Abstract

In this paper we investigate the behavior of various centroiding methods (weighted center of gravity, matched filtering, and correlation) classically used in Shack–Hartmann wavefront sensing when dealing with an elongated asymmetric spot. We study the impact of model errors on these centroiding methods at high signal-to-noise ratios, and, using a one-dimensional formalism, we show that the associated estimates all suffer from a bias uncorrelated with the actual spot displacement if its shape is not known precisely. Additionally, we show that the correlation method provides an estimate with a unitary gain whatever the parameters used, while the other two methods introduce a non-unitary gain in the estimation process. Finally, we show that the sampling of the spot structures after filtering by some convolution kernels is crucial to get an unbiased estimate of the spot displacement.

© 2010 Optical Society of America

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References

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  1. Proceedings of the First International Conference on Adaptive Optics for Extremely Large Telescopes, Y.Clénet, J.-M.Conan, T.Fusco, and G.Rousset, eds. (EDP Sciences, http://ao4elt.edpsciences.org/(2010).
    [PubMed]
  2. M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack–Hartmann wave-front sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004).
    [Crossref] [PubMed]
  3. L. A. Poyneer, “Scene-based Shack–Hartmann wave-front sensing: analysis and simulation,” Appl. Opt. 42, 5807–5815 (2003).
    [Crossref] [PubMed]
  4. L. Gilles and B. L. Ellerbroek, “Shack–Hartmann wavefront sensing with elongated sodium laser beacons: centroiding versus matched filtering,” Appl. Opt. 45, 6568–6576 (2006).
    [Crossref] [PubMed]
  5. S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
    [Crossref]
  6. R. Conan, O. Lardiere, G. Herriot, C. Bradley, and K. Jackson, “Experimental assessment of the matched filter for laser guide star wavefront sensing,” Appl. Opt. 48, 1198–1211 (2009).
    [Crossref]
  7. O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
    [Crossref]
  8. D. Gratadour, E. Gendron, and G. Rousset, “Symmetrically weighted center of gravity for Shack–Hartmann wavefront sensing on a laser guide star,” Proc. SPIE7736 (to be published).
  9. D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
    [Crossref]
  10. L. Gilles and B. L. Ellerbroek, “Constrained matched filtering for extended dynamic range and improved noise rejection for Shack–Hartmann wavefront sensing,” Opt. Lett. 33, 1159–1161 (2008).
    [Crossref] [PubMed]
  11. P. S. Argall, O. N. Vassiliev, R. J. Sica, and M. M. Mwangi, “Lidar measurements taken with a large-aperture liquid mirror. 2. Sodium resonance-fluorescence system,” Appl. Opt. 39, 2393–2400 (2000).
    [Crossref]
  12. S. J. Thomas, D. T. Gavel, and B. Kibrick, “Analysis of on-sky sodium profile data from lick observatory,” Appl. Opt. 49, 394–402 (2010).
    [Crossref] [PubMed]
  13. L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
    [Crossref]
  14. C. D’Orgeville, F. J. Rigaut, and B. L. Ellerbroek, “LGS AO photon return simulations and laser requirements for the Gemini LGS AO program,” Proc. SPIE 4007, 131–141 (2000).
    [Crossref]
  15. N. Védrenne, V. Michau, C. Robert, and J. M. Conan, “Shack–Hartmann wavefront estimation with extended sources: anisoplanatism influence,” J. Opt. Soc. Am. A 24, 2980–2993 (2007).
    [Crossref]
  16. D. Gratadour and F. J. Rigaut, “Online centroid gain determination for LGS AO systems,” in Adaptive Optics: Analysis and Methods, OSA 2007 Technical Digest Series (Optical Soceity of America, 2007), paper PMA4.

2010 (1)

2009 (3)

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

R. Conan, O. Lardiere, G. Herriot, C. Bradley, and K. Jackson, “Experimental assessment of the matched filter for laser guide star wavefront sensing,” Appl. Opt. 48, 1198–1211 (2009).
[Crossref]

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

2008 (2)

L. Gilles and B. L. Ellerbroek, “Constrained matched filtering for extended dynamic range and improved noise rejection for Shack–Hartmann wavefront sensing,” Opt. Lett. 33, 1159–1161 (2008).
[Crossref] [PubMed]

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

2007 (2)

N. Védrenne, V. Michau, C. Robert, and J. M. Conan, “Shack–Hartmann wavefront estimation with extended sources: anisoplanatism influence,” J. Opt. Soc. Am. A 24, 2980–2993 (2007).
[Crossref]

D. Gratadour and F. J. Rigaut, “Online centroid gain determination for LGS AO systems,” in Adaptive Optics: Analysis and Methods, OSA 2007 Technical Digest Series (Optical Soceity of America, 2007), paper PMA4.

