Abstract

Underlying many techniques of image restortation, quantization, and enhancement is the mathematically convenient, but visually unsuitable distortion measure of squared difference in intensity. Squared-intensity difference has an indirect phenomenological correspondence in a model of the visual system. We have undertaken, therefore, an experiment that derives a new distortion measure from an acceptable visual system model and compares it in a fair test against squared difference in intensity in an image restoration task. We start with an eye–brain system model, inferred from the works of current vision researchers, which consists of a bank of parallel spatial frequency channels and image detectors. From this model we derive a new distortion criterion that is related to changes in the per-channel detection probability and phase angle. The optimal linear (Wiener) filters for each distortion measure operate in turn on the same noisy incoherent images. The results show that the filter for the new distortion measure yields a superior restoration. It is more visually agreeable, more sharply detailed, and truer in contrast compared to the squared-difference filter, and impressive in its own right. Its mathematical properties suggest that significantly increased efficiency in the storage or communication of images may be gained by its use.

© 1978 Optical Society of America

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References

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  1. O. H. Schade, “Optical and photoelectric analog of the eye,” J. Opt. Soc. Am. 46, 721 (1956).
    [Crossref] [PubMed]
  2. T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–847 (1972).
    [Crossref]
  3. J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
    [Crossref]
  4. C. F. Hall and E. L. Hall, “A nonlinear model for the spatial characteristics of the human visual system,” IEEE Trans. Syst. Man and Cybern. 7, 161–170 (1977).
    [Crossref]
  5. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).
  6. C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517 (1966).
  7. F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).
  8. C. Blakemore and F. W. Campbell, “On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).
  9. A. Pantle and R. Sekular, “Size-detecting mechanisms in human vision,” Science 1621146–1148 (1968).
    [Crossref] [PubMed]
  10. P. Tynan and R. Sekular, “Perceived spatial frequency varies with stimulus duration,” J. Opt. Soc. Am. 64, 1251–1255 (1974).
    [Crossref] [PubMed]
  11. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. F. W. Campbell, J. Nachmias, and J. Jukes, “Spatial-frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
    [Crossref] [PubMed]
  14. H. Mostafavi and D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
    [Crossref] [PubMed]
  15. A. D. Schnitzler, “Image-detector model and parameters of the human visual system,” J. Opt. Soc. Am. 631357–1368 (1973).
    [Crossref] [PubMed]
  16. H. Richard Blackwell, “Neural theories of simple visual discrimination,” J. Opt. Sc. Am. 53, 129 (1963).
    [Crossref]
  17. H. R. Blackwell and J. H. Taylor, in Proceedings of the NATO Seminar on Detection, Recognition, and Identification of Line-of-Sight Targets (unpublished); also, University of Michigan Engineering Research Institute Report No. 2455-10-F (1958) (unpublished).
  18. J. Johnson, Image Intensity Symposium, Fort Belvoir, Va., October 6–7, 1958, AD220160 (unpublished).
  19. A. D. Schnitzler, “Analysis of noise-required contrast and modulation in image-detecting and display systems,” In Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973).
    [Crossref]
  20. A. D. Schnitzler, “Theory of spatial-frequency filtering by the human visual system. I. Performance limited by quantum noise,” J. Opt. Soc. Am. 66, 608–617 (1976).
    [Crossref] [PubMed]
  21. The spatial bandwidth of the channels is in dispute. Generally, experiments without grating adaptation produce narrow channels and, with adaptation, wide channels. We assume here that adaptation has not taken place.
  22. W. A. Pearlman, “A limit on optimum performance degradation in fixed-rate coding of the Discrete Fourier Transform,” IEEE Trans. Inf. Theory IT-22, 485–488 (1976).
    [Crossref]
  23. W. A. Pearlman, Quantization Error Bounds for Computer-Generated Holograms, Technical Report No. 6503-1, Information System Laboratory, Stanford University, Stanford, Calif. accessed August1974 (unpublished).
  24. The DFT of the Gaussian noise process alone is exactly Gaussian with statistically independent nonredundant coefficients due to the linearity of the DFT.
  25. P. D. Welch, “The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio and Electroacoust. 15, 70–73 (1967).
    [Crossref]
  26. G. Senge, Quantization of Image Transforms with Minimum Distortion, Technical Report No. ECE-77-8, Dept. of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison, Wis., May, 1977 (unpublished).

