Abstract

Changing a scene’s illuminant causes the chromatic properties of reflected lights to change. This change in the lights from surfaces provides spectral information about surface reflectances and illuminants. We examine conditions under which these properties may be recovered by using bilinear models. Necessary conditions that follow from comparing the number of equations and the number of unknowns in the recovery procedure are not sufficient for unique recovery. Necessary and sufficient conditions follow from demanding a one-to-one relationship between quantum catch data and sets of lit surfaces. We present an algorithm for determining whether spectral descriptions of lights and surfaces can be recovered uniquely from reflected lights.

© 1993 Optical Society of America

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  54. M. D’Zmura, G. Iverson, “Color constancy. II. Results for two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
    [CrossRef]
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1993 (2)

M. D’Zmura, G. Iverson, “Color constancy: feasibility and recovery,” Invest. Ophthalmol. Vis. Sci. Suppl. 34, 748 (1993).

M. D’Zmura, G. Iverson, “Color constancy. II. Results for two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
[CrossRef]

1992 (4)

1991 (1)

B. V. Funt, M. S. Drew, J. Ho, “Color constancy from mutual reflection,” Int. J. Comput. Vis. 6, 5–24 (1991).
[CrossRef]

1990 (3)

J. Ho, B. V. Funt, M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 966–977 (1990).
[CrossRef]

J. Neitz, G. H. Jacobs, “Polymorphism in normal human color vision and its mechanism,” Vision Res. 30, 621–636 (1990).
[CrossRef] [PubMed]

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” Proc. Third Int. Conf. Comput. Vis. 3, 385–393 (1990).

1989 (4)

1988 (2)

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2, 7–32 (1988).
[CrossRef]

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” Crit. Rev. Neurobiol. 3, 333–400 (1988).
[PubMed]

1987 (1)

1986 (5)

1985 (1)

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1983 (1)

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

1982 (1)

D. D. Hoffman, B. E. Flinchbaugh, “The interpretation of biological motion,” Biol. Cybern. 42, 195–204 (1982).
[PubMed]

1981 (1)

M. Alpern, “Color blind color vision,” Trends Neurosci. 4, 131–135 (1981).
[CrossRef]

1980 (1)

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

1979 (2)

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theoret. Biol. 78, 305–308 (1979).
[CrossRef]

S. Ullman, “The interpretation of structure from motion,” Proc. R. Soc. London Ser. B 203, 405–426 (1979).
[CrossRef]

1978 (1)

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theoret. Biol. 71, 473–478 (1978).
[CrossRef]

1975 (2)

G. Johansson, “Visual motion perception,” Sci. Am. 232, 76–88 (1975).
[CrossRef] [PubMed]

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

1969 (1)

1964 (2)

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,”J. Opt. Soc. Am. 54, 1031–1040 (1964).
[CrossRef]

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

1957 (2)

1952 (1)

H. Helson, D. B. Judd, M. H. Warren, “Object-color changes from daylight to incandescent filament illumination,” Ilium. Eng. 47, 221–232 (1952).

1940 (1)

1938 (1)

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of non-selective samples in chromatic illumination,”J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

1929 (1)

H. Kapferer, “Ueber Resultanten und Resultantensysteme,” Sitzungsber. Bayerisch. Akad. Munchen179–200 (1929).

1802 (1)

T. Young, “On the theory of light and colours,” Philos. Trans.12–48 (1802).

Alpern, M.

M. Alpern, “Color blind color vision,” Trends Neurosci. 4, 131–135 (1981).
[CrossRef]

Beck, J.

J. Beck, Surface Color Perception (Cornell U. Press, Ithaca, N.Y., 1972).

Bennett, B. M.

Benton, J. L.

Billthoff, H.

A. C. Hurlbert, H.-C. Lee, H. Billthoff, “Cues to the color of the illuminant,” Invest. Ophthalmol. Vis. Sci. Suppl. 30, 221 (1989).

Brainard, D.

D. Brainard, B. A. Wandell, “A bilinear model of the illuminant’s effect on color appearance,” in Computational Models of Visual Processing, M. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 171–186.

Braunstein, M. L.

M. L. Braunstein, Depth Perception through Motion (Academic, New York, 1976).

Brill, M. H.

M. H. Brill, “Further features of the illuminant-invariant trichromatic photosensor,”J. Theoret. Biol. 78, 305–308 (1979).
[CrossRef]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theoret. Biol. 71, 473–478 (1978).
[CrossRef]

Buchsbaum, G.

