Abstract

Photometric stereo is a well-known technique for recovering surface normals of a surface but requires three or more images of a surface taken under illumination from different directions. At best, one may dispense with the need for multiple images by using colored lights tuned to camera filters. But a less restrictive paradigm is available that uses the orientation-from-color approach, wherein multiple broadband illuminants impinge on a surface simultaneously. In that method, colors for a Lambertian surface lie on an ellipsoid in color space. The method has been applied mainly to single-color objects, with ellipsoid quadratic-form parameters determined from a large number of pixels. However, recently Petrov and Antonova [Color Res.Appl. 21, 97 (1996)] developed an entirely local approach, useful also for multicolored objects with color uniform in each patch. We investigate to what extent a method such as that of Petrov and Antonova can be applied in the ostensibly simpler situation in which the complex lighting environment is known, i.e., a color photometric stereo situation, with all lights in play at once with only a single image to analyze. We find that, assuming a simple model of color formation, we are able to recover the object colors along with surface normals, using only a single image. Because we immerse the object in a known lighting environment, we show that only half of the equations utilized by Petrov and Antonova are actually needed, making the method more stable. Nevertheless, solutions do not exist at every pixel; instead we may determine a best estimate of patch color, using a robust estimator, and then apply that estimate throughout a patch. Results are shown to be quite good compared with ground truth. The simple color model can often be made to hold more exactly by transforming the color space into one corresponding to spectrally sharpened sensors, which are a matrix transform away from the actual camera sensors. In our study the reliability and accuracy of the normal vector and of the surface color recovery algorithm are improved by this straightforward transformation.

© 2000 Optical Society of America

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References

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  1. L. L. Kontsevich, “Shape reconstruction for uniformly colored and illuminated object from its two-dimensional projection,” in Data Processing in Complex Information Systems, pp. 16–19 (in Russian) (available from Dr. Leonid Kontsevich, The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, Calif. 94115; e-mail, lenny@skivs.ski.org).
  2. A. P. Petrov, “Color and Grassman–Cayley coordinates of shape,” in Human Vision, Visual Processing and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 342–352 (1991).
    [CrossRef]
  3. M. S. Drew, “Shape from color,” (Simon Fraser University School of Computing Science, Vancouver, B.C., Canada, 1992), Available through ftp://fas.sfu.ca/pub/cs/techreports/1992/CSS-LCCR92-07.ps.Z.
  4. L. L. Kontsevich, A. P. Petrov, I. S. Vergelskaya, “Reconstruction of shape from shading in color images,” J. Opt. Soc. Am. A 11, 1047–1052 (1994).
    [CrossRef]
  5. A. P. Petrov, L. L. Kontsevich, “Properties of color images of surfaces under multiple illuminants,” J. Opt. Soc. Am. A 11, 2745–2749 (1994).
    [CrossRef]
  6. M. S. Drew, “Robust specularity detection from a single multi-illuminant color image,” CVGIP Image Understand. 59, 320–327 (1994).
    [CrossRef]
  7. M. S. Drew, L. L. Kontsevich, “Closed-form attitude determination under spectrally varying illumination,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 985–990.
  8. M. S. Drew, “Reduction of rank-reduced orientation-from-color problem with many unknown lights to two-image known-illuminant photometric stereo,” in Proceedings of the International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 419–424.
  9. M. S. Drew, “Direct solution of orientation-from-color problem using a modification of Pentland’s light source direction estimator,” Comput. Vision Image Understand. 64, 286–299 (1996).
    [CrossRef]
  10. A. P. Petrov, G. N. Antonova, “Resolving the color-image irradiance equation,” Color Res. Appl. 21, 97–103 (1996).
    [CrossRef]
  11. B. V. Funt, M. S. Drew, M. Brockington, “Recovering shading from color images,” in Proceedings of ECCV-92: Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, Berlin, 1992), pp. 124–132.
  12. G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
    [CrossRef]
  13. R. J. Woodham, “Gradient and curvature from the photometric-stereo method, including local confidence estimation,” J. Opt. Soc. Am. A 11, 3050–3068 (1994).
    [CrossRef]
  14. G. Healey, L. Wang, “Three-dimensional surface segmentation using multicolored illumination,” Opt. Eng. 37, 1553–1562 (1998).
    [CrossRef]
  15. M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant color object recognition via compressed chromaticity histograms of color-channel-normalized images,” in Proceedings of ICCV-98: International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 533–540.
  16. G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3019 (1994).
    [CrossRef]
  17. G. Healey, D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003–3010 (1994).
    [CrossRef]
  18. M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
    [CrossRef]
  19. C. F. Borges, “Trichromatic approximation method forsurface illumination,” J. Opt. Soc. Am. A 8, 1319–1323 (1991).
    [CrossRef]
  20. M. S. Drew, “Photometric stereo without multiple images,” in Human Vision and Electronic Imaging, B. E. Roqowitz, T. N. Pappas, eds., Proc. SPIE3016, 369–380 (1997). ftp://fas.sfu.ca/pub/cs/mark/Spie97/spie97.ps.gz.
  21. C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).
  22. Color figures may be viewed at http://www.cs.sfu.ca/people/Faculty/Drew/ftp/Josa00/ .
  23. W. Skarbek, A. Koschan, “Colour image seg-mentation—a survey, 1994,” (Technical University of Berlin, Berlin, 1994).
  24. P. J. Rousseeuw, A. M. Leroy, Robust Regression and Outlier Detection (Wiley, New York, 1987).
  25. G. D. Finlayson, M. S. Drew, “Positive Bradford curves through sharpening,” in Proceedings of the 7th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1999), pp. 227–232.
  26. M. S. Drew, G. D. Finlayson, “Spectral sharpening with positivity,” J. Opt. Soc. Am. A 17, 1361–1370 (2000).
    [CrossRef]