1977 (1)

C. F. Hall and E. L. Hall, “A nonlinear model for the spatial characteristics of the human visual system,” IEEE Trans. Syst. Man and Cybern. 7, 161–170 (1977).
[Crossref]

1976 (3)

H. Mostafavi and D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

A. D. Schnitzler, “Theory of spatial-frequency filtering by the human visual system. I. Performance limited by quantum noise,” J. Opt. Soc. Am. 66, 608–617 (1976).
[Crossref] [PubMed]

W. A. Pearlman, “A limit on optimum performance degradation in fixed-rate coding of the Discrete Fourier Transform,” IEEE Trans. Inf. Theory IT-22, 485–488 (1976).
[Crossref]

1974 (2)

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[Crossref]

P. Tynan and R. Sekular, “Perceived spatial frequency varies with stimulus duration,” J. Opt. Soc. Am. 64, 1251–1255 (1974).
[Crossref] [PubMed]

1973 (1)

1972 (1)

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–847 (1972).
[Crossref]

1971 (2)

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: A comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[Crossref] [PubMed]

1970 (1)

1969 (2)

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

C. Blakemore and F. W. Campbell, “On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

1968 (2)

A. Pantle and R. Sekular, “Size-detecting mechanisms in human vision,” Science 1621146–1148 (1968).
[Crossref] [PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).

1967 (1)

P. D. Welch, “The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio and Electroacoust. 15, 70–73 (1967).
[Crossref]

1966 (1)

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517 (1966).

1963 (1)

H. Richard Blackwell, “Neural theories of simple visual discrimination,” J. Opt. Sc. Am. 53, 129 (1963).
[Crossref]

1956 (1)

Blackwell, H. R.

H. R. Blackwell and J. H. Taylor, in Proceedings of the NATO Seminar on Detection, Recognition, and Identification of Line-of-Sight Targets (unpublished); also, University of Michigan Engineering Research Institute Report No. 2455-10-F (1958) (unpublished).

Blakemore, C.

C. Blakemore and F. W. Campbell, “On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Campbell, F. W.

F. W. Campbell, J. Nachmias, and J. Jukes, “Spatial-frequency discrimination in human vision,” J. Opt. Soc. Am. 60, 555–559 (1970).
[Crossref] [PubMed]

C. Blakemore and F. W. Campbell, “On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).

Cooper, G. F.

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

Enroth-Cugell, C.

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517 (1966).

Graham, N.

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: A comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[Crossref] [PubMed]

Hall, C. F.

C. F. Hall and E. L. Hall, “A nonlinear model for the spatial characteristics of the human visual system,” IEEE Trans. Syst. Man and Cybern. 7, 161–170 (1977).
[Crossref]

Hall, E. L.

C. F. Hall and E. L. Hall, “A nonlinear model for the spatial characteristics of the human visual system,” IEEE Trans. Syst. Man and Cybern. 7, 161–170 (1977).
[Crossref]

Johnson, J.

J. Johnson, Image Intensity Symposium, Fort Belvoir, Va., October 6–7, 1958, AD220160 (unpublished).

Jukes, J.

Mannos, J. L.

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[Crossref]

Mostafavi, H.

H. Mostafavi and D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Nachmias, J.

Pantle, A.

A. Pantle and R. Sekular, “Size-detecting mechanisms in human vision,” Science 1621146–1148 (1968).
[Crossref] [PubMed]

Pearlman, W. A.

W. A. Pearlman, “A limit on optimum performance degradation in fixed-rate coding of the Discrete Fourier Transform,” IEEE Trans. Inf. Theory IT-22, 485–488 (1976).
[Crossref]

W. A. Pearlman, Quantization Error Bounds for Computer-Generated Holograms, Technical Report No. 6503-1, Information System Laboratory, Stanford University, Stanford, Calif. accessed August1974 (unpublished).

Richard Blackwell, H.

H. Richard Blackwell, “Neural theories of simple visual discrimination,” J. Opt. Sc. Am. 53, 129 (1963).
[Crossref]

Robson, J. G.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517 (1966).

Sachs, M. B.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

Sakrison, D. J.

