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
[Crossref]

Yao, J.

ACM Trans. Graph. (1)

W. Wang, H. Pottmann, and Y. Liu, “Fitting B-spline curves to point clouds by curvature-based squared distance minimization,” ACM Trans. Graph. 25(2), 214–238 (2006).
[Crossref]

Appl. Opt. (2)

Comput. Aided Geom. Des. (1)

S. Flöry, “Fitting curves and surfaces to point clouds in the presence of obstacles,” Comput. Aided Geom. Des. 26(2), 192–202 (2009).
[Crossref]

Comput. Graph. (1)

P. Bo, R. Ling, and W. Wang, “A revisit to fitting parametric surfaces to point cloud,” Comput. Graph. 36(5), 534–540 (2012).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001).
[Crossref] [PubMed]

IMA J. Appl. Math. (1)

J. G. Hayes and J. Halliday, “The least-squares fitting of cubic spline surfaces to general data sets,” IMA J. Appl. Math. 14(1), 89–103 (1974).
[Crossref]

Int. J. Adv. Manuf. Technol. (1)

D. Rodríguez-Ibáñez, J. Alonso, and J. A. Quiroga, “Squareness error calibration of a CMM for quality control of ophthalmic lenses,” Int. J. Adv. Manuf. Technol. 68(1-4), 487–493 (2013).
[Crossref]

Invest. Ophthalmol. Vis. Sci. (1)

M. K. Smolek and S. D. Klyce, “Zernike polynomial fitting fails to represent all visually significant corneal aberrations,” Invest. Ophthalmol. Vis. Sci. 44(11), 4676–4681 (2003).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

Open Opt. J. (1)

G. M. Dai, “Wavefront reconstruction from slope data within pupils of arbitrary shapes using iterative Fourier Transform,” Open Opt. J. 1(1), 1–3 (2007).
[Crossref]

Opt. Express (1)

Proc. SPIE (1)

J. A. Gómez-Pedrero, D. Rodriguez-Ibañez, J. Alonso, and J. A. Quirgoa, “Design and development of a profilometer for the fast and accurate characterization of optical surfaces,” Proc. SPIE 9628, 96281I (2015).
[Crossref]

Surf. Topogr.: Metrol. Prop. (1)

J. Wang, R. K. Leach, and X. Jiang, “Review of the mathematical foundations of data fusion techniques in surface metrology,” Surf. Topogr.: Metrol. Prop. 3(2), 023001 (2015).
[Crossref]

Technometrics (1)

A. E. Hoerl and R. W. Kennard, “Ridge regression: biased estimation for nonorthogonal problems,” Technometrics 12(1), 55–67 (1970).
[Crossref]

Other (5)

J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE, 2004).

M. M. Lipschutz, Differential Geometry (McGraw Hill, 1970).

K. P. Murphy, Machine Learning a Probabilistic Perspective (Massachusetts Institute of Technology, 2012).

P. Dierkx, Curve and Surface Fitting with Splines (Oxford University, 1993).

F. Remondino, “From point cloud to surface: the modelling and visualization problem,” in International Workshop on Visualization and Animation of Reality-Based 3D Models, http://dx.doi.org/10.3929/ethz-a-004655782 (2003).

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Figures (9)

Fig. 1
Fig. 1 Map of the elevation of the theoretical surface generated together with the polygonal path (white line with circular dots) which defines the region of interest (ROI) where the cloud of points is defined.
Fig. 2
Fig. 2 a) Plot of the values of the first 50 coefficients corresponding to the standard fit (with λ = 0 and μ = 0) of a Zernike polynomial series to the theoretical surface of Fig. 1), defined over the entire circular region, b) values of the coefficients corresponding ot the fit of the same surface but defined over the ROI delimited by the white line of Fig. 1). In both plots the piston term have been removed for clarity.
Fig. 3
Fig. 3 a) Plot of the elevation error versus the regularization parameter λ for three different levels of noise added to the theoretical surface, b) plot of the elevation error versus the curvature smoothing parameter μ for the same noise levels and λ = 10−6. These plots are useful in order to get the optimum values of the parameters λ and μ.
Fig. 4
Fig. 4 a) Map of the elevation error along the lens surface obtained for the first training set, i.e., setting λ = 10−6 and μ = 0 and b) elevation error map obtained for the extended training set with λ = 10−6 and μ = 3·10−5. In both cases the elevation error is lower than 1 micron for the ROI. The ROI is limited by the white line and the red line delimits an inner region for computing the statistical magnitudes (mean and standard deviation) of the fitting residues. The green area outside the ROI represents the circular domain where the Zernike polynomials are defined.
Fig. 5
Fig. 5 a) Map of the spherical error in the ROI obtained for a theoretical progressive surface by setting λ = 10−6 and μ = 0 (original training set), b) map of the equivalent spherical error for the same surface obtained using the extended training set (λ = 10−6 and μ = 3·10−5), c) map of the cylindrical error obtained with the original training set and d) map of the cylindrical error obtained with the extended training set. The effect of the training set extension can be noticed as a reduction on the errors thorough the whole ROI particularly for the spherical equivalent power.
Fig. 6
Fig. 6 a) Plot of the elevation error recovered for different values of the standard deviation of the added Gaussian noise, b) plot of the spherical error and c) cylindrical error versus Gaussian noise. We have computed the value of error for three cases: no regularization and no curvature smoothing (blue curve), regularization but no curvature smoothing (red curve) and regularization and curvature smoothing (yellow curve).
Fig. 7
Fig. 7 a) Elevation map measured with the CMM of a progressive surface. b) Plot of the 50 first coefficients of the Zernike polynomial expansion corresponding to the surface defined in Fig. 7(a) for the whole circular domain. c) Equivalent sphere and b) cylindrical refractive power maps computed for the reference surface from the elevation map of Fig. 7(a). As in former cases, we have defined a ROI region whose border is given by the white line with circular markers.
Fig. 8
Fig. 8 a) Distribution of the elevation error without curvature smoothing and b) with curvature smoothing, c) difference of the equivalent spherical refractive power between the measured and reconstructed surfaces with no curvature smoothing, d) Same difference as in case c) but with curvature smoothing. Difference between the measured and reconstructed cylindrical refractive power e) with no curvature smoothing and f) with curvature smoothing.
Fig. 9
Fig. 9 Plot of the values of the first 50 coefficients corresponding to the fit of the surface to a Zernike polynomial series to the measured surface within the non regular region delimited by the white line in Figs. 7) and 8). The plot of panel a) correspond to the fitting with no curvature smoothing while that of panel b) corresponds to the fitting with curvature smoothing. In both plots the piston term have been removed for clarity.

