Abstract

We present a rigorous electromagnetic formalism for defining, evaluating, and optimizing the degrees of freedom of an optical system. The analysis is valid for the delivery of information with electromagnetic waves under arbitrary boundary conditions communicating between domains in three-dimensional space. We show that, although in principle there is an infinity of degrees of freedom, the effective number is finite owing to the presence of noise. This is in agreement with the restricted classical theories that showed this property for specific optical systems and within the scalar and paraxial approximations. We further show that the best transmitting and receiving functions are the solutions of well-defined eigenvalue equations. The present approach is useful for understanding and designing modern optical systems for which the previous approaches are not applicable, as well as for application in inverse and synthesis problems.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  2. D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.
  3. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  4. D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. 23, 1645–1647 (1998).
    [CrossRef]
  5. D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. 39, 1681–1699 (2000).
    [CrossRef]
  6. D. Gabor, “Theory of information,” J. IEE 93, 429–153 (1946).
  7. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  8. H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 157–210.
  9. D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
    [CrossRef]
  10. D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  11. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
    [CrossRef]
  12. B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 313–407.
  13. F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
    [CrossRef]
  14. F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
    [CrossRef]
  15. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  16. A. W. Lohmann, “The space–bandwidth product applied to spatial filtering and holography,” (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.
  17. M. Bendinelli, A. Consortini, L. Ronchi, B. Roy Frieden, “Degrees of freedom, and eigenfunctions, for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974).
    [CrossRef]
  18. R. Barakat, “Shannon numbers of diffraction images,” Opt. Commun. 6, 391–394 (1982).
    [CrossRef]
  19. G. Newsam, R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985).
    [CrossRef]
  20. E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).
  21. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  22. A. Starikov, “Effective number of degrees of freedom of par-tially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [CrossRef]
  23. S. A. Basinger, E. Michielssen, D. J. Brady, “Degrees of freedom of polychromatic images,” J. Opt. Soc. Am. A 12, 704–714 (1995).
    [CrossRef]
  24. I. J. Cox, C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3, 1152–1158 (1986).
    [CrossRef]
  25. O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
    [CrossRef]
  26. R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
    [CrossRef]
  27. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
    [CrossRef]
  28. R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  29. L. Felsen, N. Markuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
  30. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1961).
  31. Note that by Lebesgue’s (dominated convergence) theorem32 this is possible if there exists an integrable majorant function m(r) such that ‖ΣibiaTi(r′)‖⩽m(r) for all i. This is satisfied, for example, if the basis functions aTi are bounded almost everywhere.
  32. See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, UK, 1990).
  33. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
    [CrossRef] [PubMed]
  34. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  35. R. Piestun, D. A. B. Miller, “Ill-posedness of the three-dimensional wave-field synthesis problem,” presented at the Annual Meeting of the Optical Society of America, Santa Clara, California, September 26–October 1, 1999, paper WLL27.
  36. See, for example, R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

2000 (1)

1998 (2)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. 23, 1645–1647 (1998).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

1989 (1)

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

1986 (1)

1985 (1)

1984 (1)

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

1982 (3)

1981 (1)

1976 (1)

D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
[CrossRef]

1974 (1)

1973 (1)

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

1969 (1)

1966 (1)

1961 (2)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

1955 (1)

1946 (1)

D. Gabor, “Theory of information,” J. IEE 93, 429–153 (1946).

Barakat, R.

Basinger, S. A.

Bendinelli, M.

Bertero, M.

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

Brady, D. J.

Bucci, O. M.

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Consortini, A.

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1961).

Cox, I. J.

de Mol, C.

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

Dorsch, R. G.

Felsen, L.

L. Felsen, N. Markuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

Ferreira, C.

Franceschetti, G.

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 313–407.

Gabor, D.

D. Gabor, “Theory of information,” J. IEE 93, 429–153 (1946).

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.

Gori, F.

F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
[CrossRef]

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Harrington, R.

R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1961).

Kress, R.

See, for example, R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

Landau, H. J.