H. Mostafavi and D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[Crossref]

Schade, O. H.

Schnitzler, A. D.

Sekular, R.

Senge, G.

G. Senge, Quantization of Image Transforms with Minimum Distortion, Technical Report No. ECE-77-8, Dept. of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison, Wis., May, 1977 (unpublished).

Stockham, T. G.

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–847 (1972).
[Crossref]

Taylor, J. H.

H. R. Blackwell and J. H. Taylor, in Proceedings of the NATO Seminar on Detection, Recognition, and Identification of Line-of-Sight Targets (unpublished); also, University of Michigan Engineering Research Institute Report No. 2455-10-F (1958) (unpublished).

Tynan, P.

Welch, P. D.

P. D. Welch, “The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio and Electroacoust. 15, 70–73 (1967).
[Crossref]

IEEE Trans. Audio and Electroacoust. (1)

P. D. Welch, “The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. Audio and Electroacoust. 15, 70–73 (1967).
[Crossref]

IEEE Trans. Inf. Theory (2)

W. A. Pearlman, “A limit on optimum performance degradation in fixed-rate coding of the Discrete Fourier Transform,” IEEE Trans. Inf. Theory IT-22, 485–488 (1976).
[Crossref]

J. L. Mannos and D. J. Sakrison, “The effects of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[Crossref]

IEEE Trans. Syst. Man and Cybern. (1)

C. F. Hall and E. L. Hall, “A nonlinear model for the spatial characteristics of the human visual system,” IEEE Trans. Syst. Man and Cybern. 7, 161–170 (1977).
[Crossref]

J. Opt. Sc. Am. (1)

H. Richard Blackwell, “Neural theories of simple visual discrimination,” J. Opt. Sc. Am. 53, 129 (1963).
[Crossref]

J. Opt. Soc. Am. (6)

J. Physiol. (1)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).

J. Physiol. (London) (3)

C. Enroth-Cugell and J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. (London) 187, 517 (1966).

F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, “The spatial selectivity of visual cells of cat and the squirrel monkey,” J. Physiol. (London) 204, 120 (1969).

C. Blakemore and F. W. Campbell, “On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Proc. IEEE (1)

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828–847 (1972).
[Crossref]

Science (1)

A. Pantle and R. Sekular, “Size-detecting mechanisms in human vision,” Science 1621146–1148 (1968).
[Crossref] [PubMed]

Vision Res. (2)

N. Graham and J. Nachmias, “Detection of grating patterns containing two spatial frequencies: A comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[Crossref] [PubMed]

H. Mostafavi and D. J. Sakrison, “Structure and properties of a single channel in the human visual system,” Vision Res. 16, 957–968 (1976).
[Crossref] [PubMed]

Other (7)

The spatial bandwidth of the channels is in dispute. Generally, experiments without grating adaptation produce narrow channels and, with adaptation, wide channels. We assume here that adaptation has not taken place.

H. R. Blackwell and J. H. Taylor, in Proceedings of the NATO Seminar on Detection, Recognition, and Identification of Line-of-Sight Targets (unpublished); also, University of Michigan Engineering Research Institute Report No. 2455-10-F (1958) (unpublished).

J. Johnson, Image Intensity Symposium, Fort Belvoir, Va., October 6–7, 1958, AD220160 (unpublished).

A. D. Schnitzler, “Analysis of noise-required contrast and modulation in image-detecting and display systems,” In Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973).
[Crossref]

W. A. Pearlman, Quantization Error Bounds for Computer-Generated Holograms, Technical Report No. 6503-1, Information System Laboratory, Stanford University, Stanford, Calif. accessed August1974 (unpublished).

The DFT of the Gaussian noise process alone is exactly Gaussian with statistically independent nonredundant coefficients due to the linearity of the DFT.

G. Senge, Quantization of Image Transforms with Minimum Distortion, Technical Report No. ECE-77-8, Dept. of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison, Wis., May, 1977 (unpublished).

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Figures (9)

FIG. 1
FIG. 1

Model of eye-brain decision system.

FIG. 2
FIG. 2

Experimental setup for comparative image filtering.

FIG. 3
FIG. 3

Source images. (a) Lunar landscape. (b) Woman’s face.

