Abstract

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input–output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  2. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  3. R. A. Gonsalves, J. Opt. Soc. Am. 66, 961 (1976).
    [CrossRef]
  4. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [CrossRef] [PubMed]
  5. J. R. Fienup, Opt. Eng. 18, 529 (1979).
    [CrossRef]
  6. J. R. Fienup, Opt. Eng. 19, 297 (1980).
    [CrossRef]
  7. J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, Eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1982), to be published.
  8. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (9Nov., 1971).
  9. N. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  10. J. R. Fienup, T. R. Crimmins, W. Holsztynski, J. Opt. Soc. Am. 72, 610 (1982).
    [CrossRef]
  11. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  12. R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).
  13. M. T. Manry, J. K. Aggarwal, IEEE Trans. Circuits Syst. CAS-23, 185 (1976).
    [CrossRef]
  14. D. P. Feder, Appl. Opt. 2, 1209 (1963).
    [CrossRef]
  15. D. R. Buchele, Appl. Opt. 7, 2433 (1968).
    [CrossRef] [PubMed]
  16. B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111 (1976) (Abstract).
    [CrossRef]
  17. J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Stanford University, May, 1975 (University Microfilms No. 75-25523), Chap. 5.
  18. G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).
  19. J. R. Fienup, “Fourier Modulus Image Construction,” Report RADC-TR-81-63 (1981).
  20. A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. J. Lett. 173, L1 (1972).
    [CrossRef]
  21. J. W. Goodman, J. F. Belsher, Proc. Soc. Photo-Opt. Instrum. Eng. 75, 141 (1976).
  22. A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).
  23. A. J. Devaney, R. Chidlaw, J. Opt. Soc. Am. 68, 1352 (1978).
    [CrossRef]
  24. Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
    [CrossRef]
  25. W. Lawton, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94 (1980).
  26. A. M. J. Huiser, P. Van Toorn, Opt. Lett. 5, 499 (1980).
    [CrossRef] [PubMed]
  27. T. R. Crimmins, J. R. Fienup, J. Opt. Soc. Am. 71, 1026 (1981).
    [CrossRef]
  28. J. R. Fienup, “Image Reconstruction for Stellar Interferometry,” in Current Trends in Optics, F. T. Arecchi, F. R. Aussenegg, Eds. (Taylor & Francis, London, 1981), pp. 95–102.

1982 (1)

1981 (1)

1980 (5)

W. Lawton, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94 (1980).

A. M. J. Huiser, P. Van Toorn, Opt. Lett. 5, 499 (1980).
[CrossRef] [PubMed]

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

1979 (2)

J. R. Fienup, Opt. Eng. 18, 529 (1979).
[CrossRef]

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[CrossRef]

1978 (2)

1976 (5)

R. A. Gonsalves, J. Opt. Soc. Am. 66, 961 (1976).
[CrossRef]

M. T. Manry, J. K. Aggarwal, IEEE Trans. Circuits Syst. CAS-23, 185 (1976).
[CrossRef]

J. W. Goodman, J. F. Belsher, Proc. Soc. Photo-Opt. Instrum. Eng. 75, 141 (1976).

A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111 (1976) (Abstract).
[CrossRef]

1973 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1970 (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. J. Lett. 173, L1 (1972).
[CrossRef]

1968 (1)

1963 (1)

Aggarwal, J. K.

M. T. Manry, J. K. Aggarwal, IEEE Trans. Circuits Syst. CAS-23, 185 (1976).
[CrossRef]

Belsher, J. F.

J. W. Goodman, J. F. Belsher, Proc. Soc. Photo-Opt. Instrum. Eng. 75, 141 (1976).

Boucher, R. H.

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Bruck, Yu. M.

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[CrossRef]

Buchele, D. R.

Chidlaw, R.

Crimmins, T. R.

Currie, D. G.

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111 (1976) (Abstract).
[CrossRef]

Devaney, A. J.

Drenth, A. J. J.

A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).

Feder, D. P.

Feldkamp, G. B.

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

Ferwerda, H. A.

A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).

Fienup, J. R.

J. R. Fienup, T. R. Crimmins, W. Holsztynski, J. Opt. Soc. Am. 72, 610 (1982).
[CrossRef]

T. R. Crimmins, J. R. Fienup, J. Opt. Soc. Am. 71, 1026 (1981).
[CrossRef]

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

J. R. Fienup, Opt. Eng. 18, 529 (1979).
[CrossRef]

J. R. Fienup, Opt. Lett. 3, 27 (1978).
[CrossRef] [PubMed]

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, Eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1982), to be published.