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

A. W. Lohmann, “The space–bandwidth product applied to spatial filtering and holography,” (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.

Lukosz, W.

Markuvitz, N.

L. Felsen, N. Markuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

McWhirter, J. G.

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

Mendlovic, D.

Michielssen, E.

Miller, D. A. B.

D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. 39, 1681–1699 (2000).
[CrossRef]

D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. 23, 1645–1647 (1998).
[CrossRef]

R. Piestun, D. A. B. Miller, “Ill-posedness of the three-dimensional wave-field synthesis problem,” presented at the Annual Meeting of the Optical Society of America, Santa Clara, California, September 26–October 1, 1999, paper WLL27.

Newsam, G.

Pierri, R.

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

Piestun, R.

R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
[CrossRef]

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994).
[CrossRef] [PubMed]

R. Piestun, D. A. B. Miller, “Ill-posedness of the three-dimensional wave-field synthesis problem,” presented at the Annual Meeting of the Optical Society of America, Santa Clara, California, September 26–October 1, 1999, paper WLL27.

Pike, E. R.

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

Pollack, H. O.

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Porter, D.

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, UK, 1990).

Ronchi, L.

Roy Frieden, B.

Shamir, J.

Sheppard, C. J. R.

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Slepian, D.

D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
[CrossRef]

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Soldovieri, F.

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

Spektor, B.

Starikov, A.

Stirling, D. S. G.

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, UK, 1990).

Toraldo di Francia, G.

Wolf, E.

Wolter, H.

H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 157–210.

Zalevsky, Z.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

IEE Proc. F (1)

E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).

IEEE Trans. Antennas Propag. (1)

O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Inverse Probl. (1)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

J. IEE (1)

D. Gabor, “Theory of information,” J. IEE 93, 429–153 (1946).

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

R. Barakat, “Shannon numbers of diffraction images,” Opt. Commun. 6, 391–394 (1982).
[CrossRef]

F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
[CrossRef]

Other (12)

R. Piestun, D. A. B. Miller, “Ill-posedness of the three-dimensional wave-field synthesis problem,” presented at the Annual Meeting of the Optical Society of America, Santa Clara, California, September 26–October 1, 1999, paper WLL27.

See, for example, R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

L. Felsen, N. Markuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1961).

Note that by Lebesgue’s (dominated convergence) theorem32 this is possible if there exists an integrable majorant function m(r) such that ‖ΣibiaTi(r′)‖⩽m(r) for all i. This is satisfied, for example, if the basis functions aTi are bounded almost everywhere.

See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, UK, 1990).

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 157–210.

B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 313–407.

A. W. Lohmann, “The space–bandwidth product applied to spatial filtering and holography,” (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Coherent imaging system used in the classical theory of DOF.

Fig. 2
Fig. 2

Communication with EM waves between transmitting (VT) and receiving (VR) domains in the presence of material bodies (B).

Fig. 3
Fig. 3

1D transmitting and receiving domains used in the first example of EM DOF calculation in 3D space (λ=500 nm).

Fig. 4
Fig. 4

Calculated eigenvalues for the system of Fig. 3. Note the rapid reduction of the coupling strengths.

Fig. 5
Fig. 5

Singular functions for the system of Fig. 3. (a) Source functions ςn, (b) x component of the receiving functions ϵnx, (c) z component of the receiving functions ϵnz. The solid curves represent the real part of the functions; the dashed curves represent the imaginary part.

Fig. 6
Fig. 6

1D transmitting and 3D receiving domains used in the second example of EM DOF d=1.04λ, a=0.04λ, b=0.07λ, c=0.11λ (λ=500 nm).

Fig. 7
Fig. 7

Calculated eigenvalues for the system of Fig. 6.

Fig. 8
Fig. 8

Source singular functions ςn for the system of Fig. 6. Solid curve, real part of the function; dashed curve, imaginary part.