FIG. 4
FIG. 4

Source images in additive noise. (a) Lunar landscape, S/N = −9.2 dB. (b) Woman’s face, S/N = −2.2 dB.

FIG. 5
FIG. 5

Noisy lunar landscape (S/N = −9.2 dB) filtered by new error criterion. (a) B = 707, I0 = 135. (b) B = 1414, I0 = 135.

FIG. 6
FIG. 6

Squared-error filtering of noisy lunar landscape, S/N = −9.2 dB.

FIG. 7
FIG. 7

Lunar landscape in additive noise, S/N = −6.2 dB.

FIG. 8
FIG. 8

Comparative filterings of lunar landscape in noise (S/N = −6.2 dB, I0 = 135). (a) New error criterion, B = 707. (b) Squared-error criterion.

FIG. 9
FIG. 9

Comparative filterings of woman’s face in noise (S/N = −2.2 dB, I0 = 99). (a) New error criterion, B = 707. (b) Squared-error criterion.

Equations (43)

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

p i = T i 1 ( 2 π σ o 2 ) 1 / 2 e - ( x - C o I i ) 2 / 2 σ o 2 d x ,
p i = k T 1 ( 2 π ) 1 / 2 e - ( y - k i ) 2 / 2 d y ,
k i = n ¯ i σ o = ( C 2 C 1 ) 1 / 2 I i ( I o ) 1 / 2 = C 2 ( L o I o ) 1 / 2 L i I o = ( C 2 L o ) 1 / 2 I i I o .
d ( p i , p i ) = ( p i - p i ) 2 ,             i = 0 , 1 , 2 , , n ,
D = d ( p i , p i ) = 1 n + 1 i = 0 n d ( p i , p i ) .
( p i - p i ) p i I i ( I i - I i ) = B ( 2 π ) 1 / 2 exp [ - ( k T - B I i I o ) 2 / 2 ] I i - I i I o .
d ( p i , p i ) = d ( I i , I i ) = B 2 2 π exp [ - ( k T - B I i I o ) 2 ] ( I i - I i I o ) 2 .
A i = I i e j ϕ i and A i = I i e j ϕ i ,
I i - I i = | A i - A i | A i - A i ,
d 1 ( A i , A i ) = B 2 2 π exp [ - ( k T - B A i I o ) 2 ] A i - A i 2 I o 2
d 1 ( A i , A i ) = K I o - 2 exp [ - ( k T - B A i I o ) 2 ] A i - A i 2
D = 1 n + 1 i = 0 n d 1 ( A i , A i ) .
 i = c i A i ,             c i = E g [ A i 2 ] E g [ A i 2 ] + E g [ N i 2 ]
E g [ Z ] = z g ( x ) p ( z , x ) d z d x
g ( A i ) = K i I o - 2 exp [ - ( k T - B A i I o ) 2 ]
d i = E g [ A i 2 ] ( 1 - c i ) .
E [ N i 2 ] = σ n 2 , E g [ N i 2 ] = 0 0 n i 2 g ( a i ) p ( a i ) p ( n i ) d ( a i ) d ( n i ) = σ n 2 0 ω g ( a i ) p ( a i ) d ( a i ) = { σ n 2 0 K i I o - 2 exp [ - ( k T - B x I o ) 2 ] 2 x σ i 2 exp ( - x 2 σ i 2 ) d x ,             i 0 , N 2 σ n 2 0 K i I o - 2 exp [ - ( k T - B x I o ) 2 ] ( 2 π σ i 2 ) 1 / 2 exp ( - x 2 2 σ i 2 ) d x ,             i = 0 , N 2 ,
E [ A i 2 ] = σ i 2 E g [ A i 2 ] = 0 a i 2 g ( a i ) p ( a i ) d ( a i ) = { 2 K i I o - 2 σ i 2 0 x 3 exp [ - ( k T - B n x ) 2 ] exp ( - x 2 σ i 2 ) d x ,             i 0 , N 2 K i I o - 2 0 x 2 exp [ - ( k T - B n x ) 2 ] ( 2 π σ i 2 ) 1 / 2 exp ( - x 2 2 σ i 2 ) d x ,             i = 0 , N 2