J. R. Fienup, “Fourier Modulus Image Construction,” Report RADC-TR-81-63 (1981).

J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Stanford University, May, 1975 (University Microfilms No. 75-25523), Chap. 5.

J. R. Fienup, “Image Reconstruction for Stellar Interferometry,” in Current Trends in Optics, F. T. Arecchi, F. R. Aussenegg, Eds. (Taylor & Francis, London, 1981), pp. 95–102.

Frieden, B. R.

B. R. Frieden, D. G. Currie, J. Opt. Soc. Am. 66, 1111 (1976) (Abstract).
[CrossRef]

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, J. F. Belsher, Proc. Soc. Photo-Opt. Instrum. Eng. 75, 141 (1976).

Hirsch, P. M.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (9Nov., 1971).

Holsztynski, W.

Huiser, A. M. J.

A. M. J. Huiser, P. Van Toorn, Opt. Lett. 5, 499 (1980).
[CrossRef] [PubMed]

A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).

Jordan, J. A.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (9Nov., 1971).

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. J. Lett. 173, L1 (1972).
[CrossRef]

Lawton, W.

W. Lawton, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94 (1980).

Lesem, L. B.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (9Nov., 1971).

Liu, B.

Manry, M. T.

M. T. Manry, J. K. Aggarwal, IEEE Trans. Circuits Syst. CAS-23, 185 (1976).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Sodin, L. G.

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[CrossRef]

Van Toorn, P.

Appl. Opt. (3)

Astron. Astrophys. (1)

A. Labeyrie, Astron. Astrophys. 6, 85 (1970); D. Y. Gezari, A. Labeyrie, R. V. Stachnik, Astrophys. J. Lett. 173, L1 (1972).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

M. T. Manry, J. K. Aggarwal, IEEE Trans. Circuits Syst. CAS-23, 185 (1976).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

Yu. M. Bruck, L. G. Sodin, Opt. Commun. 30, 304 (1979).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, Opt. Eng. 18, 529 (1979).
[CrossRef]

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Opt. Lett. (2)

Optik (2)

A. M. J. Huiser, A. J. J. Drenth, H. A. Ferwerda, Optik 45, 303 (1976); A. M. J. Huiser, H. A. Ferwerda, Optik 46, 407 (1976).

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

W. Lawton, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 94 (1980).

J. W. Goodman, J. F. Belsher, Proc. Soc. Photo-Opt. Instrum. Eng. 75, 141 (1976).

G. B. Feldkamp, J. R. Fienup, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 84 (1980).

Other (7)

J. R. Fienup, “Fourier Modulus Image Construction,” Report RADC-TR-81-63 (1981).

J. R. Fienup, “Image Reconstruction for Stellar Interferometry,” in Current Trends in Optics, F. T. Arecchi, F. R. Aussenegg, Eds. (Taylor & Francis, London, 1981), pp. 95–102.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

J. R. Fienup, “Improved Synthesis and Computational Methods for Computer-Generated Holograms,” Ph.D. Thesis, Stanford University, May, 1975 (University Microfilms No. 75-25523), Chap. 5.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

J. R. Fienup, “Reconstruction and Synthesis Applications of an Iterative Algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, Eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1982), to be published.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (9Nov., 1971).

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 (10)

Fig. 1
Fig. 1

Block diagram of the error-reduction (Gerchberg-Saxton) algorithm.

Fig. 2
Fig. 2

RMS error vs the number of iterations for the problem of phase retrieval from a single intensity measurement using the error-reduction algorithm.

Fig. 3
Fig. 3

Block diagram for the gradient-search methods using the method of Fourier transforms to compute the gradient.

Fig. 4
Fig. 4

Block diagram of the system for the input–output concept.

Fig. 5
Fig. 5

RMS error vs the number of iterations using the error-reduction algorithm.

Fig. 6
Fig. 6

RMS error after a fixed number of iterations vs the algorithm parameter. Curve A: steepest-descent method (○); B: conjugate-gradient method (●); C: basic input–output algorithm (△); D: output–output algorithm (▲); E: hybrid input–output algorithm (□). The result using the error-reduction algorithm is indicated by a large circle.

