Abstract

The optimum linear combination channels for an N-receptor polarization-sensitive imaging or vision system are found by using a principal-components analysis. The channels that are derived are optimum in the sense that their information contents are uncorrelated when considered over the ensemble of possible polarization signals. For a two-receptor system, the optimum channels are shown to be the sum and the difference of the outputs of the individual receptors. As a corollary, the optimal arrangement of the two receptors is shown to be a mosaic of identical, orthogonally aligned linear polarization analyzers. The implications of these results on the development of a representational scheme for polarization information are discussed.

© 1998 Optical Society of America

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References

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  1. R. Walraven, “Polarization imagery,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. SPIE112, 164–167 (1977).
    [CrossRef]
  2. W. G. Egan, W. R. Johnson, V. S. Whitehead, “Terrestrial polarization imagery obtained from the Space Shuttle: Characterization and interpretation,” Appl. Opt. 30, 435–442 (1991).
    [CrossRef] [PubMed]
  3. L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 635–657 (1991).
    [CrossRef]
  4. L. B. Wolff, T. A. Mancini, “Liquid crystal polarization camera,” in Proceedings of the IEEE Workshop on Applications of Computer Vision (IEEE, New York, 1992), pp. 120–127.
  5. L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
    [CrossRef]
  6. G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-I–A-III-10 (1966).
  7. M. P. Rowe, E. N. Pugh, J. S. Tyo, N. Engheta, “Polarization-difference imaging: a biologically inspired technique for imaging in scattering media,” Opt. Lett. 20, 608–610 (1995).
    [CrossRef] [PubMed]
  8. S. G. Demos, R. R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarization,” Opt. Lett. 21, 161–163 (1996).
    [CrossRef] [PubMed]
  9. J. S. Tyo, M. P. Rowe, E. N. Pugh, N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt. 35, 1855–1870 (1996).
    [CrossRef] [PubMed]
  10. J. E. Solomon, “Polarization imaging,” Appl. Opt. 20, 1537–1544 (1981).
    [CrossRef] [PubMed]
  11. G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).
  12. J. S. Tyo, E. N. Pugh, N. Engheta, “Colorimetric representations for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A 15, 367–374 (1998).
    [CrossRef]
  13. G. D. Bernard, R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17, 1019–1028 (1977).
    [CrossRef] [PubMed]
  14. G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
    [CrossRef]
  15. Photoreceptors actually capture radiation in discrete quanta; however, if the intensity of the radiation is strong enough, the discretization can be ignored, and the above formulation can be applied.
  16. S. M. Kay, Modern Spectral Estimation: Theory And Application (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  17. In this section the ensemble is assumed to be uniformly distributed in angle of polarization with a fixed ellipticity described by the parameter ∊. These results can be easily generalized to include other ensembles by simply altering the pdf’s used to calculate the entries in the correlation matrix.
  18. T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1984), Chap. 11.
  19. Z. Zhao, N. H. Farhat, “Tomographic microwave diversity image reconstruction employing unitary compression,” IEEE Trans. Microwave Theory Tech. 40, 315–322 (1992).
    [CrossRef]
  20. A. Bermann, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979), Chap. 4.
  21. C. E. Shannon, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).
  22. C. H. Papas, Theory of Electromagnetic Radiation (Dover, New York, 1988), pp. 118–134.
  23. Although choosing N>3 does not seem to make sense in the context explored herein, some simple, fast, nonlinear systems may be proposed that operate optimally with N>3.
  24. D. A. Cameron, E. N. Pugh, “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature (London) 353, 161–164 (1991).
    [CrossRef]
  25. M. P. Rowe, N. Engheta, S. S. Easter, E. N. Pugh, “Graded index model of a fish double cone exhibits differential polarization sensitivity,” J. Opt. Soc. Am. A 11, 55–70 (1994).
    [CrossRef]
  26. The four Stokes parameters can be determined (up to an ambiguity in sign of one of the parameters) with only three intensity measurements for monochromatic radiation.22 Because circular polarization and unpolarized light are indistinguishable with a system that detects only linear polarization states, their effects on the final representation should be similar, and partially circularly polarized monochromatic radiation is analyzed here for mathematical clarity.
  27. This representation of the incident field is exact up to the choice of a constant. Although the linearly polarized portion of the radiation is constrained to have unit amplitude here, in general it can have arbitrary amplitude. In the general case, A is the ratio between the amplitudes of the circularly polarized and linearly polarized portions of the incident radiation.
  28. Nonbirefringent, nonoptically active photoreceptors are considered for simplicity, since most linear polarization analyzers satisfy this requirement. In a biological PVS, the photoreceptors may in fact be birefringent or optically active, but this can be accounted for by adding appropriate phase delay terms to the P and Q matrices (birefringence) or adding off-diagonal terms to these matrices (optical activity). As mentioned in Section 2, to sense the complete state of polarization, at least one birefringent receptor is needed either to yield the intensity of a circular polarization state or to give a relative phase value.
  29. G. Buchsbaum, J. L. Goldstein. “Optimum probabilistic processing in colour perception. II. Colour vision as template matching,” Proc. R. Soc. London, Ser. B 205, 249–266 (1979).
    [CrossRef]
  30. The matrix given by Eq. (19) [as well as Eq. (10)] is real symmetric and therefore an example of a self-adjoint matrix. Self-adjoint M×M matrices always produce M orthogonal eigenvectors that span the vector space operated on by the matrix.20 Correlation matrices like the ones treated here are always self-adjoint, even if the random variables are complex.16 The fact that any M×M correlation matrix necessarily spawns M orthogonal eigenvectors is the basis of principal-components analysis.18
  31. In their study Bernard and Wehner13 assumed that the output of the individual receptors was proportional to the logarithm of the input. In this investigation the receptors are assumed to respond linearly within a particular range of intensities.
  32. J. S. Tyo, “Polarization-difference imaging: a means for seeing through scattering media,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, 1997).