2000 (1)

1999 (1)

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
[CrossRef]

1998 (1)

G. Healey, L. Wang, “Three-dimensional surface segmentation using multicolored illumination,” Opt. Eng. 37, 1553–1562 (1998).
[CrossRef]

1996 (2)

M. S. Drew, “Direct solution of orientation-from-color problem using a modification of Pentland’s light source direction estimator,” Comput. Vision Image Understand. 64, 286–299 (1996).
[CrossRef]

A. P. Petrov, G. N. Antonova, “Resolving the color-image irradiance equation,” Color Res. Appl. 21, 97–103 (1996).
[CrossRef]

1994 (7)

1991 (1)

1976 (1)

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).

Antonova, G. N.

A. P. Petrov, G. N. Antonova, “Resolving the color-image irradiance equation,” Color Res. Appl. 21, 97–103 (1996).
[CrossRef]

Borges, C. F.

Brockington, M.

B. V. Funt, M. S. Drew, M. Brockington, “Recovering shading from color images,” in Proceedings of ECCV-92: Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, Berlin, 1992), pp. 124–132.

Davidson, J. G.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).

Drew, M. S.

M. S. Drew, G. D. Finlayson, “Spectral sharpening with positivity,” J. Opt. Soc. Am. A 17, 1361–1370 (2000).
[CrossRef]

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
[CrossRef]

M. S. Drew, “Direct solution of orientation-from-color problem using a modification of Pentland’s light source direction estimator,” Comput. Vision Image Understand. 64, 286–299 (1996).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3019 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

M. S. Drew, “Robust specularity detection from a single multi-illuminant color image,” CVGIP Image Understand. 59, 320–327 (1994).
[CrossRef]

M. S. Drew, “Reduction of rank-reduced orientation-from-color problem with many unknown lights to two-image known-illuminant photometric stereo,” in Proceedings of the International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 419–424.

G. D. Finlayson, M. S. Drew, “Positive Bradford curves through sharpening,” in Proceedings of the 7th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1999), pp. 227–232.

M. S. Drew, L. L. Kontsevich, “Closed-form attitude determination under spectrally varying illumination,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 985–990.

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant color object recognition via compressed chromaticity histograms of color-channel-normalized images,” in Proceedings of ICCV-98: International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 533–540.

B. V. Funt, M. S. Drew, M. Brockington, “Recovering shading from color images,” in Proceedings of ECCV-92: Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, Berlin, 1992), pp. 124–132.

