Abstract

In optical shape measurement systems, systematic errors appear as a result of imaging aberrations of the lens assemblies in the cameras and projectors. A mathematical description of this effect is intended to correct the whole measurement area with a few independent coefficients. We apply the ideas of photogrammetry to one- and two-dimensional fringe projection techniques. We also introduce some new terms for close-range applications and telecentric objectives. Further, an algorithm for distance-dependent corrections is introduced. Also, we describe a new method with which to determine coefficients of aberration with an optimization-based method.

© 2004 Optical Society of America

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References

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  1. M. A. R. Cooper, S. Robson, “Theory of close range photogrammetry,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 9–22.
  2. K. Kraus, Photogrammetrie (de Gruyter, Bonn, Germany, 1982), Vols. 1 and 2.
  3. G. Konecny, G. Lehmann, Photogrammetrie, 4th ed. (de Gruyter, Berlin, 1984).
  4. W. Schreiber, G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159–169 (2000).
    [CrossRef]
  5. T. Luhmann, Nahbereichsphotogrammetrie (Wichmann, Heidelberg, Germany, 2000).
  6. V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
    [CrossRef]
  7. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis—Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), pp. 95–140.
  8. G. Wiora, R. Malz, U. Laukenmann, “Robustes erweitertes Kalibrier- und Messverfahren für Streifenprojektionssysteme mit metrischem Projektor,” in 6. Symposium Bildverarbeitung 1999, R. Ahlers, ed. (Technische Akademie Esslingen, Esslingen, 1999), Vol. 6, pp. 215–230.
  9. S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).
  10. W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
    [CrossRef]
  11. R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
    [CrossRef]
  12. J. G. Fryer, “Camera calibration,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 156–164.
  13. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).
  14. F. L. Pedrotti, L. S. Pedrotti, Optics and Vision (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  15. H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag, Berlin, 1986).
  16. D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 38, 855–866 (1971).
  17. P. J. Scott, “The pupil in perspective,” Photogrammetric Record 49, 83–92 (1977).
  18. A. A. Magill, “Variation in distortion with magnification,” J. Res. Nat. Bureau Stand. U.S. 54, 135–142 (1955).
    [CrossRef]
  19. R. Schuhmann, T. Thöniss, “Telezentrische Systeme für die optische Mess- und Prüftechnik,” Technisches Messen 65, 131–136 (1998).
  20. W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.
  21. W. Wester-Ebbinghaus, “Bündeltriangulation mit gemeinsamer Ausgleichung photogrammetrischer und geodätischer Beobachtungen,” Zeitschrift für Vermessungswesen 3, 101–110 (1985).
  22. J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

2000 (2)

W. Schreiber, G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159–169 (2000).
[CrossRef]

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

1999 (1)

S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).

1998 (1)

R. Schuhmann, T. Thöniss, “Telezentrische Systeme für die optische Mess- und Prüftechnik,” Technisches Messen 65, 131–136 (1998).

1985 (1)

W. Wester-Ebbinghaus, “Bündeltriangulation mit gemeinsamer Ausgleichung photogrammetrischer und geodätischer Beobachtungen,” Zeitschrift für Vermessungswesen 3, 101–110 (1985).

1977 (1)

P. J. Scott, “The pupil in perspective,” Photogrammetric Record 49, 83–92 (1977).

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 38, 855–866 (1971).

1955 (1)

A. A. Magill, “Variation in distortion with magnification,” J. Res. Nat. Bureau Stand. U.S. 54, 135–142 (1955).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Brown, D. C.

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 38, 855–866 (1971).

Cooper, M. A. R.

M. A. R. Cooper, S. Robson, “Theory of close range photogrammetry,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 9–22.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis—Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), pp. 95–140.

de Freitas Caetano, F.

S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).

Dennis, J. E.

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Fryer, J. G.

J. G. Fryer, “Camera calibration,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 156–164.

Gerber, J.

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

Haferkorn, H.

H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag, Berlin, 1986).

Kirschner, V.

W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.

V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
[CrossRef]

Konecny, G.

G. Konecny, G. Lehmann, Photogrammetrie, 4th ed. (de Gruyter, Berlin, 1984).

Kowarschik, R.

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
[CrossRef]

W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.

Kraus, K.

K. Kraus, Photogrammetrie (de Gruyter, Bonn, Germany, 1982), Vols. 1 and 2.

Kühmstedt, P.

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

Laukenmann, U.

G. Wiora, R. Malz, U. Laukenmann, “Robustes erweitertes Kalibrier- und Messverfahren für Streifenprojektionssysteme mit metrischem Projektor,” in 6. Symposium Bildverarbeitung 1999, R. Ahlers, ed. (Technische Akademie Esslingen, Esslingen, 1999), Vol. 6, pp. 215–230.

