Abstract

A combination of phase-shift with gray-code light projection into a three-dimensional vision system based on the projection of structured light is presented. The gray-code method is exploited to detect without ambiguity even marked surface discontinuities, whereas the phase-shift technique allows the measurement of fine surface details. The system shows excellent linearity. An overall mean value of the measurement error equal to 40 µm, with a variability of approximately ±35 µm, corresponding to 0.06% of full scale, has been estimated. The implementation of the technique is discussed, the analysis of the systematic errors is presented in detail, and the calibration procedure designed to determine the optimal setting of the measurement parameters is illustrated.

© 1999 Optical Society of America

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  1. F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.
  2. S. F. El-Hakim, N. Pizzi, “Multicamera vision-based approach to flexible feature measurement for inspection and reverse engineering,” Opt. Eng. 32, 2201–2215 (1993).
    [CrossRef]
  3. D. Poussart, D. Laurendeau, “3-D sensing for industrial computer vision,” in Advances in Machine Vision, J. L. C. Sanz, ed. (Springer-Verlag, New York, 1989), Chap. 3.
    [CrossRef]
  4. M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
    [CrossRef]
  5. R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-5, 122–139 (1983).
    [CrossRef]
  6. S. Kuwamura, I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 36, 4473–4482 (1997).
    [CrossRef] [PubMed]
  7. T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
    [CrossRef]
  8. M. Rioux, “Laser range finder based on synchronized scanners,” Appl. Opt. 23, 3837–3843 (1984).
    [CrossRef] [PubMed]
  9. Q. Fang, S. Zheng, “Linearly coded profilometry,” Appl. Opt. 36, 2401–2407 (1997).
    [CrossRef] [PubMed]
  10. T. G. Stahs, F. M. Wahl, “Fast and robust range data acquisition in a low-cost environment,” in Close-Range Photogrammetry Meets Machine Vision, E. P. Baltsavias, A. Gruen, eds., Proc. SPIE1395, 496–503 (1990).
  11. G. Sansoni, S. Corini, S. Lazzari, R. Rodella, F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
    [CrossRef] [PubMed]
  12. G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.
  13. W. Krattenthaler, K. J. Mayer, H. P. Duwe, “3D-surface measurement with coded light approach,” in Fourth International Workshop for Digital Image Processing and Computer Graphics, Proceedings of Österreichische Arbeitsgem. MustererKennung (ÖCG Schriftenreihe, Oldenburg, Germany, 1993), Vol. 12, pp. 103–114.
  14. G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
    [CrossRef]
  15. M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
    [CrossRef]

1999 (1)

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

1997 (3)

1996 (1)

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

1993 (1)

S. F. El-Hakim, N. Pizzi, “Multicamera vision-based approach to flexible feature measurement for inspection and reverse engineering,” Opt. Eng. 32, 2201–2215 (1993).
[CrossRef]

1984 (1)

1983 (1)

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

Andersen, P.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Baribeau, R.

M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
[CrossRef]

Blais, F.

M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
[CrossRef]

Bonardi, M.

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

Bormann, F.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Burrows, M. D.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Carocci, M.

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
[CrossRef]

Corini, S.

Docchio, F.

G. Sansoni, S. Corini, S. Lazzari, R. Rodella, F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
[CrossRef] [PubMed]

G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

Duwe, H. P.

W. Krattenthaler, K. J. Mayer, H. P. Duwe, “3D-surface measurement with coded light approach,” in Fourth International Workshop for Digital Image Processing and Computer Graphics, Proceedings of Österreichische Arbeitsgem. MustererKennung (ÖCG Schriftenreihe, Oldenburg, Germany, 1993), Vol. 12, pp. 103–114.

El-Hakim, S. F.

S. F. El-Hakim, N. Pizzi, “Multicamera vision-based approach to flexible feature measurement for inspection and reverse engineering,” Opt. Eng. 32, 2201–2215 (1993).
[CrossRef]

Fang, Q.

Godin, G.

M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
[CrossRef]

Jarvis, R. A.

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

Krattenthaler, W.

W. Krattenthaler, K. J. Mayer, H. P. Duwe, “3D-surface measurement with coded light approach,” in Fourth International Workshop for Digital Image Processing and Computer Graphics, Proceedings of Österreichische Arbeitsgem. MustererKennung (ÖCG Schriftenreihe, Oldenburg, Germany, 1993), Vol. 12, pp. 103–114.

Kuwamura, S.

Laurendeau, D.

D. Poussart, D. Laurendeau, “3-D sensing for industrial computer vision,” in Advances in Machine Vision, J. L. C. Sanz, ed. (Springer-Verlag, New York, 1989), Chap. 3.
[CrossRef]

Lazzari, S.

