Abstract

In computational terms we can solve the color constancy problem if device red, green, and blue sensor responses, or RGB’s, for surfaces seen under an unknown illuminant can be mapped to corresponding RGB’s under a known reference light. In recent years almost all authors have argued that this three-dimensional problem is too hard. It is argued that because a bright light striking a dark surface results in the same physical spectra as those of a dim light incident on a light surface, the magnitude of RGB’s cannot be recovered. Consequently, modern color constancy algorithms attempt only to recover image chromaticities under the reference light: They solve a two-dimensional problem. While significant progress has been made toward achieving chromaticity constancy, recent work has shown that the most advanced algorithms are unable to render chromaticity stable enough so that it can be used as a cue for object recognition [B. V. Funt, K. Bernard, and L. Martin, in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), Vol. II, p. 445.] We take this reductionist approach a little further and look at the one-dimensional color constancy problem. We ask, Is there a single color coordinate, a function of image chromaticities, for which the color constancy problem can be solved? Our answer is an emphatic yes. We show that there exists a single invariant color coordinate, a function of R, G, and B, that depends only on surface reflectance. Two corollaries follow. First, given an RGB image of a scene viewed under any illuminant, we can trivially synthesize the same gray-scale image (we simply code the invariant coordinate as a gray scale). Second, this result implies that we can solve the one-dimensional color constancy problem at a pixel (in scenes with no color diversity whatsoever). We present experiments that show that invariant gray-scale histograms are a stable feature for object recognition. Indexing on invariant distributions supports almost perfect recognition for a dataset of 11 objects viewed under five colored lights. In contrast, object recognition based on chromaticity histograms (post-color constancy preprocessing) delivers much poorer recognition.

© 2001 Optical Society of America

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    [Crossref]
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2000 (1)

1999 (2)

G. Finlayson, S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comput. 17, 597–604 (1999).
[Crossref]

G. D. Finlayson, G. Y. Tian, “Color normalization for color object recognition,” Int. J. Pattern Recogn. Artif. Intell. 13, 1271–1285 (1999).
[Crossref]

1997 (1)

1996 (2)

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[Crossref]

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
[Crossref]

1995 (3)

G. Healey, L. Wang, “The illumination-invariant recognition of texture in color images,” J. Opt. Soc. Am. A 12, 1877–1883 (1995).
[Crossref]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[Crossref]

J. Hafner, H. S. Sawhney, W. Equitz, M. Flickner, W. Niblack, “Efficient color histogram indexing for quadratic form distance functions,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 729–735 (1995).
[Crossref]

1994 (5)

1993 (1)

S. K. Nayar, R. Bolle, “Computing reflectance ratios from an image,” Pattern Recogn. 26, 1529–1542 (1993).
[Crossref]

1992 (3)

D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
[Crossref] [PubMed]

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vis. Image Underst. 56, 139–151 (1992).
[Crossref]

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

1991 (1)

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(11), 11–32 (1991).
[Crossref]

1990 (1)

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

1988 (1)

G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vis. Graph. Image Process. 41, 57–79 (1988).
[Crossref]

1987 (1)

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

1986 (2)

1977 (1)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237(6), 108–129 (1977).
[Crossref] [PubMed]

1976 (1)

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2(3), 95–99 (1976).

1971 (1)

Ballard, D. H.

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(11), 11–32 (1991).
[Crossref]

Barber, R.

W. Niblack, R. Barber, “The QBIC project: querying images by content using color, texture and shape,” in Storage and Retrieval for Image and Video Databases, W. Niblack, ed., Proc. SPIE1908, 175–187 (1993).

Barnard, K.

K. Barnard, “Computational color constancy: taking theory into practice,” M.Sc. thesis (Simon Fraser University, School of Computing Science, 1995).

B. V. Funt, K. Barnard, L. Martin, “Is machine colour constancy good enough,” in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), Vol. II, pp. 445–459.

V. Cardei, B. V. Funt, K. Barnard, “Learning color constancy,” in Proceedings of the Fourth IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1996), pp. 58–60.

Berens, J.

G. D. Finlayson, J. Berens, “Log-opponent chromaticity coding of colour space,” (University of East Anglia School of Information Systems, Norwich, NR4 7TJ Norwich, UK, 2000). Available from http://www.sys.uea.ac.uk/Research/colourgroup/Col-00-01.ps.Z .

Bolle, R.

S. K. Nayar, R. Bolle, “Computing reflectance ratios from an image,” Pattern Recogn. 26, 1529–1542 (1993).
[Crossref]

Brainard, D. H.

Brill, M. H.

Cardei, V.