2006 (1)

2005 (1)

D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
[Crossref]

2004 (1)

2003 (1)

2000 (2)

P. S. Argall, O. N. Vassiliev, R. J. Sica, and M. M. Mwangi, “Lidar measurements taken with a large-aperture liquid mirror. 2. Sodium resonance-fluorescence system,” Appl. Opt. 39, 2393–2400 (2000).
[Crossref]

C. D’Orgeville, F. J. Rigaut, and B. L. Ellerbroek, “LGS AO photon return simulations and laser requirements for the Gemini LGS AO program,” Proc. SPIE 4007, 131–141 (2000).
[Crossref]

Adkins, S.

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

Argall, P. S.

Bradley, C.

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

R. Conan, O. Lardiere, G. Herriot, C. Bradley, and K. Jackson, “Experimental assessment of the matched filter for laser guide star wavefront sensing,” Appl. Opt. 48, 1198–1211 (2009).
[Crossref]

Conan, J. M.

Conan, J. -M.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

Conan, R.

R. Conan, O. Lardiere, G. Herriot, C. Bradley, and K. Jackson, “Experimental assessment of the matched filter for laser guide star wavefront sensing,” Appl. Opt. 48, 1198–1211 (2009).
[Crossref]

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

D’Orgeville, C.

C. D’Orgeville, F. J. Rigaut, and B. L. Ellerbroek, “LGS AO photon return simulations and laser requirements for the Gemini LGS AO program,” Proc. SPIE 4007, 131–141 (2000).
[Crossref]

Diolaiti, E.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

Ellerbroek, B. L.

Foppiani, I.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

Fusco, T.

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack–Hartmann wave-front sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004).
[Crossref] [PubMed]

Gavel, D. T.

S. J. Thomas, D. T. Gavel, and B. Kibrick, “Analysis of on-sky sodium profile data from lick observatory,” Appl. Opt. 49, 394–402 (2010).
[Crossref] [PubMed]

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

Gendron, E.

D. Gratadour, E. Gendron, and G. Rousset, “Symmetrically weighted center of gravity for Shack–Hartmann wavefront sensing on a laser guide star,” Proc. SPIE7736 (to be published).

Gilles, L.

Gratadour, D.

D. Gratadour and F. J. Rigaut, “Online centroid gain determination for LGS AO systems,” in Adaptive Optics: Analysis and Methods, OSA 2007 Technical Digest Series (Optical Soceity of America, 2007), paper PMA4.

D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
[Crossref]

D. Gratadour, E. Gendron, and G. Rousset, “Symmetrically weighted center of gravity for Shack–Hartmann wavefront sensing on a laser guide star,” Proc. SPIE7736 (to be published).

Hampton, P.

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

Herriot, G.

Jackson, K.

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

R. Conan, O. Lardiere, G. Herriot, C. Bradley, and K. Jackson, “Experimental assessment of the matched filter for laser guide star wavefront sensing,” Appl. Opt. 48, 1198–1211 (2009).
[Crossref]

Kibrick, B.

Lardiere, O.

Lardière, O.

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

Lombini, M.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

Michau, V.

Mugnier, L. M.

D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
[Crossref]

Mwangi, M. M.

Nicolle, M.

Poyneer, L. A.

Rigaut, F. J.

D. Gratadour and F. J. Rigaut, “Online centroid gain determination for LGS AO systems,” in Adaptive Optics: Analysis and Methods, OSA 2007 Technical Digest Series (Optical Soceity of America, 2007), paper PMA4.

C. D’Orgeville, F. J. Rigaut, and B. L. Ellerbroek, “LGS AO photon return simulations and laser requirements for the Gemini LGS AO program,” Proc. SPIE 4007, 131–141 (2000).
[Crossref]

Robert, C.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

N. Védrenne, V. Michau, C. Robert, and J. M. Conan, “Shack–Hartmann wavefront estimation with extended sources: anisoplanatism influence,” J. Opt. Soc. Am. A 24, 2980–2993 (2007).
[Crossref]

Rouan, D.

D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
[Crossref]

Rousset, G.

M. Nicolle, T. Fusco, G. Rousset, and V. Michau, “Improvement of Shack–Hartmann wave-front sensor measurement for extreme adaptive optics,” Opt. Lett. 29, 2743–2745 (2004).
[Crossref] [PubMed]

D. Gratadour, E. Gendron, and G. Rousset, “Symmetrically weighted center of gravity for Shack–Hartmann wavefront sensing on a laser guide star,” Proc. SPIE7736 (to be published).

