Abstract

Most color-acquisition devices capture spectral signals by acquiring only three samples, critically undersampling the spectral information. We analyze the problem of estimating high-dimensional spectral signals from low-dimensional device responses. We begin with the theory and geometry of linear estimation methods. These methods use linear models to characterize the likely input signals and reduce the number of estimation parameters. Next, we introduce two submanifold estimation methods. These methods are based on the observation that for many data sets the deviation between the signal and the linear estimate is systematic; the methods incorporate knowledge of these systematic deviations to improve upon linear estimation methods. We describe the geometric intuition of these methods and evaluate the submanifold method on hyperspectral image data.

© 2003 Optical Society of America

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  40. P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstrche,” Z. Tech. Phys. 12, 593–601 (1931).
  41. H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbendruckes,” Z. Wiss. Photogr. 36, 73–89 (1937).
  42. G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (1991).
    [CrossRef]
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    [CrossRef]
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2002

K. Barnard, L. Martin, B. Funt, A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

M. Shi, G. Healey, “Using reflectance models for color scanner calibration,” J. Opt. Soc. Am. A 19, 645–656 (2002).
[CrossRef]

2001

1999

M. Vrhel, J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

1997

1994

M. Vrhel, R. Gershon, L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 9, 4–9 (1994).

1993

P. Hung, “Colorimetric calibrations in electronic imaging devices using a look-up-table model and interpolations,” J. Electron. Imaging 2, 53–61 (1993).
[CrossRef]

1992

M. Vrhel, H. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

M. S. Drew, B. V. Funt, “Natural metamers,” CVGIP: Image Understand. 56, 139–151 (1992).
[CrossRef]

1991

G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (1991).
[CrossRef]

1990

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
[CrossRef]

1987

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

1986

1984

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
[CrossRef]

1980

G. Buchsbaum, “A spatial processor model for object color perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

1977

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

1976

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. 48, 777–784 (1976).

1964

1937

H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbendruckes,” Z. Wiss. Photogr. 36, 73–89 (1937).

1931

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstrche,” Z. Tech. Phys. 12, 593–601 (1931).

1895

H. Poincaré, “Analysis situs,” J. Ec. Polytech Series 2 1, 1–123 (1895).

Arvidson, R. E.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Barnard, K.

K. Barnard, L. Martin, B. Funt, A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

K. Barnard, L. Martin, B. Funt, “Colour by correlation in a three-dimensional colour space,” in Sixth European Conference on Computer Vision (Springer-Verlag, Berlin, 2000), pp. 275–289.

Benton, W. D.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Berns, R. S.

F. H. Imai, R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of the International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives (Chiba, Japan, 1999), pp. 42–49.

Bibby, J. M.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979).

Brainard, D.

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object color perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Catrysse, P.

P. Catrysse, A. E. Gamal, B. Wandell, “Color architectures for CMOS sensor imaging,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 26–35 (1999).
[CrossRef]

Chen, J.

J. Chen, K. Huang, “Adaptive color correction by high-order CMAC neural network,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 182–186.

Coath, A.

K. Barnard, L. Martin, B. Funt, A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychonomic Sci. 1, 369–370 (1964).
[CrossRef]

Davidson, J. G.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. 48, 777–784 (1976).

DiCarlo, J. M.

J. M. DiCarlo, B. A. Wandell, “Spectral estimation examples: beyond linear but before Bayesian,” (manuscript in preparation), contact authors for information: dicarlo@white.stanford.edu.

J. M. DiCarlo, B. A. Wandell, “Illuminant estimation: beyond the bases,” in Proceedings of the Eighth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 2000), pp. 91–96.

Drew, M.

G. Finlayson, M. Drew, “The maximum ignorance assumption with positivity,” in Proceedings of the Fourth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1996), pp. 202–205.

Drew, M. S.

M. S. Drew, B. V. Funt, “Natural metamers,” CVGIP: Image Understand. 56, 139–151 (1992).
[CrossRef]

Ebuisi, S.

Farrell, J. E.