V. Cardei, B. V. Funt, K. Barnard, “Learning color constancy,” in Proceedings of the Fourth IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1996), pp. 58–60.

Chatterjee, S. S.

G. D. Finlayson, S. S. Chatterjee, B. V. Funt, “Color angular indexing,” in Proceedings of the Fourth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1996), Vol. II, pp. 16–27.

Crowley, J. L.

G. D. Finlayson, B. Schiele, J. L. Crowley, “Comprehensive color image normalization,” in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), pp. 475–490.

D’Zmura, M.

M. D’Zmura, G. Iverson, “Probabalistic color constancy,” in R. D. Luce, M. M. D’Zmura, D. Hoffman, G. Iverson, K. Romney, eds., Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday (Erlbaum, Hillsdale, N.J., 1994).

Davidson, J. G.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2(3), 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]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: generalized diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3020 (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, B. V. Funt, “Natural metamers,” Comput. Vis. Image Underst. 56, 139–151 (1992).
[Crossref]

G. D. Finlayson, M. S. Drew, “Positive Bradford curves through sharpening,” in Proceedings of the Seventh IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1999), pp. 227–232.

Ebuisi, S.

S. Tominaga, S. Ebuisi, B. Wandell, “Color temperature estimation of scene illumination,” in Proceedings of the Seventh IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1999), pp. 42–48.

Equitz, W.

J. Hafner, H. S. Sawhney, W. Equitz, M. Flickner, W. Niblack, “Efficient color histogram indexing for quadratic form distance functions,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 729–735 (1995).
[Crossref]

Finlayson, G.

G. Finlayson, S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comput. 17, 597–604 (1999).
[Crossref]

Finlayson, G. D.

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

G. D. Finlayson, G. Y. Tian, “Color normalization for color object recognition,” Int. J. Pattern Recogn. Artif. Intell. 13, 1271–1285 (1999).
[Crossref]

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[Crossref]

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
[Crossref]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[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]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: generalized diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3020 (1994).
[Crossref]

G. D. Finlayson, M. S. Drew, “Positive Bradford curves through sharpening,” in Proceedings of the Seventh IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1999), pp. 227–232.

G. D. Finlayson, J. Berens, “Log-opponent chromaticity coding of colour space,” (University of East Anglia School of Information Systems, Norwich, NR4 7TJ Norwich, UK, 2000). Available from http://www.sys.uea.ac.uk/Research/colourgroup/Col-00-01.ps.Z .

G. D. Finlayson, B. Schiele, J. L. Crowley, “Comprehensive color image normalization,” in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), pp. 475–490.

G. D. Finlayson, S. S. Chatterjee, B. V. Funt, “Color angular indexing,” in Proceedings of the Fourth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1996), Vol. II, pp. 16–27.

P. M. Hubel, J. Holm, G. D. Finlayson, “Illuminant estimation and colour correction,” in Proceedings of The Colour in Multimedia Conference (Colour and Imaging Institute, Derby, UK, 1998), pp. 97–105.

G. D. Finlayson, S. D. Hordley, “The theory and practice of gamut mapping color constancy,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Institute of Electrical and Electronics Engineers, New York, 1998), pp. 60–65.

G. D. Finlayson, S. D. Hordley, P. M. Hubel, “Colour by correlation: a simple unifying theory of colour constancy,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 835–842.

Flickner, M.

J. Hafner, H. S. Sawhney, W. Equitz, M. Flickner, W. Niblack, “Efficient color histogram indexing for quadratic form distance functions,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 729–735 (1995).
[Crossref]

Forsyth, D.

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

Freeman, W. T.

Funt, B. V.

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[Crossref]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[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]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Color constancy: generalized diagonal transforms suffice,” J. Opt. Soc. Am. A 11, 3011–3020 (1994).
[Crossref]

M. S. Drew, B. V. Funt, “Natural metamers,” Comput. Vis. Image Underst. 56, 139–151 (1992).
[Crossref]

G. D. Finlayson, S. S. Chatterjee, B. V. Funt, “Color angular indexing,” in Proceedings of the Fourth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1996), Vol. II, pp. 16–27.

B. V. Funt, K. Barnard, L. Martin, “Is machine colour constancy good enough,” in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), Vol. II, pp. 445–459.

V. Cardei, B. V. Funt, K. Barnard, “Learning color constancy,” in Proceedings of the Fourth IS&T and SID Color Imaging Conference (Society for Imaging Science and Technology, Springfield, Va., 1996), pp. 58–60.

Gershon, R.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Gevers, T.

T. Gevers, Color Image Invariant Segmentation and Retrieval (University of Amsterdam, Amsterdam, The Netherlands, 1996).

Hafner, J.