Lehmann, G.

G. Konecny, G. Lehmann, Photogrammetrie, 4th ed. (de Gruyter, Berlin, 1984).

Luhmann, T.

T. Luhmann, Nahbereichsphotogrammetrie (Wichmann, Heidelberg, Germany, 2000).

Magill, A. A.

A. A. Magill, “Variation in distortion with magnification,” J. Res. Nat. Bureau Stand. U.S. 54, 135–142 (1955).
[CrossRef]

Malz, R.

G. Wiora, R. Malz, U. Laukenmann, “Robustes erweitertes Kalibrier- und Messverfahren für Streifenprojektionssysteme mit metrischem Projektor,” in 6. Symposium Bildverarbeitung 1999, R. Ahlers, ed. (Technische Akademie Esslingen, Esslingen, 1999), Vol. 6, pp. 215–230.

Notni, G.

W. Schreiber, G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159–169 (2000).
[CrossRef]

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
[CrossRef]

W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Optics and Vision (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Optics and Vision (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Robson, S.

M. A. R. Cooper, S. Robson, “Theory of close range photogrammetry,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 9–22.

Schreiber, W.

W. Schreiber, G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159–169 (2000).
[CrossRef]

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
[CrossRef]

W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.

Schuhmann, R.

R. Schuhmann, T. Thöniss, “Telezentrische Systeme für die optische Mess- und Prüftechnik,” Technisches Messen 65, 131–136 (1998).

Scott, P. J.

P. J. Scott, “The pupil in perspective,” Photogrammetric Record 49, 83–92 (1977).

Thöniss, T.

R. Schuhmann, T. Thöniss, “Telezentrische Systeme für die optische Mess- und Prüftechnik,” Technisches Messen 65, 131–136 (1998).

Viana, S. A. A.

S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).

Waldmann, J.

S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).

Wester-Ebbinghaus, W.

W. Wester-Ebbinghaus, “Bündeltriangulation mit gemeinsamer Ausgleichung photogrammetrischer und geodätischer Beobachtungen,” Zeitschrift für Vermessungswesen 3, 101–110 (1985).

Wiora, G.

G. Wiora, R. Malz, U. Laukenmann, “Robustes erweitertes Kalibrier- und Messverfahren für Streifenprojektionssysteme mit metrischem Projektor,” in 6. Symposium Bildverarbeitung 1999, R. Ahlers, ed. (Technische Akademie Esslingen, Esslingen, 1999), Vol. 6, pp. 215–230.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

J. Res. Nat. Bureau Stand. U.S. (1)

A. A. Magill, “Variation in distortion with magnification,” J. Res. Nat. Bureau Stand. U.S. 54, 135–142 (1955).
[CrossRef]

Opt. Eng. (2)

R. Kowarschik, P. Kühmstedt, J. Gerber, W. Schreiber, G. Notni, “Adaptive optical three-dimensional measurement with structured light,” Opt. Eng. 39, 150–158 (2000).
[CrossRef]

W. Schreiber, G. Notni, “Theory and arrangements of self-calibrating whole-body three-dimensional measurement systems using fringe projection technique,” Opt. Eng. 39, 159–169 (2000).
[CrossRef]

Photogram. Eng. (1)

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 38, 855–866 (1971).

Photogrammetric Record (1)

P. J. Scott, “The pupil in perspective,” Photogrammetric Record 49, 83–92 (1977).

Sociedade Brasileira de Automática Controle Automação (1)

S. A. A. Viana, J. Waldmann, F. de Freitas Caetano, “Non-linear optimization-based batch calibration with accuracy evaluation,” Sociedade Brasileira de Automática Controle Automação 10, 89–99 (1999).

Technisches Messen (1)

R. Schuhmann, T. Thöniss, “Telezentrische Systeme für die optische Mess- und Prüftechnik,” Technisches Messen 65, 131–136 (1998).

Zeitschrift für Vermessungswesen (1)

W. Wester-Ebbinghaus, “Bündeltriangulation mit gemeinsamer Ausgleichung photogrammetrischer und geodätischer Beobachtungen,” Zeitschrift für Vermessungswesen 3, 101–110 (1985).

Other (14)

J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

W. Schreiber, V. Kirschner, R. Kowarschik, G. Notni, “Managing some calibration problems in fringe projection shape measurement systems,” in Fringe ’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 443–450.

J. G. Fryer, “Camera calibration,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 156–164.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

F. L. Pedrotti, L. S. Pedrotti, Optics and Vision (Prentice-Hall, Englewood Cliffs, N.J., 1993).

H. Haferkorn, Bewertung Optischer Systeme (VEB Deutscher Verlag, Berlin, 1986).

W. Schreiber, G. Notni, P. Kühmstedt, J. Gerber, R. Kowarschik, “Optical 3D-coordinate measuring system using structured light,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 620–627 (1996).
[CrossRef]

M. A. R. Cooper, S. Robson, “Theory of close range photogrammetry,” in Close Range Photogrammetry and Machine Vision, K. B. Atkinson, ed. (Whittles, Caithness, Scotland, 1996), pp. 9–22.