Equations (22)

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

y (i) =h( x (i) ).
h( x (i) ,θ )= j=0 N θ j x j (i) x (i) θ .
J(θ)= 1 2M i=1 M ( y (i) h( x (i) ,θ ) ) 2 +λ j=1 N θ j ,
θ opt = ( λ I D + X T X ) 1 · X T · y T ,
X=[ x (1) x (2) x (M) ],
I D =[ 0 0 0 0 1 0 0 0 1 ].
θ opt = θ ls λW θ ls ,
Form>0 Z n m ( ξ,η )= 2(n+1) s=0 n| m | 2 j=0 n| m | 2 s k=0 n| m | 2 ( 1 ) s+k ( ns )! s![ n+| m | 2 s ]![ n+| m | 2 s ]! ( nm 2 s j )( m 2k ) ξ n2( s+j+k ) η 2(j+k) Form<0 Z n m ( ξ,η )= 2(n+1) s=0 n| m | 2 j=0 n| m | 2 s k=0 n| m | 2 ( 1 ) s+k ( ns )! s![ n+| m | 2 s ]![ n+| m | 2 s ]! ( nm 2 s j )( m 2k+1 ) ξ n2( s+j+k ) η 2(j+k)+1 Form=0 Z n 0 ( ξ,η )= n+1 s=0 n/2 j=0 n/2 s ( 1 ) s ( ns )! s!( n 2 s )!( n 2 s )! ( n 2 s j ) ξ n2( s+j ) η 2j .
P ^ (i) =( ξ (i) R , η (i) R ),
x (i) = ( Z 0 ( P ^ (i) ), Z 1 ( P ^ (i) ), Z 2 ( P ^ (i) ),, Z N ( P ^ (i) ) ) i=1,2,,M ,
h( x (i) ,θ )= θ 0 Z 0 ( P ^ (i) )+ θ 1 Z 1 ( P ^ (i) )+...+ θ N Z N ( P ^ (i) ),
κ 1 =H H 2 K , κ 2 =H+ H 2 K .
H= ( 1+ ( s x ) 2 ) 2 s x 2 2 s x s y 2 s xy +( 1+ ( s y ) 2 ) 2 s y 2 ( 1+ ( s x ) 2 + ( s y ) 2 ) 3/2 , K= 2 s x 2 2 s y 2 ( 2 s xy ) 2 1+ ( s x ) 2 + ( s y ) 2 .
( 2 s( x,y ) x 2 )=0, ( 2 s( x,y ) y 2 )=0,
P ^ ' (i) = ( ξ ' (i) R , η ' (i) R ) i=1,2,,M' .
x ' (i) = ( 3 Z 0 ( P ^ ' (i) ) x 3 , 3 Z 1 ( P ^ ' (i) ) x 3 ,, 3 Z N ( P ^ ' (i) ) x 3 ) i=1,2,,M' , x' ' (i) = ( 3 Z 0 ( P ^ ' (i) ) x y 2 , 3 Z 1 ( P ^ ' (i) ) x y 2 ,, 3 Z N ( P ^ ' (i) ) x y 2 ) i=1,2,,M' , x'' ' (i) = ( 3 Z 0 ( P ^ ' (i) ) y x 2 , 3 Z 1 ( P ^ ' (i) ) y x 2 ,, 3 Z N ( P ^ ' (i) ) y x 2 ) i=1,2,,M' , x''' ' (i) = ( 3 Z 0 ( P ^ ' (i) ) y 3 , 3 Z 1 ( P ^ ' (i) ) y 3 ,, 3 Z N ( P ^ ' (i) ) y 3 ) i=1,2,,M' ,
X ext ={ x (1) ,, x (M) ,μx ' (1) ,,μx ' (M') ,μx' ' (1) ,,μx' ' (M') , μx'' ' (1) ,,μx'' ' (M') ,μx''' ' (1) ,,μx''' ' (M') },
y ext =[ y 0 1×M' 0 1×M' 0 1×M' 0 1×M' ],
X ext ( μ )=[ x (1) x (M) μx ' (1) μx ' (M') μx' ' (1) μx' ' (M') μx'' ' (1) μx'' ' (M') μx''' ' (1) μx''' ' (M') ],
θ opt ( λ,μ )= ( λ I D + X ext T ( μ ) X ext ( μ ) ) 1 X ext T ( μ )· y ext T ,
S=( n1 )( κ 2 + κ 1 )/2,
C=( n1 )( κ 2 κ 1 ),

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