J. Hafner, H. S. Sawhney, W. Equitz, M. Flickner, W. Niblack, “Efficient color histogram indexing for quadratic form distance functions,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 729–735 (1995).
[Crossref]

Healey, G.

Holm, J.

P. M. Hubel, J. Holm, G. D. Finlayson, “Illuminant estimation and colour correction,” in Proceedings of The Colour in Multimedia Conference (Colour and Imaging Institute, Derby, UK, 1998), pp. 97–105.

Hordley, S.

G. Finlayson, S. Hordley, “Selection for gamut mapping colour constancy,” Image Vis. Comput. 17, 597–604 (1999).
[Crossref]

Hordley, S. D.

G. D. Finlayson, S. D. Hordley, “The theory and practice of gamut mapping color constancy,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Institute of Electrical and Electronics Engineers, New York, 1998), pp. 60–65.

G. D. Finlayson, S. D. Hordley, P. M. Hubel, “Colour by correlation: a simple unifying theory of colour constancy,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 835–842.

Hubel, P. M.

G. D. Finlayson, S. D. Hordley, P. M. Hubel, “Colour by correlation: a simple unifying theory of colour constancy,” in Proceedings of the IEEE International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 835–842.

P. M. Hubel, J. Holm, G. D. Finlayson, “Illuminant estimation and colour correction,” in Proceedings of The Colour in Multimedia Conference (Colour and Imaging Institute, Derby, UK, 1998), pp. 97–105.

Hunt, R. W. G.

R. W. G. Hunt, The Reproduction of Color, 5th ed. (Fountain, Kingston-Upon-Thames, UK, 1995).

Iverson, G.

M. D’Zmura, G. Iverson, “Probabalistic color constancy,” in R. D. Luce, M. M. D’Zmura, D. Hoffman, G. Iverson, K. Romney, eds., Geometric Representations of Perceptual Phenomena: Papers in Honor of Tarow Indow’s 70th Birthday (Erlbaum, Hillsdale, N.J., 1994).

Iwan, L. S.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Jolliffe, I. T.

I. T. Jolliffe, Principal Component Analysis (Springer-Verlag, Berlin, 1986).

Land, E. H.

Maloney, L. T.

Marcus, H.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2(3), 95–99 (1976).

Marimont, D. H.

Martin, L.

B. V. Funt, K. Barnard, L. Martin, “Is machine colour constancy good enough,” in Proceedings of the Fifth European Conference on Computer Vision (European Vision Society, Springer-Verlag, Berlin, 1998), Vol. II, pp. 445–459.

McCamy, C. S.

C. S. McCamy, H. Marcus, J. G. Davidson, “A color-rendition chart,” J. Appl. Photogr. Eng. 2(3), 95–99 (1976).

McCann, J. J.

Meyer, G. W.

G. W. Meyer, “Wavelength selection for synthetic image generation,” Comput. Vis. Graph. Image Process. 41, 57–79 (1988).
[Crossref]

Nayar, S. K.

S. K. Nayar, R. Bolle, “Computing reflectance ratios from an image,” Pattern Recogn. 26, 1529–1542 (1993).
[Crossref]

Niblack, W.

J. Hafner, H. S. Sawhney, W. Equitz, M. Flickner, W. Niblack, “Efficient color histogram indexing for quadratic form distance functions,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 729–735 (1995).
[Crossref]

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Figures (15)

Fig. 1
Fig. 1

Normalized 2500-K blackbody radiator: exact equation (solid curve) and approximation (dashed curve). Note in this case that the two curves are on top of each other.

Fig. 2
Fig. 2

Normalized 5500-K blackbody radiator: exact equation (solid curve) and approximation (dashed curve).

Fig. 3
Fig. 3

Normalized 10,000-K blackbody radiator: exact equation (solid curve) and approximation (dashed curve).

Fig. 4
Fig. 4

Normalized 5500-K CIE D55 standard daylight (solid curve) and Planckian 5500-K illuminant (dashed curve).

Fig. 5
Fig. 5

CIE xy chromaticity diagram. The solid curve is the chromaticity locus for all typical Planckian blackbody lights. From left to right illuminants begin bluish, become whitish and then yellowish, and end in reddish light. Crosses (+) denote chromaticities of typical natural and man-made lights.

Fig. 6
Fig. 6

Perfect Dirac delta camera data (sensitivities anchored at 450, 540, and 610 nm). Log-chromaticity differences (LCD’s) for seven surfaces (green, yellow, white, blue, purple, orange, and red) under ten Planckian lights (with increasing temperature from 2800 to 10,000 K). Variation that is due to illumination is along a single direction.