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

Burnham, R. W.

Canny, J. E.

J. E. Canny, The Complexity of Robot Motion Planning (MIT Press, Cambridge, Mass., 1988).

Cohen, J.

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

D’Zmura, M.

Drew, M. S.

M. S. Drew, B. V. Funt, “Variational approach to interreflection in color images,” J. Opt. Soc. Am. A 9, 1255–1265 (1992).
[CrossRef]

B. V. Funt, M. S. Drew, J. Ho, “Color constancy from mutual reflection,” Int. J. Comput. Vis. 6, 5–24 (1991).
[CrossRef]

J. Ho, B. V. Funt, M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 966–977 (1990).
[CrossRef]

Evans, R. M.

Eves, H.

H. Eves, Elementary Matrix Theory (Dover, New York, 1966).

Fairchild, M.

M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
[CrossRef] [PubMed]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (Wiley, New York, 1968), Vol. 1.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Flinchbaugh, B. E.

D. D. Hoffman, B. E. Flinchbaugh, “The interpretation of biological motion,” Biol. Cybern. 42, 195–204 (1982).
[PubMed]

Funt, B. V.

M. S. Drew, B. V. Funt, “Variational approach to interreflection in color images,” J. Opt. Soc. Am. A 9, 1255–1265 (1992).
[CrossRef]

B. V. Funt, M. S. Drew, J. Ho, “Color constancy from mutual reflection,” Int. J. Comput. Vis. 6, 5–24 (1991).
[CrossRef]

J. Ho, B. V. Funt, M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 966–977 (1990).
[CrossRef]

Gelb, A.

A. Gelb, “Die Farbenkonstanz der Sehdinge,” in Handbuch der normalen und pathologischen Physiologie, A. Bethe, ed. (Springer, Berlin, 1929), Vol. XII, pp. 594–678.
[CrossRef]

Grzywacz, N. M.

Hallikainen, J.

Helson, H.

H. Helson, D. B. Judd, M. H. Warren, “Object-color changes from daylight to incandescent filament illumination,” Ilium. Eng. 47, 221–232 (1952).

H. Helson, “Fundamental problems in color vision. I. The principle governing changes in hue, saturation and lightness of non-selective samples in chromatic illumination,”J. Exp. Psychol. 23, 439–476 (1938).
[CrossRef]

Hildreth, E. C.

Ho, J.

B. V. Funt, M. S. Drew, J. Ho, “Color constancy from mutual reflection,” Int. J. Comput. Vis. 6, 5–24 (1991).
[CrossRef]

J. Ho, B. V. Funt, M. S. Drew, “Separating a color signal into illumination and surface reflectance components: theory and applications,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 966–977 (1990).
[CrossRef]

Hoffman, D. D.

Hunter, R. S.

R. S. Hunter, The Measurement of Appearance (Wiley, New York, 1975).

Hurlbert, A. C.

A. C. Hurlbert, H.-C. Lee, H. Billthoff, “Cues to the color of the illuminant,” Invest. Ophthalmol. Vis. Sci. Suppl. 30, 221 (1989).

Hurvich, L. M.

L. M. Hurvich, D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384–404 (1957).
[CrossRef] [PubMed]

Iverson, G.

M. D’Zmura, G. Iverson, “Color constancy: feasibility and recovery,” Invest. Ophthalmol. Vis. Sci. Suppl. 34, 748 (1993).

M. D’Zmura, G. Iverson, “Color constancy. II. Results for two-stage linear recovery of spectral descriptions for lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
[CrossRef]

Jaaskelainen, T.

Jacobs, G. H.

J. Neitz, G. H. Jacobs, “Polymorphism in normal human color vision and its mechanism,” Vision Res. 30, 621–636 (1990).
[CrossRef] [PubMed]

Jameson, D.

L. M. Hurvich, D. Jameson, “An opponent-process theory of color vision,” Psychol. Rev. 64, 384–404 (1957).
[CrossRef] [PubMed]

Johansson, G.

G. Johansson, “Visual motion perception,” Sci. Am. 232, 76–88 (1975).
[CrossRef] [PubMed]

Judd, D. B.

Kanade, T.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2, 7–32 (1988).
[CrossRef]

Kapferer, H.

H. Kapferer, “Ueber Resultanten und Resultantensysteme,” Sitzungsber. Bayerisch. Akad. Munchen179–200 (1929).

Katz, D.