M. S. Drew, “Photometric stereo without multiple images,” in Human Vision and Electronic Imaging, B. E. Roqowitz, T. N. Pappas, eds., Proc. SPIE3016, 369–380 (1997). ftp://fas.sfu.ca/pub/cs/mark/Spie97/spie97.ps.gz.

M. S. Drew, “Shape from color,” (Simon Fraser University School of Computing Science, Vancouver, B.C., Canada, 1992), Available through ftp://fas.sfu.ca/pub/cs/techreports/1992/CSS-LCCR92-07.ps.Z.

Finlayson, G. D.

Funt, B. V.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3019 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

B. V. Funt, M. S. Drew, M. Brockington, “Recovering shading from color images,” in Proceedings of ECCV-92: Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, Berlin, 1992), pp. 124–132.

Healey, G.

Kontsevich, L. L.

L. L. Kontsevich, A. P. Petrov, I. S. Vergelskaya, “Reconstruction of shape from shading in color images,” J. Opt. Soc. Am. A 11, 1047–1052 (1994).
[CrossRef]

A. P. Petrov, L. L. Kontsevich, “Properties of color images of surfaces under multiple illuminants,” J. Opt. Soc. Am. A 11, 2745–2749 (1994).
[CrossRef]

L. L. Kontsevich, “Shape reconstruction for uniformly colored and illuminated object from its two-dimensional projection,” in Data Processing in Complex Information Systems, pp. 16–19 (in Russian) (available from Dr. Leonid Kontsevich, The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, Calif. 94115; e-mail, lenny@skivs.ski.org).

M. S. Drew, L. L. Kontsevich, “Closed-form attitude determination under spectrally varying illumination,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 985–990.

Koschan, A.

W. Skarbek, A. Koschan, “Colour image seg-mentation—a survey, 1994,” (Technical University of Berlin, Berlin, 1994).

Leroy, A. M.

P. J. Rousseeuw, A. M. Leroy, Robust Regression and Outlier Detection (Wiley, New York, 1987).

Li, Z. N.

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
[CrossRef]

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant color object recognition via compressed chromaticity histograms of color-channel-normalized images,” in Proceedings of ICCV-98: International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 533–540.

Marcus, H.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).

McCamy, C. S.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).

Petrov, A. P.

A. P. Petrov, G. N. Antonova, “Resolving the color-image irradiance equation,” Color Res. Appl. 21, 97–103 (1996).
[CrossRef]

L. L. Kontsevich, A. P. Petrov, I. S. Vergelskaya, “Reconstruction of shape from shading in color images,” J. Opt. Soc. Am. A 11, 1047–1052 (1994).
[CrossRef]

A. P. Petrov, L. L. Kontsevich, “Properties of color images of surfaces under multiple illuminants,” J. Opt. Soc. Am. A 11, 2745–2749 (1994).
[CrossRef]

A. P. Petrov, “Color and Grassman–Cayley coordinates of shape,” in Human Vision, Visual Processing and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 342–352 (1991).
[CrossRef]

Rousseeuw, P. J.

P. J. Rousseeuw, A. M. Leroy, Robust Regression and Outlier Detection (Wiley, New York, 1987).

Skarbek, W.

W. Skarbek, A. Koschan, “Colour image seg-mentation—a survey, 1994,” (Technical University of Berlin, Berlin, 1994).

Slater, D.

Vergelskaya, I. S.

Wang, L.

G. Healey, L. Wang, “Three-dimensional surface segmentation using multicolored illumination,” Opt. Eng. 37, 1553–1562 (1998).
[CrossRef]

Wei, J.

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
[CrossRef]

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant color object recognition via compressed chromaticity histograms of color-channel-normalized images,” in Proceedings of ICCV-98: International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 533–540.

Woodham, R. J.