Fig. 7
Fig. 7

Perfect Dirac delta camera data (sensitivities anchored at 450, 540, and 610 nm). LCD’s for seven surfaces (green, yellow, white, blue, purple, orange, and red) under ten Planckian lights (with increasing temperature from 2800 to 10,000 K). LCD’s (from Fig. 6) have been rotated so that the x coordinate depends only on surface reflectance; the y coordinate depends strongly on illumination.

Fig. 8
Fig. 8

CIE xy chromaticity diagram. The solid curve is the chromaticity locus for all typical Planckian blackbody lights. From left to right illuminants begin bluish, become whitish and then yellowish, and end in reddish light. Dots (⋅) denote chromaticities of weighted averages of Planckian locus lights. It is clear that though these new illuminants do not fall on the locus, they do fall close to the locus.

Fig. 9
Fig. 9

SONY DXC-930 normalized camera sensitivities.

Fig. 10
Fig. 10

SONY DXC-930 camera data. LCD’s for seven surfaces (green, yellow, white, blue, purple, orange, and red) under ten Planckian lights (with increasing temperature from 2800 to 10,000 K). Variation that is due to illumination is along a single direction.

Fig. 11
Fig. 11

SONY DXC-930 camera data. LCD’s for seven surfaces (green, yellow, white, blue, purple, orange, and red) under ten Planckian lights (with increasing temperature from 2800 to 10,000 K). LCD’s (from Fig. 9) have been rotated so that the x coordinate depends only on surface reflectance; the y coordinate depends strongly on illumination.

Fig. 12
Fig. 12

Normalized XYZ color-matching curves.

Fig. 13
Fig. 13

XYZ tristimuli are transformed to corresponding SONY camera RGB’s by using a linear transform. LCD’s for seven surfaces (green, yellow, white, blue, purple, orange, and red) under ten Planckian lights (with increasing temperature from 2800 to 10,000 K) are calculated and rotated according to the SONY rotation matrix. The x coordinate depends weakly on surface reflectance; the y coordinate depends strongly on illumination.

Fig. 14
Fig. 14

LCD’s calculated for XYZ tristimuli that have been transformed to corresponding SONY camera RGB’s by using a linear transform are plotted as circles. Lines linking these approximate coordinates to the actual SONY LCD coordinates.

Fig. 15
Fig. 15

The first and third columns show luminance gray-scale images of a ball and a detergent packet under five lights (top to bottom). The second and fourth columns show corresponding invariant images.

Equations (29)

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

pk=ωE(λ)S(λ)Rk(λ)dλ(k=R, G, B),
pk=ωE(λ)S(λ)δ(λ-λk)dλ=E(λk)S(λk).
E1(λR)S(λR)E1(λG)S(λG)E1(λB)S(λB)
=E1(λR)/E2(λR)000E1(λG)/E2(λG)000E1(λB)/E2(λB)
×E2(λR)S(λR)E2(λG)S(λG)E2(λB)S(λB).
pr,1pg,1pb,1=α000β000γpr,2pg,2pb,2,
E(λ, T)=c1λ-5expc2Tλ-1-1.
E(λ, T)=Ic1λ-5[exp(c2/Tλ)-1]-1.
c2Tλ=1.4388×10-2x×10-7*t*103=1.4388×102x*t.
E(λ, T)Ic1λ-5exp(-c2/Tλ).
pk=ωE(λ)S(λ)δ(λ-λk)dλ=Ic1λk-5exp(-c2/Tλk)S(λk).
ln pk=ln I+ln[S(λk)λk-5c1]-c2Tλk.
pR=ln pR-ln pG=SR-SG+1T (ER-EG),
pB=ln pB-ln pG=SB-SG+1T (EB-EG).
pR-ER-EGEB-EG pB=SR-SR-ER-EGEB-EG (SB-SG)=f(SR, SG, SB),
ax+b(mx+c)=(ab)  (xmx+c)
ab-baxmx+c=xy
μi=1mk=1mQki,
qji=Qji-μi
M=(q11, q21,, qm1, q12, q22,, qm2,q1n, q2n,, qmn).
Σ(M)=1nm MMt.
ifRqki=×RQki=Rμ i+×,
Σ(M)=UtDU,
Σ(UM)=D.
Tpdpc,
f(Tpd)f(pc).
E(λ)=I1c1λ-5exp(-c2/T1λ)+I2c1λ-5exp(-c2/T2λ).
 
I3c1λ-5exp(-c2/T3λ)I1c1λ-5exp(-c2/T1λ)+I2c1λ-5exp(-c2/T2λ).

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