B. A. Wandell, J. E. Farrell, “Water into wine: converting scanner RGB to tristimulus XYZ,” in Device-Independent Color Imaging and Imaging Systems Integration, R. J. Motta, H. A. Berberian, eds., Proc. SPIE1909, 92–100 (1993).
[CrossRef]

Finlayson, G.

G. Finlayson, M. Drew, “The maximum ignorance assumption with positivity,” in Proceedings of the Fourth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1996), pp. 202–205.

Finlayson, G. D.

G. D. Finlayson, P. M. Hubel, S. Hordley, “Color by cor-relation,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 6–11.

Forsyth, D. A.

D. A. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
[CrossRef]

Freeman, W.

Friedman, J.

T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer-Verlag, Berlin, 2001).

Funt, B.

K. Barnard, L. Martin, B. Funt, A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

K. Barnard, L. Martin, B. Funt, “Colour by correlation in a three-dimensional colour space,” in Sixth European Conference on Computer Vision (Springer-Verlag, Berlin, 2000), pp. 275–289.

Funt, B. V.

M. S. Drew, B. V. Funt, “Natural metamers,” CVGIP: Image Understand. 56, 139–151 (1992).
[CrossRef]

Gamal, A. E.

P. Catrysse, A. E. Gamal, B. Wandell, “Color architectures for CMOS sensor imaging,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 26–35 (1999).
[CrossRef]

Gershon, R.

M. Vrhel, R. Gershon, L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 9, 4–9 (1994).

Haneishi, H.

N. Tsumura, M. Kawabuchi, H. Haneishi, Y. Miyake, “Mapping pigmentation in human skin by multi-visible-spectral imaging by inverse optical scattering technqiue,” in Proceedings of the Eighth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 2000), pp. 81–84.

Hardeberg, J.

J. Hardeberg, F. Schmitt, “Color printer characterization using a computational geometry approach,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 97–99.

Hassibi, B.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Englewood Cliffs., N.J., 2000), p. 854.

Hastie, T.

T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer-Verlag, Berlin, 2001).

Healey, G.

Hong, G.

G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (1991).
[CrossRef]

Hordley, S.

G. D. Finlayson, P. M. Hubel, S. Hordley, “Color by cor-relation,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 6–11.

Horn, B. K. P.

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graph. Image Process. 26, 135–167 (1984).
[CrossRef]

Huang, K.

J. Chen, K. Huang, “Adaptive color correction by high-order CMAC neural network,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 182–186.

Hubel, P. M.

G. D. Finlayson, P. M. Hubel, S. Hordley, “Color by cor-relation,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 6–11.

Huck, F. O.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Hung, P.

P. Hung, “Colorimetric calibrations in electronic imaging devices using a look-up-table model and interpolations,” J. Electron. Imaging 2, 53–61 (1993).
[CrossRef]

Imai, F. H.

F. H. Imai, R. S. Berns, “Spectral estimation using trichromatic digital cameras,” in Proceedings of the International Symposium on Multispectral Imaging and Color Reproduction for Digital Archives (Chiba, Japan, 1999), pp. 42–49.

Iwan, L.

M. Vrhel, R. Gershon, L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 9, 4–9 (1994).

Jobson, D. J.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Johnson, R. A.

R. A. Johnson, D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice-Hall, Upper Saddle River, N.J., 2002).

Judd, D. B.

Kailath, T.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Englewood Cliffs., N.J., 2000), p. 854.

Kang, H. R.

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE Press, Bellingham, Wash., 1997).

Kawabuchi, M.

N. Tsumura, M. Kawabuchi, H. Haneishi, Y. Miyake, “Mapping pigmentation in human skin by multi-visible-spectral imaging by inverse optical scattering technqiue,” in Proceedings of the Eighth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 2000), pp. 81–84.

Kent, J. T.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979).

Krinov, E. L.

E. L. Krinov, “Surface reflectance properties of natural formations,” Technical TranslationTT-439 (National Research Council of Canada, Ottawa, 1947).

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstrche,” Z. Tech. Phys. 12, 593–601 (1931).

Luo, M. R.