Color Res. Appl. (1)

A. P. Petrov, G. N. Antonova, “Resolving the color-image irradiance equation,” Color Res. Appl. 21, 97–103 (1996).
[CrossRef]

Comput. Vision Image Understand. (1)

M. S. Drew, “Direct solution of orientation-from-color problem using a modification of Pentland’s light source direction estimator,” Comput. Vision Image Understand. 64, 286–299 (1996).
[CrossRef]

CVGIP Image Understand. (1)

M. S. Drew, “Robust specularity detection from a single multi-illuminant color image,” CVGIP Image Understand. 59, 320–327 (1994).
[CrossRef]

J. Appl. Photogr. Eng. (1)

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2, 95–99 (1976).

J. Opt. Soc. Am. A (8)

Opt. Eng. (1)

G. Healey, L. Wang, “Three-dimensional surface segmentation using multicolored illumination,” Opt. Eng. 37, 1553–1562 (1998).
[CrossRef]

Pattern Recogn. (1)

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant image retrieval and video segmentation,” Pattern Recogn. 32, 1369–1388 (1999).
[CrossRef]

Other (12)

M. S. Drew, “Photometric stereo without multiple images,” in Human Vision and Electronic Imaging, B. E. Roqowitz, T. N. Pappas, eds., Proc. SPIE3016, 369–380 (1997). ftp://fas.sfu.ca/pub/cs/mark/Spie97/spie97.ps.gz.

Color figures may be viewed at http://www.cs.sfu.ca/people/Faculty/Drew/ftp/Josa00/ .

W. Skarbek, A. Koschan, “Colour image seg-mentation—a survey, 1994,” (Technical University of Berlin, Berlin, 1994).

P. J. Rousseeuw, A. M. Leroy, Robust Regression and Outlier Detection (Wiley, New York, 1987).

G. D. Finlayson, M. S. Drew, “Positive Bradford curves through sharpening,” in Proceedings of the 7th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1999), pp. 227–232.

M. S. Drew, J. Wei, Z. N. Li, “Illumination-invariant color object recognition via compressed chromaticity histograms of color-channel-normalized images,” in Proceedings of ICCV-98: International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 533–540.

M. S. Drew, L. L. Kontsevich, “Closed-form attitude determination under spectrally varying illumination,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 985–990.

M. S. Drew, “Reduction of rank-reduced orientation-from-color problem with many unknown lights to two-image known-illuminant photometric stereo,” in Proceedings of the International Symposium on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1995), pp. 419–424.

B. V. Funt, M. S. Drew, M. Brockington, “Recovering shading from color images,” in Proceedings of ECCV-92: Second European Conference on Computer Vision, G. Sandini, ed. (Springer-Verlag, Berlin, 1992), pp. 124–132.

L. L. Kontsevich, “Shape reconstruction for uniformly colored and illuminated object from its two-dimensional projection,” in Data Processing in Complex Information Systems, pp. 16–19 (in Russian) (available from Dr. Leonid Kontsevich, The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, Calif. 94115; e-mail, lenny@skivs.ski.org).

A. P. Petrov, “Color and Grassman–Cayley coordinates of shape,” in Human Vision, Visual Processing and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 342–352 (1991).
[CrossRef]

M. S. Drew, “Shape from color,” (Simon Fraser University School of Computing Science, Vancouver, B.C., Canada, 1992), Available through ftp://fas.sfu.ca/pub/cs/techreports/1992/CSS-LCCR92-07.ps.Z.

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Figures (6)

Fig. 1
Fig. 1

(a) Radar range image. (b) Lambertian surface with surface reflectance of 16 Macbeth patches, simultaneously illuminated by three illuminants. (c) White Lambertian sphere in same lighting. Symbols r, g, b, mark location of maxima in R, G, and B channels.

Fig. 2
Fig. 2

(a) Three illuminants. Equi-energy white light is also shown by the dashed line across the top. (b) Camera sensitivities. (c) Sharpened camera sensitivities.

Fig. 3
Fig. 3

(a) Pixels self-shadowed from at least one light. (b) pixels with solutions for ratios e.

Fig. 4
Fig. 4

(a) Recovered normals, shaded from (0, 0, 1). (b) Ground truth normal vectors, also shaded from (0, 0, 1).

Fig. 5
Fig. 5

(a) Recovered surface with recovered color as seen under white light. (b) Ground truth surface and colors under white light.