Schreiber, L.

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

Sica, R. J.

Thomas, S. J.

S. J. Thomas, D. T. Gavel, and B. Kibrick, “Analysis of on-sky sodium profile data from lick observatory,” Appl. Opt. 49, 394–402 (2010).
[Crossref] [PubMed]

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

Vassiliev, O. N.

Védrenne, N.

Appl. Opt. (5)

Astron. Astrophys. (1)

D. Gratadour, L. M. Mugnier, and D. Rouan, “Sub-pixel image registration with a maximum likelihood estimator. Application to the first adaptive optics observations of arp 220 in the l’ band,” Astron. Astrophys. 443, 357–365 (2005).
[Crossref]

J. Opt. Soc. Am. A (1)

Mon. Not. R. Astron. Soc. (3)

L. Schreiber, I. Foppiani, C. Robert, E. Diolaiti, J.-M. Conan, and M. Lombini, “Laser guide stars for extremely large telescopes: efficient Shack–Hartmann wavefront sensor design using the weighted centre-of-gravity algorithm,” Mon. Not. R. Astron. Soc. 396, 1513–1521 (2009).
[Crossref]

O. Lardière, R. Conan, C. Bradley, K. Jackson, and P. Hampton, “Radial thresholding to mitigate laser guide star aberrations on centre-of-gravity-based Shack–Hartmann wavefront sensors,” Mon. Not. R. Astron. Soc. 398, 1461–1467 (2009).
[Crossref]

S. J. Thomas, S. Adkins, D. T. Gavel, T. Fusco, and V. Michau, “Study of optimal wavefront sensing with elongated laser guide stars,” Mon. Not. R. Astron. Soc. 387, 173–187 (2008).
[Crossref]

Opt. Lett. (2)

Proc. SPIE (1)

C. D’Orgeville, F. J. Rigaut, and B. L. Ellerbroek, “LGS AO photon return simulations and laser requirements for the Gemini LGS AO program,” Proc. SPIE 4007, 131–141 (2000).
[Crossref]

Other (3)

D. Gratadour and F. J. Rigaut, “Online centroid gain determination for LGS AO systems,” in Adaptive Optics: Analysis and Methods, OSA 2007 Technical Digest Series (Optical Soceity of America, 2007), paper PMA4.

D. Gratadour, E. Gendron, and G. Rousset, “Symmetrically weighted center of gravity for Shack–Hartmann wavefront sensing on a laser guide star,” Proc. SPIE7736 (to be published).

Proceedings of the First International Conference on Adaptive Optics for Extremely Large Telescopes, Y.Clénet, J.-M.Conan, T.Fusco, and G.Rousset, eds. (EDP Sciences, http://ao4elt.edpsciences.org/(2010).
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

Whole LIDAR sequence (left) and average sodium layer profile (right) from the data set of the University of Western Ontario (black curve) along with several synthetic profiles: a single Gaussian centered at 90 km and with a FWHM of 9.4 km (red curve); an asymmetric double Gaussian centered at 87 and 92 km both with a FWHM of 4.7 km (blue curve); and a triple Gaussian centered at 85, 88, and 92 km with FWHMs of 3.5, 4.7, and 4.7 km, respectively (green curve).

Fig. 2
Fig. 2

Images illustrating the various steps in the computation of the spot image from a specified profile. From left to right, top to bottom: the profile (one of the individual profiles of the previously presented sequence), the high resolution version of the object (i.e., the projected spot profile convolved by the beam function and the upward turbulence), the turbulence PSF, the final high resolution image of the spot, and the degraded version of this image to the resolution of the WFS sub-apertures.

Fig. 3
Fig. 3

Error in nanometer RMS against pixel scale in arcseconds when using the WCOG for centroid estimation on an on-axis spot with the simulation parameters from Table 1 and additional photon and readout noise as described in Subsection 3C.

Fig. 4
Fig. 4

Multipanel figure showing the behavior of each method (from left to right: WCOG, correlation, and constrained MF) when the spot image has been simulated using the average sodium profile and the reference function has been created using the average sodium profile itself (top panel) or a single Gaussian (bottom panel). The measure spot displacement values in the non-elongated direction (N-El, light blue) and in the elongated direction without correction (El-NC, black) and corrected from gains and biases (El-C, red) are plotted against the true values all given in arcseconds.