G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (1991).
[CrossRef]

MacAdam, D. L.

Maloney, L. T.

Marcus, H.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. 48, 777–784 (1976).

Mardia, K. V.

K. V. Mardia, J. T. Kent, J. M. Bibby, Multivariate Analysis (Academic, London, 1979).

Martin, L.

K. Barnard, L. Martin, B. Funt, A. Coath, “A data set for colour research,” Color Res. Appl. 27, 147–151 (2002).
[CrossRef]

K. Barnard, L. Martin, B. Funt, “Colour by correlation in a three-dimensional colour space,” in Sixth European Conference on Computer Vision (Springer-Verlag, Berlin, 2000), pp. 275–289.

McCamy, C. S.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. 48, 777–784 (1976).

Miyake, Y.

N. Tsumura, M. Kawabuchi, H. Haneishi, Y. Miyake, “Mapping pigmentation in human skin by multi-visible-spectral imaging by inverse optical scattering technqiue,” in Proceedings of the Eighth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 2000), pp. 81–84.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag sur Optik der Farbanstrche,” Z. Tech. Phys. 12, 593–601 (1931).

Neugebauer, H. E. J.

H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mehrfarbendruckes,” Z. Wiss. Photogr. 36, 73–89 (1937).

Park, S. K.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Patterson, W. R.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Poincaré, H.

H. Poincaré, “Analysis situs,” J. Ec. Polytech Series 2 1, 1–123 (1895).

Rhodes, P. A.

G. Hong, M. R. Luo, P. A. Rhodes, “A study of digital camera colorimetric characterization based on polynomial modeling,” Color Res. Appl. 26, 76–84 (1991).
[CrossRef]

Ribes, A.

A. Ribes, F. Schmit, “Reconstructing spectral reflectances with mixture density networks,” in Proceedings of the CGIV (Poitiers, France, 2002), pp. 486–491.

Sayed, A. H.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Englewood Cliffs., N.J., 2000), p. 854.

Schmit, F.

A. Ribes, F. Schmit, “Reconstructing spectral reflectances with mixture density networks,” in Proceedings of the CGIV (Poitiers, France, 2002), pp. 486–491.

Schmitt, F.

J. Hardeberg, F. Schmitt, “Color printer characterization using a computational geometry approach,” in Proceedings of the Fifth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1997), pp. 97–99.

Shi, M.

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts andMethods, Quantitative Data and Formulae (Wiley, New York, 1982).

Tibshirani, R.

T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer-Verlag, Berlin, 2001).

Tominaga, S.

Trussell, H.

M. Vrhel, H. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

Trussell, J.

M. Vrhel, J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

Tsumura, N.

N. Tsumura, M. Kawabuchi, H. Haneishi, Y. Miyake, “Mapping pigmentation in human skin by multi-visible-spectral imaging by inverse optical scattering technqiue,” in Proceedings of the Eighth Color Imaging Conference: Color Science, Systems and Applications (Society for Imaging Science and Technology, Springfield, Va., 2000), pp. 81–84.

Viggiano, J. A. S.

J. A. S. Viggiano, “Minimal-knowledge assumptions in digital still camera characterization. I.: Uniform distribution, Toeplitz correlation,” in Proceedings of the Ninth Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 332–336.

Vrhel, M.

M. Vrhel, J. Trussell, “Color device calibration: a mathematical formulation,” IEEE Trans. Image Process. 8, 1796–1806 (1999).
[CrossRef]

M. Vrhel, R. Gershon, L. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 9, 4–9 (1994).

M. Vrhel, H. Trussell, “Color correction using principal components,” Color Res. Appl. 17, 328–338 (1992).
[CrossRef]

Wall, S. D.

F. O. Huck, D. J. Jobson, S. K. Park, S. D. Wall, R. E. Arvidson, W. R. Patterson, W. D. Benton, “Spectrophotometric color estimates of the Viking Lander sites,” J. Geophys. Res. 82, 4401–4411 (1977).
[CrossRef]

Wandell, B.