Fig. 7
Fig. 7

RMS error after a fixed number of iterations (using the alternative strategy) vs the algorithm parameter. Curve A: steepest-descent method (○); B: basic input–output algorithm (●); C: output–output algorithm (△); D: hybrid input–output algorithm (▲); E: the algorithm of Eq. (40) (□).

Fig. 8
Fig. 8

RMS error vs the number of iterations for the error-reduction algorithm (curve A), and for the hybrid input–output algorithm (curve B—see text).

Fig. 9
Fig. 9

Image reconstruction experiments. (a) Undegraded object; (b), (c) examples of degraded images simulated to include the effects of atmospheric turbulence and photon noise; (d) Fourier modulus estimate computed from the degraded images; (e) image reconstructed using the iterative algorithm.

Fig. 10
Fig. 10

RMS error vs the number of iterations.

Equations (62)

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

F ( u ) = F ( u ) exp [ i ψ ( u ) ] = F [ f ( x ) ] = - f ( x ) exp ( - i 2 π u · x ) d x ,
F ( u ) = x = 0 N - 1 f ( x ) exp ( - i 2 π u · x / N )
f ( x ) = N - 2 u = 0 N - 1 F ( u ) exp ( i 2 π u · x / N ) ,
f ( x ) = f ( x ) exp [ i η ( x ) ]
f ( x ) 0.
G k ( u ) = G k ( u ) exp [ i ϕ k ( u ) ] = F [ g k ( x ) ] ,
G k ( u ) = F ( u ) exp [ i ϕ k ( u ) ] ,
g k ( x ) = g k ( x ) exp [ i ϕ k ( x ) ] = F - 1 [ G k ( u ) ] ,
g k + 1 ( x ) = f ( x ) exp [ i θ k + 1 ( x ) ] = f ( x ) exp [ i θ k ( x ) ] ,
g k + 1 ( x ) = { g k ( x ) , x ɛ̸ γ , 0 , x ɛ γ ,
B k = E F k 2 = N - 2 u G k ( u ) - G k ( u ) 2 ,
B k = E F k 2 = N - 2 u [ G k ( u ) - F ( u ) ] 2 .
E 0 k 2 = x g k + 1 ( x ) - g k ( x ) 2 ,
E 0 k 2 = x [ f ( x ) - g k ( x ) ] 2
E 0 k 2 = x ɛ γ [ g k ( x ) ] 2 ,
E F k 2 = N - 2 u G k ( u ) - G k ( u ) 2 = x g k ( x ) - g k ( x ) 2 .
g k + 1 ( x ) - g k ( x ) g k ( x ) - g k ( x ) ,
E 0 2 E F k 2 .
E 0 k 2 = x g k + 1 ( x ) - g k ( x ) 2 = N - 2 u G k + 1 ( u ) - G k ( x ) 2 .
G k + 1 ( u ) - G k + 1 ( u ) G k + 1 ( u ) - G k ( u ) .
E F , k + 1 2 E 0 k 2 ,
E F , k + 1 2 E 0 k 2 E F k 2 .
g B B g ( x ) = 2 N - 2 u [ G ( u ) - F ( u ) ] G ( u ) g ( x ) .
G ( u ) g ( x ) = g ( x ) y g ( y ) exp [ - i 2 π u · y / N ] , = exp [ - i 2 π u · x / N ] ,
G ( u ) g ( x ) = [ G ( u ) 2 ] 1 / 2 g ( x ) = 1 2 G ( u ) G ( u ) 2 g ( x ) , = G ( u ) exp [ i 2 π u · x / N ] + G * ( u ) exp [ - i 2 π u · x / N ] 2 G ( u ) .
g B = N - 2 u [ G ( u ) - F ( u ) G ( u ) / G ( u ) ] exp [ i 2 π u · x / N ] + N - 2 u [ G * ( u ) - F ( u ) G * ( u ) / G ( u ) ] × exp [ - i 2 π u · x / N ] .
G ( u ) = F ( u ) G ( u ) / G ( u ) ,