E g [ N i 2 ] = { 2 K i σ n 2 I o 2 ( γ i B n k T σ i ) 2 exp ( - k T 2 + γ i 2 ) G 1 ( γ i ) , i 0 , N 2 ( 2 π ) 1 / 2 K i σ n 2 I o 2 ( δ i B n k T σ i ) exp ( - k T 2 + δ i 2 ) G o ( δ i ) , i = 0 , N 2
E g [ A i 2 ] = { 2 K i I o 2 σ i 2 ( γ i B n k T ) 4 exp ( - k T 2 + γ i 2 ) G 3 ( γ i ) , i 0 , N 2 ( 2 π ) 1 / 2 K i I o 2 ( δ i B n k T ) 3 ( 1 σ i ) exp ( - k T 2 + δ i 2 ) G 2 ( δ i ) , i = 0 , N 2
G m ( a ) 0 x m e - ( x - a ) 2 d x ,             m = 0 , 1 , 2 , 3.
2 G 0 ( a ) = π [ 1 + erf ( a ) ] , 2 G 1 ( a ) = exp { - a } 2 + π a [ 1 + erf ( a ) ] , 2 G 2 ( a ) = 3 a exp { - a 2 } + π ( a 2 + ½ ) [ 1 + erf ( a ) ] , 2 G 3 ( a ) = ( 1 + 7 a 2 ) exp { - a 2 } + π a ( a 2 + ³ / ) [ 1 + erf ( a ) ] .
 i = c i A i , c i = { γ i 2 G 3 ( γ i ) γ i 2 G 3 ( γ i ) + ( B n k T σ n ) 2 G 1 ( γ i ) , i 0 , N 2 δ i 2 G 2 ( δ i ) δ i 2 G 2 ( δ i ) + ( B n k T σ n ) 2 G 0 ( δ i ) , i = 0 , N 2 .
c i ( u ) = σ i 2 / ( σ i 2 + σ n 2 ) ,
σ i 2 = σ N - i 2 ,             c i = c N - i ,             c i ( u ) = c N - i ( u ) .
E [ g ( X ) ( X - X ˆ ) 2 ]
g ( x ) [ x - h ( y ) ] 2 p ( x , y ) d x d y = p ( y ) { g ( x ) [ x - h ( y ) ] 2 p ( x / y ) d x } d y .
g ( x ) ( x - a ) 2 p ( x / y ) d x .
- 2 g ( x ) ( x - a ) p ( x / y ) d x = 0.
x ˆ = h ( y ) = x g ( x ) p ( x / y ) d x g ( x ) p ( x / y ) d x .
2 g ( x ) p ( x / y ) d x ,
E g [ Z ] = z g ( x ) p ( z , x ) d z d x
X ˆ = h ( Y ) = E g [ X / Y ] E g [ 1 / Y ] ,
E g [ b ] = b g ( x ) p ( x ) d x = b ,
g ( x ) p ( x ) d x = 1.
d = E [ g ( X ) X - c Y 2 ] = E [ g ( X ) X 2 ] - c E [ g ( X ) X Y ] - c * E [ g ( X ) X Y * ] + c 2 E [ g ( X ) Y 2 ] = E g [ X 2 ] + c 2 E g [ Y 2 ] - c E g [ X * Y ] - c * E g [ X Y * ] .
d a = 2 a E g [ Y 2 ] - α * - α = 0 , d b = 2 b E g [ Y 2 ] - j α * + j α = 0 ,
a = ( α * + α ) / 2 E g [ Y 2 ] = Re [ α ] / E g [ Y 2 ] , b = j ( α * - α ) / 2 E g [ Y 2 ] = Im [ α 0. E g [ Y 2 ] .
c = a + j b = E g [ X Y * ] / E g [ Y 2 ]
E g [ ( X - c Y ) Y * ] = 0.
d min = E g [ X - c Y 2 ] = E g [ ( X - c Y ) X * ] = E g [ X 2 ] - E g [ X Y * ] 2 / E g [ Y 2 ] .
c = E g [ X ( X + N ) * ] E g [ X + N 2 ] = E g [ X 2 ] E g [ X 2 ] + E g [ N 2 ] ,
d min = E g [ X 2 ] ( 1 - c ) = E g [ X 2 ] E g [ N 2 ] E g [ X 2 ] + E g [ N 2 ] .