D. Katz, The World of Color, R. B. MacLeod, C. W. Fox, trans. (Kegan Paul, Trench, Trubner, London, 1935).

Klinker, G. J.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2, 7–32 (1988).
[CrossRef]

Land, E. H.

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
[CrossRef] [PubMed]

E. H. Land, “Recent advances in retinex theory and some implications for cortical computations: color vision and the natural image,” Proc. Natl. Acad. Sci. USA 80, 5163–5169 (1983).
[CrossRef] [PubMed]

Lee, H.-C.

A. C. Hurlbert, H.-C. Lee, H. Billthoff, “Cues to the color of the illuminant,” Invest. Ophthalmol. Vis. Sci. Suppl. 30, 221 (1989).

H.-C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
[CrossRef] [PubMed]

Lennie, P.

M. Fairchild, P. Lennie, “Chromatic adaptation to natural and incandescent illuminants,” Vision Res. 32, 2077–2085 (1992).
[CrossRef] [PubMed]

P. Lennie, M. D’Zmura, “Mechanisms of color vision,” Crit. Rev. Neurobiol. 3, 333–400 (1988).
[PubMed]

M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
[CrossRef]

MacAdam, D. L.

MacLeod, D. I. A.

D. I. A. MacLeod, “Receptoral constraints on colour appearance,” in Central and Peripheral Mechanisms of Colour Vision, D. Ottoson, S. Zeki, eds. (Macmillan, London, 1985), pp. 103–116.

Maloney, L. T.

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]

L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
[CrossRef] [PubMed]

B. A. Wandell, L. T. Maloney, “Color imaging process,” U.S. patent4,648,051 (March3, 1987).

L. T. Maloney, “Computational approaches to color constancy,” Stanford Applied Psychology Lab. Tech. Rep. 1985–01 (Stanford University, Stanford, Calif., 1985).

Marimont, D. H.

McCann, J. J.

J. J. McCann, J. L. Benton, “Interaction of the long-wave cones and the rods to produce color sensations,”J. Opt. Soc. Am. 59, 103–107 (1969).
[CrossRef] [PubMed]

J. J. McCann, “The role of simple nonlinear operations in modeling human lightness and color sensations,” in Human Vision, Visual Processing, and Visual Display, B. E. Rogowitz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1077, 355–363 (1989).
[CrossRef]

Neitz, J.

J. Neitz, G. H. Jacobs, “Polymorphism in normal human color vision and its mechanism,” Vision Res. 30, 621–636 (1990).
[CrossRef] [PubMed]

Newhall, S. M.

Nicola, J. E.

Ohta, Y.

M. Tsukada, Y. Ohta, “An approach to color constancy using multiple images,” Proc. Third Int. Conf. Comput. Vis. 3, 385–393 (1990).

Parkkinen, J. P. S.

Pokorny, J.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Prakash, C.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Sälström, P.

P. Sälström, “Colour and physics: some remarks concerning the physical aspects of human colour vision,” University of Stockholm Institute of Physics Rep. 73-09 (University of Stockholm, Stockholm, 1973).

Shafer, S.

S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Shafer, S. A.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vis. 2, 7–32 (1988).
[CrossRef]

Smith, V. C.

V. C. Smith, J. Pokorny, “Spectral sensitivity of the foveal cone photopigments between 400 and 500 nm,” Vision Res. 15, 161–171 (1975).
[CrossRef] [PubMed]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science. Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C. The Art of Scientific Computing (Cambridge U. Press, New York, 1988).

Tominaga, S.

Tsukada, M.

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Figures (6)

Fig. 1
Fig. 1

Linear components of a bilinear model: an example. A, Model for illumination represented by three basis functions that describes well the phases of daylight14; B, three-dimensional model for surface reflectance that describes well Munsell chips13 C, linear model for human trichromatic photoreception.56 See text for discussion.

Fig. 2
Fig. 2

Formulation of necessary and sufficient conditions for the unique recovery of illuminant and reflectance descriptors from quantum catch data, up to an arbitrary positive scalar: an example provided by the case (p m n v s) = (33333). The data dt from the three surfaces rt, t = 1, 2, 3, when viewed sequentially under the three illuminants aw, w = 1, 2, 3, must be equal to the data xt from the surfaces st viewed under the illuminants zw if and only if the illuminants are identical up to a single scalar [zw = λaw for w = 1, 2, 3] and the reflectances are identical up to the reciprocal scalar [st = (1/λ)rt for t = 1, 2, 3]. See text for discussion.