g B = 2 [ g ( x ) - g ( x ) ] ,
B B k + x g B k [ g ( x ) - g k ( x ) ] .
g k ( x ) - g k ( x ) = - B k g B k y ( g B k ) 2 ,
y ( g B k ) 2 = 4 y [ g k ( y ) - g k ( y ) ] 2 = 4 B k .
g k ( x ) - g k ( x ) = - ( ¼ ) g B k = ( ½ ) [ g k ( x ) - g k ( x ) ] .
g k ( x ) - g k ( x ) = [ g k ( x ) - g k ( x ) ]
g k ( x ) = g k ( x ) .
g k ( x ) = g k ( x ) - h k g B k ,
g k + 1 ( x ) = { g k ( x ) , x ɛ̸ γ , 0 , x ɛ γ ,
g k ( x ) = g k ( x ) + h k D k ( x ) ,
D k ( x ) = - ( ½ ) g B k + [ y ( g B k ) 2 / y ( g B k - 1 ) 2 ] D k - 1 ( x ) ,
D k ( x ) = g k ( x ) - g k ( x ) + ( B k / B k - 1 ) D k - 1 ( x ) ,
D k ( x ) = g k ( x ) - g k - 1 ( x ) .
g k ( x ) = g k ( x ) + h k [ g k ( x ) - g k - 1 ( x ) ] ,
Δ g k ( x ) = { 0 , x ɛ̸ γ , - g k ( x ) , x ɛ γ ,
g k + 1 ( x ) = g k ( x ) + β Δ g k ( x ) = { g k ( x ) , x ɛ̸ γ , g k ( x ) - β g k ( x ) , x ɛ γ .
g k + 1 ( x ) = g k ( x ) + β Δ g k ( x ) = { g k ( x ) , x ɛ̸ γ , g k ( x ) - β g k ( x ) , x ɛ γ .
g k + 1 ( x ) = { g k ( x ) , x ɛ̸ γ , g k ( x ) - β g k ( x ) , x ɛ γ .
F ^ ( u ) = W ( u ) [ m = 1 M I m ( u ) 2 - N p m = M + 1 2 M S m ( u ) 2 ] 1 / 2 ,
θ B = B θ ( x ) = 2 N - 2 u [ G ( u ) - F ( u ) ] G ( u ) θ ( x ) .
G ( u ) θ ( x ) = θ ( x ) y f ( y ) exp [ i θ ( y ) ] exp [ - i 2 π u · y / N ] = i f ( x ) exp [ i θ ( x ) ] exp [ - i 2 π u · x / N ] = i g ( x ) exp [ - i 2 π u · x / N ] ,
G ( u ) θ ( x ) = G ( u ) ( - i ) g * ( x ) exp [ i 2 π u · x / N ] + c . c . 2 G ( u ) ,
θ B = i g * ( x ) [ g ( x ) - g ( x ) ] + c . c . = i g * ( x ) g ( x ) + c . c . = - 2 Im [ g * ( x ) g ( x ) ] = - 2 f ( x ) g ( x ) sin [ θ ( x ) - θ ( x ) ] ,
B B k + x θ B k [ θ ( x ) - θ k ( x ) ] ,
θ ( x ) - θ k ( x ) = - B k θ B k y ( θ B k ) 2 ,
y ( θ B k ) 2 = 4 y f ( y ) 2 g k ( y ) 2 × sin 2 [ θ k ( y ) - θ k ( y ) ] ,
θ ( x ) - θ k ( x ) = B k f ( x ) g k ( x ) sin [ θ k ( x ) - θ k ( x ) ] 2 y f ( y ) 2 g k ( y ) 2 sin 2 [ θ k ( y ) - θ k ( y ) ] .
sin [ θ k ( x ) - θ k ( x ) ] θ k ( x ) - θ k ( x ) .
B k = x f ( x ) exp [ i θ k ( x ) ] - g k ( x ) exp [ i θ k ( x ) ] 2 = x { f ( x ) 2 + g k ( x ) 2 - 2 f ( x ) g k ( x ) cos [ θ k ( x ) - θ k ( x ) ] } x [ f ( x ) - g k ( x ) ] 2 + x f ( x ) g k ( x ) × sin 2 [ θ k ( x ) - θ k ( x ) ] .
f ( x ) - g k ( x ) f ( x ) .
θ ( x ) - θ k ( x ) 1 2 ( 1 + B k r B k t ) [ θ k ( x ) - θ k ( x ) ] ,
B k r = x [ f ( x ) - g k ( x ) ] 2 ,
B k t = x g k ( x ) 2 [ θ k ( x ) - θ k ( x ) ] 2
B k t B k - B k r = E F k 2 - E 0 k 2 E 0 k 2 = B k r .
½ [ 1 + B k r B k t ] 1.

Metrics