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

G. Sansoni, S. Corini, S. Lazzari, R. Rodella, F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
[CrossRef] [PubMed]

M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
[CrossRef]

G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

Mayer, K. J.

W. Krattenthaler, K. J. Mayer, H. P. Duwe, “3D-surface measurement with coded light approach,” in Fourth International Workshop for Digital Image Processing and Computer Graphics, Proceedings of Österreichische Arbeitsgem. MustererKennung (ÖCG Schriftenreihe, Oldenburg, Germany, 1993), Vol. 12, pp. 103–114.

Nielsen, T.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Peli, S.

G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.

Pizzi, N.

S. F. El-Hakim, N. Pizzi, “Multicamera vision-based approach to flexible feature measurement for inspection and reverse engineering,” Opt. Eng. 32, 2201–2215 (1993).
[CrossRef]

Poussart, D.

D. Poussart, D. Laurendeau, “3-D sensing for industrial computer vision,” in Advances in Machine Vision, J. L. C. Sanz, ed. (Springer-Verlag, New York, 1989), Chap. 3.
[CrossRef]

Rioux, M.

M. Rioux, “Laser range finder based on synchronized scanners,” Appl. Opt. 23, 3837–3843 (1984).
[CrossRef] [PubMed]

M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
[CrossRef]

Rodella, R.

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

G. Sansoni, S. Corini, S. Lazzari, R. Rodella, F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
[CrossRef] [PubMed]

M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
[CrossRef]

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

Sansoni, G.

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

G. Sansoni, S. Corini, S. Lazzari, R. Rodella, F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. 36, 4463–4472 (1997).
[CrossRef] [PubMed]

G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.

M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
[CrossRef]

Spiecker, H.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Stahs, T. G.

T. G. Stahs, F. M. Wahl, “Fast and robust range data acquisition in a low-cost environment,” in Close-Range Photogrammetry Meets Machine Vision, E. P. Baltsavias, A. Gruen, eds., Proc. SPIE1395, 496–503 (1990).

Wahl, F. M.

T. G. Stahs, F. M. Wahl, “Fast and robust range data acquisition in a low-cost environment,” in Close-Range Photogrammetry Meets Machine Vision, E. P. Baltsavias, A. Gruen, eds., Proc. SPIE1395, 496–503 (1990).

Wolbeck, S.

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Yamaguchi, I.

Zheng, S.

Zorzella, E.

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. A. Jarvis, “A perspective on range finding techniques for computer vision,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-5, 122–139 (1983).
[CrossRef]

J. Opt. A (1)

G. Sansoni, M. Carocci, S. Lazzari, R. Rodella, “A 3D imaging system for industrial applications with improved flexibility and robustness,” J. Opt. A 1, 83–93 (1999).
[CrossRef]

Opt. Eng. (1)

S. F. El-Hakim, N. Pizzi, “Multicamera vision-based approach to flexible feature measurement for inspection and reverse engineering,” Opt. Eng. 32, 2201–2215 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

T. Nielsen, F. Bormann, S. Wolbeck, H. Spiecker, M. D. Burrows, P. Andersen, “Time-of-light analysis of flight pulses with a temporal resolution of 100ps,” Rev. Sci. Instrum. 67, 1721–1724 (1996).
[CrossRef]

Other (7)

T. G. Stahs, F. M. Wahl, “Fast and robust range data acquisition in a low-cost environment,” in Close-Range Photogrammetry Meets Machine Vision, E. P. Baltsavias, A. Gruen, eds., Proc. SPIE1395, 496–503 (1990).

D. Poussart, D. Laurendeau, “3-D sensing for industrial computer vision,” in Advances in Machine Vision, J. L. C. Sanz, ed. (Springer-Verlag, New York, 1989), Chap. 3.
[CrossRef]

M. Rioux, G. Godin, F. Blais, R. Baribeau, “High resolution digital 3D imaging of large structures,” in Three-Dimensional Image Capture, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3023, 109–118 (1997).
[CrossRef]

F. Docchio, M. Bonardi, S. Lazzari, R. Rodella, E. Zorzella, “Electro-optical sensors for mechanical applications,” in Optical Sensors and Microsystems, A. N. Chester, S. Martellucci, A. G. Mignani, eds. (Plenum, New York, London, 1999), Chap. 1.