1998 (1)

1996 (2)

1995 (1)

1994 (2)

1992 (1)

Z. Zhao, N. H. Farhat, “Tomographic microwave diversity image reconstruction employing unitary compression,” IEEE Trans. Microwave Theory Tech. 40, 315–322 (1992).
[CrossRef]

1991 (3)

D. A. Cameron, E. N. Pugh, “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature (London) 353, 161–164 (1991).
[CrossRef]

W. G. Egan, W. R. Johnson, V. S. Whitehead, “Terrestrial polarization imagery obtained from the Space Shuttle: Characterization and interpretation,” Appl. Opt. 30, 435–442 (1991).
[CrossRef] [PubMed]

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

1983 (1)

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

1981 (1)

1979 (1)

G. Buchsbaum, J. L. Goldstein. “Optimum probabilistic processing in colour perception. II. Colour vision as template matching,” Proc. R. Soc. London, Ser. B 205, 249–266 (1979).
[CrossRef]

1977 (1)

G. D. Bernard, R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef] [PubMed]

Alfano, R. R.

Anderson, T. W.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1984), Chap. 11.

Bermann, A.

A. Bermann, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979), Chap. 4.

Bernard, G. D.

G. D. Bernard, R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef] [PubMed]

Boult, T. E.

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

Buchsbaum, G.

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

G. Buchsbaum, J. L. Goldstein. “Optimum probabilistic processing in colour perception. II. Colour vision as template matching,” Proc. R. Soc. London, Ser. B 205, 249–266 (1979).
[CrossRef]

Cameron, D. A.

D. A. Cameron, E. N. Pugh, “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature (London) 353, 161–164 (1991).
[CrossRef]

Demos, S. G.

Easter, S. S.

Egan, W. G.

Engheta, N.

Farhat, N. H.

Z. Zhao, N. H. Farhat, “Tomographic microwave diversity image reconstruction employing unitary compression,” IEEE Trans. Microwave Theory Tech. 40, 315–322 (1992).
[CrossRef]

Garlick, G. F. J.

G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).

Gilbert, G. D.

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-I–A-III-10 (1966).

Goldstein, J. L.