Fig. 5
Fig. 5

Results of the simulation described in Section 4. Left, case of the average profile from experimental data and right, case of the triple Gaussian profile. Light blue (N.El), results in the non-elongated dimension; black (El-NC), results in the elongated dimension.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (37)

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

w = 1 D + x g ( x ) i ( x + δ x ) d x ,
D = + g ( x ) i ( x + δ x ) d x .
+ x g ( x ) d x = + x i ( x ) d x = 0.
w = 1 D + x g ( x ) i ( x ) d x + δ x 1 D + x g ( x ) d i d x ( x ) d x ,
D = + g ( x ) i ( x ) d x + δ x + g ( x ) d i d x ( x ) d x ,
c ( u ) = + g ( x ) i ( x + u + δ x ) d x ,
d c d u ( u 0 ) = 0 ,
u 0 δ x .
d c d u ( u ) = + g ( x ) d i d u ( x + u + δ x ) d x = + d g d x ( x ) i ( x + u + δ x ) d x .
d c d u ( u 0 ) = 0 = + d g d x ( x ) i ( x ) d x ( u 0 + δ x ) + d g d x ( x ) d i d x ( x ) d x ,
u 0 = 1 D + d g d x ( x ) i ( x ) d x δ x ,
D = + d g d x ( x ) d i d x ( x ) d x ,
i ( x + δ x ) g ( x + δ x ) g ( x ) + δ x d g d x ( x ) ,
m = + [ i ( x + δ x ) g ( x ) ] ( d g d x ( x ) ) 1 d x .
m c = 1 D + d g d x ( x ) i ( x + δ x ) d x ,
D = + [ d g d x ( x ) ] 2 d x .
m c = B ( g , d g d x , i ) + δ x G ( g , d g d x , d i d x ) ,
B ( g , d g d x , i ) = 1 D + d g d x ( x ) i ( x ) d x ,
G ( g , d g d x , d i d x ) = 1 D + d g d x ( x ) d i d x ( x ) d x .
+ d g d x ( x ) i ( x ) d x = FT ( d g d x i ) ( f = 0 ) = FT ( d g d x ) FT ( i ) ( f = 0 ) .
G ( f ) = | G ( f ) | exp ( j ϕ G ( f ) ) ,
+ d g d x ( x ) i ( x ) d x = + 2 π j f | G ( f ) | | I ( f ) | exp ( j ϕ G ( f ) + j ϕ I ( f ) ) d f .
+ d g d x ( x ) i ( x ) d x = 2 π + f | G ( f ) | | I ( f ) | sin ( ϕ G ( f ) ϕ I ( f ) ) d f
+ d g d x ( x ) d i d x ( x ) d x = 4 π 2 + f 2 | G ( f ) | | I ( f ) | cos ( ϕ G ( f ) ϕ I ( f ) ) d f .
+ x g ( x ) d i d x ( x ) d x = + f | d G d f ( f ) | | I ( f ) | sin ( π 2 + ϕ d G / d f ( f ) ϕ I ( f ) ) d f ,
+ x g ( x ) i ( x ) d x = 1 2 π + f | d G d f ( f ) | | I ( f ) | sin ( π 2 + ϕ d G / d f ( f ) ϕ I ( f ) ) d f ,
+ g ( x ) i ( x ) d x = + | G ( f ) | | I ( f ) | cos ( ϕ G ( f ) ϕ I ( f ) ) d f ,
+ g ( x ) d i d x ( x ) d x = 2 π + f | G ( f ) | | I ( f ) | sin ( ϕ G ( f ) ϕ I ( f ) ) d f .
i ( x + δ x ) g ( x + δ x ) g ( x ) + δ x d g d x ( x ) .
G = [ d g d x ( x ) , g ( x ) ] .
G t G = ( [ d g d x ( x ) ] 2 [ d g d x ( x ) g ( x ) ] [ d g d x ( x ) g ( x ) ] [ g ( x ) ] 2 ) .
G t G = ( [ d g d x ( x ) ] 2 0 0 [ g ( x ) ] 2 ) ,
( G t G ) 1 = 1 det ( G t G ) ( [ g ( x ) ] 2 0 0 [ d g d x ( x ) ] 2 ) .
F C M F = M ( G t G ) 1 G t ,     M = [ 1 , 0 ] ,
F C M F = [ 1 , 0 ] 1 det ( G t G ) ( [ g ( x ) ] 2 d g d x ( x ) [ d g d x ( x ) ] 2 g ( x ) ) ,
F C M F = 1 det ( G t G ) [ ( [ g ( x ) ] 2 d x ) d g d x ( x ) ] .
m c = F C M F [ i ( x + δ x ) ] ,

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