Fig. 3
Fig. 3

Conditions for recovering spectral descriptions from chromatic change: trichromacy. A, Square problems where p = m = 3 or, by transposition, p = n = 3; B, rectangular problems where p = 3, m = 2 or, by transposition, p = 3, n = 2. In both diagrams, the horizontal axis marks the dimension n of the reflectance model, taken equal to the number s of surfaces, while the vertical axis marks the number v of views. The solid lines in both diagrams that start at the lower left and work toward the upper right separate problems that satisfy the necessary condition svpsn + vm − 1 [inequality (14)] from those that do not. The number of quantum catch data and the number of descriptors to be recovered are indicated for each case by the bracketed pair [Q/D] beneath the appropriate lattice point. Points that lie beneath and to the right of the solid lines, where Q < D − 1, fail the feasibility condition svpsn + vm − 1 and so represent problems for which unique recovery is impossible. The dotted lines divide problems that satisfy the necessary condition for the test provided by the model check algorithm to be performed, namely, that E ≥ U [Eqs. (A10) below and (72)]. The pair E/U is shown directly beneath each point. In cases where m > v, such tests are sufficient; in problems where m = v, they are necessary and sufficient. By transposition, each problem (p m n v s) also represents the transposed problem (p n m s v), and the transposed parameters are indicated in parentheses at the tops of the diagrams and along their axes. See text for further discussion.

Fig. 4
Fig. 4

Conditions for recovering spectral descriptions from chromatic change: dichromacy. Problems are shown where p = m = 2 or p = n = 2. The X marks the problem (p m n v s)= (2 2 2 2 2) where recovery fails totally (see Subsection 4.C). See the caption for Fig. 3 and text for further discussion.

Fig. 5
Fig. 5

Conditions for recovering spectral descriptions from chromatic change: tetrachromacy. A, Square problems where p = m = 4 or p = n = 4; B, rectangular problems where p = 4, m = 3 or p = 4, n = 3; C, rectangular problems where p = 4, m = 2 or p = 4, n = 2. See the caption for Fig. 3 and text for further discussion.

Fig. 6
Fig. 6

Limits on the recovery of spectral descriptions from chromatic change. The horizontal axis marks the number p of photoreceptoral types, and the vertical axis marks the dimension m of the linear model for illumination. At each point is marked (1) the maximum number Ns of surface reflectance descriptors that may be recovered by a bilinear model that can be tested with the model check algorithm and (2) the maximum number Ns of reflectance descriptors that may be recovered by a bilinear model that satisfies the necessary condition pv > n. These two quantities are shown in the format Ns/Nn for values of the number v of views indicated in the legends at the right. See text for further discussion.

Tables (1)

Tables Icon

Table 1 List of Symbols

Equations (95)