G. Buchsbaum, J. L. Goldstein. “Optimum probabilistic processing in colour perception. II. Colour vision as template matching,” Proc. R. Soc. London, Ser. B 205, 249–266 (1979).
[CrossRef]

Gottschalk, A.

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

Johnson, W. R.

Kay, S. M.

S. M. Kay, Modern Spectral Estimation: Theory And Application (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Lamb, W. E.

G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).

Mancini, T. A.

L. B. Wolff, T. A. Mancini, “Liquid crystal polarization camera,” in Proceedings of the IEEE Workshop on Applications of Computer Vision (IEEE, New York, 1992), pp. 120–127.

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Radiation (Dover, New York, 1988), pp. 118–134.

Pernicka, J. C.

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-I–A-III-10 (1966).

Plemmons, R. J.

A. Bermann, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979), Chap. 4.

Pugh, E. N.

Rowe, M. P.

Shannon, C. E.

C. E. Shannon, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).

Solomon, J. E.

Steigman, C. A.

G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).

Tyo, J. S.

Walraven, R.

R. Walraven, “Polarization imagery,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. SPIE112, 164–167 (1977).
[CrossRef]

Wehner, R.

G. D. Bernard, R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef] [PubMed]

Whitehead, V. S.

Wolff, L. B.

L. B. Wolff, “Polarization camera for computer vision with a beam splitter,” J. Opt. Soc. Am. A 11, 2935–2945 (1994).
[CrossRef]

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

L. B. Wolff, T. A. Mancini, “Liquid crystal polarization camera,” in Proceedings of the IEEE Workshop on Applications of Computer Vision (IEEE, New York, 1992), pp. 120–127.

Zhao, Z.

Z. Zhao, N. H. Farhat, “Tomographic microwave diversity image reconstruction employing unitary compression,” IEEE Trans. Microwave Theory Tech. 40, 315–322 (1992).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Microwave Theory Tech. (1)

Z. Zhao, N. H. Farhat, “Tomographic microwave diversity image reconstruction employing unitary compression,” IEEE Trans. Microwave Theory Tech. 40, 315–322 (1992).
[CrossRef]

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

L. B. Wolff, T. E. Boult, “Constraining object features using a polarization reflectance model,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 635–657 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature (London) (1)

D. A. Cameron, E. N. Pugh, “Double cones as a basis for a new type of polarization vision in vertebrates,” Nature (London) 353, 161–164 (1991).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. London, Ser. B (2)

G. Buchsbaum, A. Gottschalk, “Trichromacy, opponent colours coding and optimum colour information transmission in the retina,” Proc. R. Soc. London, Ser. B 220, 89–113 (1983).
[CrossRef]

G. Buchsbaum, J. L. Goldstein. “Optimum probabilistic processing in colour perception. II. Colour vision as template matching,” Proc. R. Soc. London, Ser. B 205, 249–266 (1979).
[CrossRef]

Vision Res. (1)

G. D. Bernard, R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17, 1019–1028 (1977).
[CrossRef] [PubMed]

Other (18)

R. Walraven, “Polarization imagery,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. SPIE112, 164–167 (1977).
[CrossRef]

The four Stokes parameters can be determined (up to an ambiguity in sign of one of the parameters) with only three intensity measurements for monochromatic radiation.22 Because circular polarization and unpolarized light are indistinguishable with a system that detects only linear polarization states, their effects on the final representation should be similar, and partially circularly polarized monochromatic radiation is analyzed here for mathematical clarity.

This representation of the incident field is exact up to the choice of a constant. Although the linearly polarized portion of the radiation is constrained to have unit amplitude here, in general it can have arbitrary amplitude. In the general case, A is the ratio between the amplitudes of the circularly polarized and linearly polarized portions of the incident radiation.

Nonbirefringent, nonoptically active photoreceptors are considered for simplicity, since most linear polarization analyzers satisfy this requirement. In a biological PVS, the photoreceptors may in fact be birefringent or optically active, but this can be accounted for by adding appropriate phase delay terms to the P and Q matrices (birefringence) or adding off-diagonal terms to these matrices (optical activity). As mentioned in Section 2, to sense the complete state of polarization, at least one birefringent receptor is needed either to yield the intensity of a circular polarization state or to give a relative phase value.