Equations (61)

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

-11k(x-y)ψn(y)dy=λnψn(x),
o(x)=n=0αnψnxX,
i(y)=n=0λnαnψnyX.
-×E=jωμH+M,
×H=(σ+jωϵ)E+J,
E(r)=VTΓ11(r, r)J(r)dr+VTΓ12(r, r)M(r)dr,
H(r)=VTΓ21(r, r)J(r)dr+VTΓ22(r, r)M(r)dr,
E(r)=VTΓ(r, r)J(r)dr.
aT1(r), aT2(r) , aTi(r) ,,
aR1(r), aR2(r), aRj(r) ,,
J(r)=ibiaTi(r),E(r)=jdjaRj(r),
jdjaRj(r)=ibiVTΓ(r, r)aTi(r)dr
dj=igjibi,
gji=VRVTaRj*T(r)Γ(r, r)aTi(r)drdr=g[aTi, aRj]
Γ(r, r)=ijgjiaRj(r)aTi*T(r),
S=ij|gji|2=VTVRΓ(r, r)2drdr.
|g[JN, aR]|2|g[JN, EN]|2,
aR:aRR(VR),aR=1.
|g|2=|g[JN,EN]|2=(E, E)VR=VRET(r)E*(r)dr=VTVTJNT(r)K(r,r)JN*(r)drdr,
K(r, r)=VRΓT(r, r)Γ*(r, r)dr.
KJ=VTK*(r, r)J(r)dr.
|g|2=(J, KJ)=(KJ, J),
Kςn=|gn|2ςn.
K(r, r)=n|gn|2ςn(r)ςn*T(r).
|gn|ϵn(r)=En(r)=VTΓ(r, r)ςn(r)dr
ϵn=En(En, En)VR1/2=En|gn|.
VR Γ*T(r, r)|gn|ϵn(r)dr=VTK*(r,r)ςn(r)dr,
VR Γ*T(r, r)ϵn(r)dr=|gn|ςn(r),
|gn|2ϵn(r)=VRL*(r1, r)ϵn(r1)dr1=Lϵn(r),
L(r1, r)=VTΓ*(r, r)ΓT(r1, r)dr,
n|gn|2=VTVRΓ(r, r)2drdr.
GJ=VTΓ(r, r)J(r)dr,
G+E=VRΓ*T(r, r)E(r)dr.
Gςn(r)=|gn|ϵn(r),G+ϵn(r)=|gn|ςnn,
J=n=1(J, ςn)ςn+PJ,GJ=n=1|gn|(J, ςn)ϵn,
G˜=VTΓ21(r, r)J(r)dr,
×Γ21-jωϵ0Γ11=δ(r-r)I,
×Γ11+jωμΓ21=0,
×Γ22-jωϵ0Γ12=0,
×Γ12+jωμ0Γ22=-δ(r-r)I,
γkk=-jωμ+1jωϵ2k2G,
γkl=1jωϵ2Gkl,kl,
G=exp(-jk|r-r|)4π|r-r|.
Γ(r, r)=ici(r)aTi*T(r).
VT Γ(r, r)2dr=i|ci(r)|2.
ci(r)=jgjiaRj(r)
VR |ci(r)|2dr=j|gji|2.
VRVT Γ(r, r)2drdr=VR i |ci(r)|2dr.
VRVT Γ(r, r)2drdr=iVR |ci(r)|2dr=ij|gji|2,
|(EN, aR)VR|2(EN, EN)VR(aR, aR)VR=1,
|(E, aR)VR|2|(E, EN)VR|2,
|g[JN, aR]|2|g[JN, EN]|2.
K(n)-K=supJ1(K(n)-K)Jp,q=13supJ1VT|[kpq(n)-kpq]jq|2dr,
p,q=13supjq1[kpq(n)-kpq]jq=p,q=13[kpq(n)-kpq]n0,
Axn=μnyn,A+yn=μnx(n=1, 2, ).
x=n=1(x, xn)xn+Px,
Ax=n=1μn(x, xn)yn.
Ax=μx(xH)
PNAy=vy,
n=1N(Mnm-vInm)αn=0,m=1  N,
det(Mnm-vInm)=0.

Metrics