M. Carocci, S. Lazzari, R. Rodella, G. Sansoni, “3D range optical sensor: analysis of the measurement errors and development of procedures for their compensation,” in Three-Dimensional Image Capture and Applications, R. N. Ellson, J. H. Nurre, eds., Proc. SPIE3313, 178–188 (1998).
[CrossRef]

G. Sansoni, S. Lazzari, S. Peli, F. Docchio, “3D imager for dimensional gauging of industrial workpieces: state of the art of the development of a robust and versatile system,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling, G. Roth, M. Rioux, eds., (IEEE Computer Society, Los Alamitos, Calif., 1997), pp. 19–26.

W. Krattenthaler, K. J. Mayer, H. P. Duwe, “3D-surface measurement with coded light approach,” in Fourth International Workshop for Digital Image Processing and Computer Graphics, Proceedings of Österreichische Arbeitsgem. MustererKennung (ÖCG Schriftenreihe, Oldenburg, Germany, 1993), Vol. 12, pp. 103–114.

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Figures (13)

Fig. 1
Fig. 1

Optical geometry of the system.

Fig. 2
Fig. 2

Example of the pattern sequence that combines gray-code and phase-shift projection.

Fig. 3
Fig. 3

Model of the system geometry used to derive the mapping equation.

Fig. 4
Fig. 4

Plot of the systematic error Δz L /z as a function of ΔL/ L.

Fig. 5
Fig. 5

Plot of the systematic error Δz d /z as a function of Δd/ d.

Fig. 6
Fig. 6

Plot of the systematic error Δz α/z as a function of Δα/α.

Fig. 7
Fig. 7

Photograph of the prototype used for the measurements.

Fig. 8
Fig. 8

Example of the GCPS sequence projected on the face model: shown are patterns (a) GC_0, (b) GC_3, (c) GC_7, (d) PS_1.

Fig. 9
Fig. 9

3D shape of the face model.

Fig. 10
Fig. 10

Image of a flowerpot.

Fig. 11
Fig. 11

3D shape of the flowerpot recovered with (a) the gray-code technique, (b) the combined GCPS technique.

Fig. 12
Fig. 12

In–out characteristic curves of the system.

Fig. 13
Fig. 13

Evaluation of the accuracy of the system. The mean value Δz¯ and standard deviation σ[Δz] of the set of values of error Δz along a section parallel to X- are shown as functions of Z in.

Tables (2)

Tables Icon

Table 1 Gray-Code Sequence for n = 4

Tables Icon

Table 2 Evaluation of the Correction Term ΔΦ(j, k)

Equations (27)

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

Iij, k=Aj, k+Bj, k2cosΦj, k-i π2.
Φj, k=tan-1I1j, k-I3j, kI0j, k-I2j, k
lj, k=lˆj, k+2πΦj, k+ΔΦj, k.
zH=LAB¯d+AB¯.
AB¯=xB-xA.
xB=PQ¯ tanγ+δx=L tanγ+δx.
γ=tan-1d-FW2L.
SB¯sin δx=SP¯sin θx.
δx=tan-1SB¯ cosγ-αSP¯+SB¯ sinγ-α.
SB¯=lPBnlpST¯,
ST¯=sin δtotsin θtotSP¯=sin δtotsin θtotLcosγ.
δtot =tan-1L tanγ + FWL-γ,
θtot=π2+α-tan-1L tanγ + FWL.
xB=Ltanγ+tan-1CllPB cosγ-α1+CllPB sinγ-α.
C1=sinδtotnlpsinθtot.
DRj, k=xOBJj, k-xREFj, k=Ltanγ+tan-1C1OBJj, kcosγ-α1+C1OBJj, ksinγ-α-tanγ+tan-1C1REFj, kcosγ-α1+C1REFj, ksinγ-α
zj, k=LDRj, kd+DRj, k
Δzj, kzj, k=ΔzLj, k+Δzdj, k+Δzαj, kzj, k,
ΔzLj, k=LeDRj, kd+DRj, k-LDRj, kd+DRj, k.
DRj, k=LC2,
C2=tanγ-tan-1C1OBJj, kcosγ-α1+C1OBJj, ksinγ-α-tanγ-tan-1C1REFj, kcosγ-α1+C1REFj, ksinγ-α.
ΔzLj, kzj, k=C2Le2d+C2LeC2L2d+C2L-1.
ΔzLj, kzj, k=Le2L2-1=L+ΔL2-L2L2=2 ΔLL+ΔLL2.
Δzdj, k=LDRj, kde+DRj, k-LDRj, kd+DRj, k.
Δzdj, kzj, k=LDRj, kde+DRj, kLDRj, kd+DRj, k-1=d-dede+DRj, k=-Δdd11+Δdd+DRj, kd.
-11+DRj, kd
αe=1+wmαe,

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