The matrix given by Eq. (19) [as well as Eq. (10)] is real symmetric and therefore an example of a self-adjoint matrix. Self-adjoint M×M matrices always produce M orthogonal eigenvectors that span the vector space operated on by the matrix.20 Correlation matrices like the ones treated here are always self-adjoint, even if the random variables are complex.16 The fact that any M×M correlation matrix necessarily spawns M orthogonal eigenvectors is the basis of principal-components analysis.18

In their study Bernard and Wehner13 assumed that the output of the individual receptors was proportional to the logarithm of the input. In this investigation the receptors are assumed to respond linearly within a particular range of intensities.

J. S. Tyo, “Polarization-difference imaging: a means for seeing through scattering media,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, 1997).

Photoreceptors actually capture radiation in discrete quanta; however, if the intensity of the radiation is strong enough, the discretization can be ignored, and the above formulation can be applied.

S. M. Kay, Modern Spectral Estimation: Theory And Application (Prentice-Hall, Englewood Cliffs, N.J., 1988).

In this section the ensemble is assumed to be uniformly distributed in angle of polarization with a fixed ellipticity described by the parameter ∊. These results can be easily generalized to include other ensembles by simply altering the pdf’s used to calculate the entries in the correlation matrix.

T. W. Anderson, An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1984), Chap. 11.

A. Bermann, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic, New York, 1979), Chap. 4.

C. E. Shannon, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).

C. H. Papas, Theory of Electromagnetic Radiation (Dover, New York, 1988), pp. 118–134.

Although choosing N>3 does not seem to make sense in the context explored herein, some simple, fast, nonlinear systems may be proposed that operate optimally with N>3.

G. F. J. Garlick, C. A. Steigman, W. E. Lamb, “Differential optical polarization detectors,” U.S. patent3,992,571 (November16, 1976).

G. D. Gilbert, J. C. Pernicka, “Improvement of underwater visibility by reduction of backscatter with a circular polarization technique,” in Underwater Photo Optics I, A. B. Dember, ed., Proc. SPIE7, A-III-I–A-III-10 (1966).

L. B. Wolff, T. A. Mancini, “Liquid crystal polarization camera,” in Proceedings of the IEEE Workshop on Applications of Computer Vision (IEEE, New York, 1992), pp. 120–127.

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Figures (4)

Fig. 1
Fig. 1

Hypothetical N-channel PVS. The incident intensity is characterized by an intensity I, a polarization angle θ, and a polarization ellipse described by the parameter . The incident intensity is captured by N photoreceptors, each with a sensitivity function given by Pi. The processor computes the optimum linear combination channels given by Si. These optimum linear combination channels are then passed on for further processing.

Fig. 2
Fig. 2

Optimum linear combination strategy for a two-channel PVS. A: The individual receptor sensitivity functions. The individual receptors are assumed to be ideal linear polarization analyzers oriented at 0° and 90°. The incident radiation is assumed to be linearly polarized [A=0 in Eq. (7)]. These curves indicate the fraction of the incident intensity that is absorbed by each receptor as a function of the incident polarization angle. Even with orthogonally oriented linear polarization analyzers, there is still a nonzero correlation between the relative sensitivity functions. B: Optimum linear combination channels of the individual receptor sensitivity functions given in A. These functions are given by the sum (line) and the difference of the receptor sensitivity functions. These two functions are uncorrelated. The sum and difference channels are the optimum linear combination pair for any two-channel PVS with rotated identical photoreceptors regardless of relative orientation.

Fig. 3
Fig. 3

Optimum linear combination strategy for a three-channel PVS with receptors oriented at 0°, 60°, and -60°. A: Individual receptor sensitivity functions versus angle of incident polarization. These three sinusoidal curves are equally spaced throughout the angular band and should be the optimum arrangement of receptors.13 B: Optimum linear combination strategy made up of one sum channel (S) and two difference channels (D1, D2). These three channels are mutually orthogonal, have uncorrelated outputs, and can be used to define the basis directions in a polarization perceptual space. Note that the sum channel is independent of polarization angle, as was the case for the two-channel PVS with orthogonal receptors. This angularly independent channel is optimum for estimating the intensity of incident light, as shown in Ref. 29 in relation to color vision. The two difference channels are sinusoidal curves 90° out of phase. The sum of the squares of the two difference channels is a constant, also independent of angle of polarization. This angular uniformity may allow for a natural metric in the three-channel PVS perceptual space.