K. Kraus, Photogrammetrie (de Gruyter, Bonn, Germany, 1982), Vols. 1 and 2.

G. Konecny, G. Lehmann, Photogrammetrie, 4th ed. (de Gruyter, Berlin, 1984).

T. Luhmann, Nahbereichsphotogrammetrie (Wichmann, Heidelberg, Germany, 2000).

V. Kirschner, W. Schreiber, R. Kowarschik, G. Notni, “Self-calibrating shape-measuring system based on fringe projection,” in Rapid Prototyping and Flexible Manufacturing, R.-J. Ahlers, G. Reinhart, eds., Proc. SPIE3102, 5–13 (1997).
[CrossRef]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis—Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Bristol, UK, 1993), pp. 95–140.

G. Wiora, R. Malz, U. Laukenmann, “Robustes erweitertes Kalibrier- und Messverfahren für Streifenprojektionssysteme mit metrischem Projektor,” in 6. Symposium Bildverarbeitung 1999, R. Ahlers, ed. (Technische Akademie Esslingen, Esslingen, 1999), Vol. 6, pp. 215–230.

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Figures (4)

Fig. 1
Fig. 1

Illustration of a changed image of projected fringes. The image was changed by image aberrations in both the projection and the camera lens assemblies.

Fig. 2
Fig. 2

Top, principle of the triangulation method (shown for 2D case). Object point A and two stations P1,2 form a triangle. The position of A is defined when two angles (α1, α2) and the baseline are known. Bottom, the measurement of angles is often replaced by the measurement of distances ξ in cameras or projectors, for which angles β1 and β2 and principal distance c to the baseline must be known.

Fig. 3
Fig. 3

Illustration for 1D (left) and 2D (right) fringe projection. One obtains one or two measurement values from the actual direction of view of each object point.

Fig. 4
Fig. 4

Image formation in telecentric systems. Illustration of the error that occurs if the aperture stop is not situated in the focal plane. Top, common image formation of an object in two different distances. Middle, principle of a telecentric objective: The aperture stop is in the back focal plane. The shift of the object from 3f to 2f leads to a little unsharpness in the image plane but not to a change of the image size. Bottom, the aperture stop is displaced from the focal plane, which leads to a change in image size in the image planes of objects that are not located in 3f.

Tables (1)

Tables Icon

Table 1 Aberration Coefficients and Related Types of Seidel Aberrationsa

Equations (25)

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

ξij-ξPηij-ηP-c=1κijRjTxi-xjPyi-yjPzi-zjP,
ξij=ξP-c×r11jxi-xjP+r21jyi-yjP+r31jzi-zjPr13jxi-xjP+r23jyi-yjP+r33jzi-zjP,
ηij=ηP-c×r12jxi-xjP+r22jyi-yjP+r32jzi-zjPr13jxi-xjP+r23jyi-yjP+r33jzi-zjP.
Φ˜ξ=2πΛ Φξ,
Φξcorr=Φξmeas-δΦξ,Φηcorr=Φηmeas-δΦη,
δΦξ=Φξmeas-Φξcorr=cn1dqdQx,δΦη=Φηmeas-Φηcorr=cn1dqdQy,
q=AlnmΦη2l+mρn cosm θ,
q=A111Φη3ρ cos θ+A120Φη2ρ2+A022Φη2ρ2 cos2 θ+A031Φηρ3 cos θ+A040ρ4,
ρ=Qx2+Qy21/2,
cos θ=Qy/ρ.
δΦηdqdQy=qρρQy.
ρQy=Qyρ=cos θ,
δΦη=cn1A111Φη3+A120Φη22ρ cos θ+A022Φη22ρ cos θ+A031Φηρ2cos 2θ+1+A0404ρ3 cos θ,
δΦη=Br3 cos θ=Br2Φη
δΦη=Φηδrr.
q=AlnmΦη2l+mRnmrcos mθ.
q=A000+A111r cos θ+A120122r2-1+A022r22 cos2 θ-1+A0313r3-2rcos θ+A040126r4-6r2+1.
δr=γ2B2r2+γ4B4r4+,
γ=s-fs-fss,
δΦξ,η=atΦξ,ηmeasΦζmeas-Φζ0,
δΦ=at1Φζmeas-Φζ0+at2Φζmeas-Φζ03.
Φξ=RTx-xPy-yPz-zP+ΦξP,
Φζ=RTx-xPy-yPz-zP+c.
δΦ=δΦξ, η, ζ, ai,
minan σa:n.

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