Fig. 6
Fig. 6

Ellipsoid in e1, e2, e3 space associated with a particular RGB point, with four solutions for normal vector equal to a given vector divided by the ellipsoid coordinates.

Tables (1)

Tables Icon

Table 1 Patches Intersecting the Figure in “Possible Pixels” Points for a 128×128 Image a

Equations (64)

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

ρx=i=1L[(ai)Tnx]Ii(λ)Sx(λ)q(λ)dλ,
biIi(λ)S(λ)q(λ)dλ.
ρ=BAn,
A=(a1, a2 , aL)T,B=(b1, b2 ,bL).
FBA,
ρ=Fn.
bkiskki/σk,k=13,
sk=S(λ)qk(λ)dλ,
ki=Ii(λ)qk(λ)dλ.
σk=qk(λ)dλ.
1nTn=ρTGTGρ=ρTHρ,
G=F-1,H=GTG.
F0=(f1)T(f2)T(f3)T,
max(pixelvalue)=fk,k=13.
ρk=(fk)Tn=fk(f^k)Tnfk.
F=DF0,D=diag(dk)diag(sk/σk),
ρ=Fn=DF0n,
(F0)kj=ikiaji.
H=GTG=D-1G0TG0D-1=EH0E,
ED-1,H0=G0TG0.
ρTEH0Eρ=1.
ρTEH0Eρx=0,
ρTEH0Eρy=0.
ρxTEH0Eρx+ρTEH0Eρxx=0,
ρyTEH0Eρx+ρTEH0Eρxy=0,
ρyTEH0Eρy+ρTEH0Eρyy=0.
E=diag(e1, e2, e3)=diag(1/d1, 1/d2, 1/d3)=diag(σk/sk)
ρTEH0Eρ=0.
EH0Eρ=ρx×ρyρT(ρx×ρy)v.
ρTv=1.
ek1.
EH0Eρ=v.
EH0Eρx-vx2+EH0Eρy-vy2.
nG0Eρ,G0=F0-1.
Q=MQ,
ρ=Mρ.
b=Mb,σ=Mσ,s=Ms,B=MB,F0=MF0,F=MF,
ρ=Mρ.
nG0Eρ.
D=diag(dk)diag(sk/σk),
diag(sk)=Ddiag(σk).
diag(sk)=M-1diag(sk).
errorb-bˆ/b,
R=diag(ρ1, ρ2, ρ3).
eT(RH0R)e=1.
E(RH0R)e=Rv.
(RH0R)e=D(Rv).
v3=(1-ρ1v1-ρ2v2)/ρ3,
e12H11ρ1+e1e2H12ρ2+e1e3H13ρ3=v1,
e1e2H12ρ1+e22H22ρ2+e2e3H23ρ3=v2,
e1e3H13ρ1+e2e3H23ρ2+e32H33ρ3=v3;
α1x+α2y+α3z=1/x,
β1x+β2y+β3z=1/y,
γ1x+γ2y+γ3z=1/z.
γ1p+γ2q+γ3=α1p2+α2pq+α3p,
γ1p+γ2q+γ3=β1pq+β2q2+β3q.
ϕ(p)=d0+d1p+d2p2+d3p3+d4p4=0,
d0=-β2(γ3)2+β3γ2γ3,
d1=-α2β3γ3-α2γ2γ3+2α3β2γ3-α3β3γ2+α3(γ2)2+β1γ2γ3-2β2γ1γ3+β3γ1γ2,
d2=2α1β2γ3-α1β3γ2+α1(γ2)2+(α2)2γ3+α2α3β3-α2α3γ2-α2β1γ3-α2β3γ1-α2γ1γ2-(α3)2β2-α3β1γ2+2α3β2γ1+β1γ1γ2-β2(γ1)2,
d3=α1α2β3-α1α2γ2-2α1α3β2-α1β1γ2+2α1β2γ1+(α2)2γ1+α2α3β1-α2β1γ1,
d4=-(α1)2β2+α1α2β1.
ρTρxTρyTHρx=0ρxxTvρxyTv,
ρTρxTρyTHρy=0ρxyTvρyyTv,

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