Fig. 4
Fig. 4

Optimum linear combination strategies for several angular distributions of the polarization-sensitive photoreceptors. The uniform angular distribution of 0° and ±60° shown in Fig. 3 has many beneficial properties, as discussed in the text and by Bernard and Wehner.13 The distributions here are not uniformly distributed across the angular band (0°–180°), so certain features like the uniformity of the sum channel with respect to angle and the equal weight of the two difference channels will be sacrificed. In each of the plots, the angular distributions of photoreceptors are 0° and ±ϕ°, where ϕ is shown at the top of each plot. Notice that the sum channel varies the most and that the maximum magnitude of the difference channels is approaching zero for small angular separations. If all three receptors are in the same direction (ϕ=0°), the sum channel is proportional to the sensitivity function of a single receptor, and the difference channel is identically zero.

Equations (36)

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

CP=C11C1NCN1CNN,
Cij=E[I2]π-π/2π/2Pi(θ)Pj(θ)dθ.
q1qN=WTp1pN
s1sN=XTp1pN
CS=XTCPX=λ1000000λN.
I(ϕ)=A sin(2ϕ+B)+C,
E(t)=A cos(±ωt+θ)+cos (θ) cos(ωt)A sin(±ωt+θ)+sin (θ) cos(ωt),
P=α00β,
Q=α cos2 ϕ+β sin2 ϕ12(α-β)sin(2ϕ)12(α-β)sin(2ϕ)β cos2 ϕ+α sin2 ϕ.
P(θ, A)=1E0212A2(α2+β2)+Aβ2+β22+A+12(α2-β2)cos2 θ,
Q(θ, A)=1E0212A2(α2+β2)+Aβ2+β22+A+12(α2-β2)cos2(θ-ϕ).
C=XYYX
X=E[I2]8E04[2(α2+β2)(A4+2A3+2A2+A+14)+(α2-β2)(A2+A+14)],
Y=E[I2]8E04[2(α2+β2)(A4+2A3+2A2+A+14)+(α2-β2)(A2+A+14)(cos2 ϕ-sin2 ϕ)].
S(θ, A)=ν1·P(θ, A)Q(θ, A)=P(θ, A)+Q(θ, A),
D(θ, A)=ν2·P(θ, A)Q(θ, A)=P(θ, A)-Q(θ, A)
λ1=E[I2]4E04[2(α2+β2)2(A4+2A3+A2+A+14)+(cos2 ϕ)(α2-β2)2(A2+A+14)],
λ2=E[I2]4E04(sin2 ϕ)(α2-β2)2(A2+A+14).
λ2λ1=OA2A4A0,
λ2λ1=(α2-β2)22(α2+β2)2β012.
Pi(θ, A)=1E0212A2(α2+β2)+A+12β2+A+12(α2-β2)cos2(θ-ϕi),
Pi(θ, A)=A1+A2 cos2(θ-ϕi),
Cij=E[pipj]=A12+A1A2+A224+A228[cos2(ϕi-ϕj)-sin2(ϕi-ϕj)].
ϕ1=0°,ϕ2=-60°,ϕ3=60°.
C=XYYYXYYYX,
X=A12+A1A2+3A22/8,
Y=A12+A1A2+3A22/16
λ1=X+2Y,νˆ1=13[111]T,
λ2=X-Y,νˆ2=a|a|(wherea·νˆ1=0),
λ3=X-Y,νˆ3=νˆ1×νˆ2.
νˆ2=1210-1,νˆ3=161-21,
S=13(P1+P2+P3),
D1=12(P1-P3),
D2=16(P1-2P2+P3),
ϕi=180N(i-1)°,
ϕ1=0°,ϕ2=45°,ϕ3=-45°.

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