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A ( λ ) A ^ ( λ ) = i = 1 m a i A i ( λ ) ,
a i = A ( λ ) A i ( λ ) d λ             for i = 1 , , m .
A ( λ ) = i = 1 m a i A i ( λ ) .
R ( λ ) = j = 1 n r j R j ( λ ) ,
r j = R ( λ ) R j ( λ ) d λ             for j = 1 , , n .
L ( λ ) = A ( λ ) R ( λ ) = i = 1 m j = 1 n a i r j A i ( λ ) R j ( λ ) .
q k = Q k ( λ ) L ( λ ) d λ = Q k ( λ ) { i = 1 m j = 1 n a i r j A i ( λ ) R j ( λ ) } d λ .
( B j ) k i = Q k ( λ ) A i ( λ ) R j ( λ ) d λ ,             j = 1 , , n .
q k = j = 1 n i = 1 m r j ( B j ) k i a i .
q t w k = j = 1 n i = 1 m r t j ( B j ) k i a w i .
d t w = j = 1 n r t j B j a w ,
C j = diag [ B j , , B j ] ,
d t = j = 1 n r t j C j a             t = 1 , , s .
s v p s n + v m - 1 ,
t = 1 n ρ j t C j - 1 d t = a             for j = 1 , , n .
D = [ d 1 d n ] ,
ρ j = [ ρ j 1 ρ j n ] T ;
C j - 1 D ρ j = a             for j = 1 , , n .
F j = C j - 1 D             for j = 1 , , n ,
F 1 ρ 1 = = F n ρ n = a ,
F 1 ρ 1 - F 2 ρ 2 = = F 1 ρ 1 - F n ρ n = 0 .
[ F 1 - F 2 0 0 F 1 0 - F 3 0 F 1 0 - F n ] [ ρ 1 ρ 2 ρ n ] = 0 ,
F ρ = 0 ,
ρ = [ ρ 1 T ρ n T ] T .
dim [ ker ( F ) ] = 1.
p n v n 2 + p v - 1
p v ( n - 1 ) n 2 - 1.
p v n + 1
p v > n .
B j * = [ B j b j , 1 * b j , d m * ] .
( B i ) k j = Q k ( λ ) A i ( λ ) R j ( λ ) d λ
q w t k = i = 1 m j = 1 n a w i ( B i ) k j r t j .
p s > m .
d t = j = 1 3 r t j C j a ,
x t = j = 1 3 s t j C j z ,
d t = x t = i = 1 3 s t i C i z = j = 1 3 r t j C j a             for t = 1 , 2 , 3.
C i z = j = 1 3 e i j C j a for i = 1 , 2 , n = 3 ,
e i j = t = 1 3 σ i t r t j             for i = 1 , 2 , n = 3 ,
e 11 = e 22 = e 33 ,             e 12 = e 21 = e 13 = e 31 = e 23 = e 32 = 0.
z = j = 1 3 e 1 j G 1 j a = j = 1 3 e 2 j G 2 j a = j = 1 3 e 3 j G 3 j a .
Z = j = 1 3 e 1 j Γ 1 j A = j = 1 3 e 2 j Γ 2 j A = j = 1 3 e 3 j Γ 3 j A .
Γ i j = B i - 1 B j
Γ i j = Γ j i - 1 ,
Γ i i = I ,
Z A - 1 = j = 1 3 e 1 j Γ 1 j = j = 1 3 e 2 j Γ 2 j = j = 1 3 e 3 j Γ 3 j .
j = 1 3 e 1 j Γ 1 j - j = 1 3 e 2 j Γ 2 j = 0 , j = 1 3 e 1 j Γ 1 j - j = 1 3 e 3 j Γ 3 j = 0 .
L i = [ γ i 1             γ i 2 γ i 3 ]             for i = 1 , 2 , 3.
e = [ e 1 T             e 2 T             e 3 T ] T ,             e i = [ e i 1             e i 2             e i 3 ] T .
L = [ L 1 - L 2 0 L 1 0 - L 3 ] ,
Le = 0 .
[ z 1 z v ] = ( j = 1 n e 1 j Γ 1 j ) [ a 1 a v ] = = ( j = 1 n e n j Γ n j ) [ a 1 a v ] .
( h = 1 n e 1 h Γ 1 h - h = 1 n e j h Γ j h ) = 0             for j = 2 , , n .
Le = 0 ,
L = [ L 1 - L 2 0 0 L 1 0 - L 3 0 L 1 0 - L n ] ,
e 11 - e j j = 0 for j = 2 , , n , e i j = 0 for i j .            
e 11 Γ 11 + e 12 Γ 12 - e 21 Γ 21 - e 22 Γ 22 = 0 .
( e 11 - e 22 ) I + e 12 Γ 12 - e 21 Γ 12 - 1 = 0 .
- e 21 I + ( e 11 - e 22 ) Γ 12 + e 12 Γ 12 2 = 0 .
( Γ 12 - λ 1 I ) ( Γ 12 - λ 2 I ) = 0 ,
λ 1 λ 2 I - ( λ 1 + λ 2 ) Γ 12 + Γ 12 2 = 0 .
e 12 = c , - e 21 = c λ 1 λ 2 = c det ( Γ 12 ) , e 22 - e 11 = c ( λ 1 + λ 2 ) = c trace ( Γ 12 ) ,