P. Catrysse, A. E. Gamal, B. Wandell, “Color architectures for CMOS sensor imaging,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 26–35 (1999).
[CrossRef]

Wandell, B. A.

S. Tominaga, S. Ebuisi, B. A. Wandell, “Scene illumination classification: brighter is better,” J. Opt. Soc. Am. A 18, 55–64 (2001).
[CrossRef]

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
[CrossRef] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

Sample measurement and estimation. The gray circles represent sample reflectance functions. The black arrow represents a sensor responsivity function, Te. A sample, s, when measured produces a sensor response, r. An estimate of s from r may fall anywhere in the null space of the sensors, denoted by the dashed vector.

Fig. 2
Fig. 2

Pseudoinverse estimation. The estimation function of the pseudoinverse method, the thin black line, is embedded in the sensor vector. The estimates for a subset of samples are denoted by the black ×’s. The estimation error for a sample is the length of the black line connecting the estimate with the sample. Other details as in Fig. 1.

Fig. 3
Fig. 3

Linear model estimation. The estimation function of the linear model method, the thin black line, is embedded in the data. Other details as in Fig. 1.

Fig. 4
Fig. 4

Linear estimation limitation. The reflectance samples have a nonlinear relationship. The estimation function of the best linear estimator, the thin black line, cannot follow the data. Other details as in Fig. 1.

Fig. 5
Fig. 5

Submanifold absolute-scale estimation example. The estimation function of the submanifold method, the thin black curve, is embedded in the data even with the nonlinear relationship among the samples. Other details as in Fig. 1.

Fig. 6
Fig. 6

Submanifold estimation flow-chart overview. See text for details.

Fig. 7
Fig. 7

Submanifold absolute-scale estimation method. The method estimates a spectral signal from a sensor response (open black square). Response values in the training data set that are similar to the measured response are identified (solid black circles). The samples producing these responses are identified in the training data (open black circles). The final estimate (black ×) is a weighted linear fit of these samples. Other details as in Fig. 1.

Fig. 8
Fig. 8

Submanifold relative-scale estimation example. The gray spheres represent reflectance functions with three wavelength samples. The two black vectors denote sensor responsivity functions that define the gray sensor response plane. The black spheres represent the sensor responses associated with each of the reflectance functions (gray spheres). The semitransparent surface is a homogeneous submanifold estimation surface. See text for details.

Fig. 9
Fig. 9

Hyperspectral images used to evaluate the submanifold algorithms.

Fig. 10
Fig. 10

Submanifold estimation performance. The abscissa represents the linear estimator error, and the ordinate represents the fractional change in error produced by using the relative-scale submanifold method. The horizontal black line indicates fractional change of 1; in this case the linear and submanifold errors are equal. Points plotted above this line indicate smaller linear error, and points plotted below indicate smaller submanifold error. The gray shading indicates the number of image pixels with fractional change in error for each level of linear error.

Fig. 11
Fig. 11

Estimation limitation. The reflectance data are not a function of the sensor responses. The solid black line represents the submanifold estimation function; The dashed black line represents the linear estimation function. Other details as in Fig. 1.

Equations (15)

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r=αTt diag(e)sΔλ.
r=Tets.
Esse=i=1Nssi-s^i2.
sˆ=γ+Γr.
Γ=Te(TetTe)-1,
γ=s¯-ΓTets¯.
Γ=ΣsTe(TetΣsTe)-1.
sˆ=γA+ΓAr+δA(r).
aj=sj-γA-ΓATetsj.
g(d, dL)=[1-(d/dL)3]3ifd/dL10otherwise.
δA(r)=ϕA(r)+ΦA(r)r.
ΦA(r)=[A-a˜(r)1Ntt]W(r)2[TetS-Tets˜(r)1Ntt]t×{[TetS-Tets˜(r)1Ntt]W(r)2×[TetS-Tets˜(r)1Ntt]t}-1,
ϕA(r)=a˜(r)-ΦA(r)Tets˜(r).
ΓR=SStTe[TetSStTe]-1.
ΦR(r)=AW(r)2StTe[TetSW(r)2StTe]-1.

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