[ z 1             z 2             0 ] = j = 2 3 e 1 j Γ 1 j [ a 1             a 2             0 ] = j = 1 3 e 2 j Γ 2 j [ a 1             a 2             0 ] = j = 1 3 e 3 j Γ 3 j [ a 1             a 2             0 ] .
( j = 1 3 e 1 j Γ 1 j - j = 1 3 e 2 j Γ 2 j ) [ a 1             a 2             0 ] = 0 , ( j = 1 3 e 1 j Γ 1 j - j = 1 3 e 3 j Γ 3 j ) [ a 1             a 2             0 ] = 0 ,
W 1 - 2 = ( j = 1 3 e 1 j Γ 1 j - j = 1 3 e 2 j Γ 2 j ) , W 1 - 3 = ( j = 1 3 e 1 j Γ 1 j - j = 1 3 e 3 j Γ 3 j ) ,
W 1 - 2 [ a 1             a 2             0 ] = W 1 - 3 [ a 1             a 2             0 ] = 0
[ W 1 - 2 , 1 W 1 - 2 , 2 W 1 - 2 , 3 W 1 - 3 , 1 W 1 - 3 , 2 W 1 - 3 , 3 ] [ a 1             a 2             0 ] = 0 ,
W 1 - j , k × W 1 - g , h = 0             for j , g = 2 , 3 , and k , h = 1 , 2 , 3.
w 1 - j , k 2 w 1 - g , h 3 - w 1 - j , k 3 w 1 - g , h 2 = 0 , w 1 - j , k 3 w 1 - g , h 1 - w 1 - j , k 1 w 1 - g , h 3 = 0 , w 1 - j , k 1 w 1 - g , h 2 - w 1 - j , k 2 w 1 - g , h 1 = 0 ,
q , r = 1 3 { [ e 1 q ( Γ 1 q ) k x - e j q ( Γ j q ) k x ] [ e 1 r ( Γ 1 r ) h y - e g r ( Γ g r ) h y ] - [ e 1 q ( Γ 1 q ) k y - e j q ( Γ j q ) k y ] [ e 1 r ( Γ 1 r ) h x - e g r ( Γ g r ) h x ] } = 0 ,
( 8 + 2 - 1 2 ) = ( 9 2 ) = 36.
[ z 1 z v 0 0 d v ] = ( j = 1 n e 1 j Γ 1 j ) [ a 1 a v 0 0 d v ] = = ( j = 1 n e n j Γ n j ) [ a 1 a v 0 0 d v ] .
W 1 - j = ( q = 1 n e 1 q Γ 1 q - q = 1 n e j q Γ j q )             for j = 2 , , n ,
W 1 - j [ a 1 a v 0 0 d v ] = 0             for j = 2 , , n .
[ W 1 - 2 , 1 W 1 - 2 , p W 1 - n , 1 W 1 - 2 , p ] [ a 1 a v 0 0 d v ] = 0 .
Number of Equations ( p = m = v + d v , n = s ) = = ( ( n - 1 ) p d v + 1 ) ( p d v + 1 ) = ( n p - m m - v + 1 ) ( m m - v + 1 ) .
Number of Unknowns ( p = m = v + d v , n = s ) = = ( ( n 2 - 1 ) + ( d v + 1 ) - 1 d v + 1 ) = ( n 2 + m - v - 1 m - v + 1 ) .
e 11 - e j j = 0 for j = 2 , , n , e i j = 0 for i j .
( p m n v s ) ( p + 1 m n v s ) ,
( p m n v s ) ( p m n v + 1 s ) ,
( p m n v s ) ( p m n v s + 1 ) .
( m > v ) ,             ( p m n v s ) ( p m - 1 n v s ) .
( p m n v s ) ( p n m s v ) .
( n 2 + m - v + 1 m - v + 1 )
B i [ z 1 z v 0 0 d v ] = ( j = 1 n e i j B j ) [ a 1 a v 0 0 d v ]             for i = 1 , , n .
B j * = [ B j b j , 1 * b j , d m * ] .
Z * = [ Z             0 0 d m 0 0 0 0 0 0 } d m ] ,
Z = [ z 1 z v             0 0 d v ] ,
A * = [ A             0 0 d m 0 0 0 0 0 0 } d m ] ,
A = [ a 1 a v             0 0 d v ] ,
Z * = ( j = 1 n e 1 j Γ 1 j * ) A * = = ( j = 1 n e n j Γ n j * ) A * ,
Γ i j * = B i * - 1 B j * .
W 1 - j * = ( q = 1 n e 1 q Γ 1 q * - q = 1 n e j q Γ j q * )             for j = 2 , , n ,
W 1 - j * A * = 0 .
Z * = ( j = 1 n e i j Γ i j * ) A *             for i = 1 , , n .
Number of Equations ( p m = v + d v , n = s ) = = ( n p - m d v + 1 ) ( m d v + 1 ) = ( n p - m m - v + 1 ) ( m m - v + 1 ) .

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