Abstract

Coherent optical power spectra of random scene imagery can become a powerful image evaluation technique when properly analyzed. A normalized low order moment of power spectra is set forth as a good image quality merit factor, which also has the advantageous property of taking out the spectrum analyzer’s aperture effect. In addition to the relevant theory, experimental results with this merit factor of power spectra are given that demonstrate a high degree of correlation with subjective quality rankings, as well as the accurate assessment of optimum focus.

© 1976 Optical Society of America

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  1. N. Jensen, Photogramm. Eng. 39, 1321 (1973).
  2. At present there is no standard terminology in this area. The term power spectrum will be used in this paper, although other terms used in the literature are spectral power density, spectral density function, energy spectrum, Wiener-Khinchin spectrum, and Wiener spectrum. A good start at standardization is in the distinctly separate definitions given to power spectrum and Wiener spectrum by Thiry, J. Photogr. Sci. 11, 69, 1963.
  3. G. G. Lendaris, G. L. Stanley, IEEE Proc. 58, 198 (1970).
  4. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).
  5. A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
  6. H. Stark, W. R. Bennett, M. Arm, Appl. Opt. 8, 2165 (1969).
  7. H. Stark, B. DimiTriadis, J. Opt. Soc. Am. 65, 425 (1975).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. The film’s light amplitude transmittance is actually a complex function, e.g., t(x,y) = h(x,y) exp[jθ(x,y)], where h(x,y) is the positive square root of the light intensity transmittance, and θ(x,y) describes a phase function caused by the emulsion relief image (signal-dependent), as well as film base thickness and index of refraction variations (signal-independent) [A. L. Ingalls, Photogr. Sci. Eng. 4, 135 (1960); H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972)]. The use of an index of refraction matching solution surrounding the film would minimize the first two phase components mentioned. However, this is an impractical technique to use for film that often must remain of archival quality. The emulsion relief image can be considered as a semisignal source since it nonlinearly tracks h(x,y). Experimental results demonstrate that the other two phase components have a relatively minimal deleterious effect upon the measured power spectra. For these reasons, liquid immersion of the film was not used.
  10. See Ref. 7, Eq. (5).
  11. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 140.
  12. By evaluating the following Fourier transform relation [given in R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 245](2πju)m(2πjv)nP(u,v)↔F.T.(∂∂x)m(∂∂y)np(x,y)at x = y = 0, we have the relation for two-dimensional rectangular coordinate joint moments of order m + n:∫-∞∞∫-∞∞umvnP(u,v)dudv=1(2πj)m+n(∂∂x)m (∂∂y)np(0,0).Solving this moment equation for (m = 0, n = 2) and (m = 2, n = 0) and combining results give∫-∞∞∫-∞∞(u2+v2)P(u,v)dudv=-14π2[∂2p(x,y)dx2+∂2p(x,y)∂y2]x=y0,which is often called the moment of inertia and is the two-dimensional analog of Eq. (6).
  13. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1957).
  14. W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 311.
  15. Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).
  16. T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C, 82 (1968).
  17. D. R. Cox, P. A. W. Lewis, Statistical Analysis of Series of Events (Wiley, New York, 1966).
  18. D. Halford, IEEE Proc. 56, 251 (1968).
  19. See Ref. 16 and also note the following. If the scene spectrum falls off only as ν−2, the theoretical existence of∫0∞ν3P(ν)dνat first appears to be less assured (see Bracewell, Ref. 12, p. 141). However, for this case P(ν) will actually fall off faster than ν−2 because of the effect of the sensor system’s spatial frequency response. In any case, over the finite limits actually used, the moment will exist in practice.
  20. K. Stumpf, R. Warren, “Wiener Spectra Flight Test Program,” Itek Corp. Technical Report, Lexington, Mass. (March1974), Vols. 1, 2.
  21. J. J. DePalma, E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).
  22. E. M. Granger, K. N. Cupery, Photogr. Sci. Eng. 16, 221 (1972) [substituting f−1df for d(Log f) in the author’s second equation and using the relation (input power spectrum) |τ(f,θ)|2 = output power spectrum, where input power spectrum = f−m, m a constant].
  23. C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), p. 4.
  24. A. H. Katz, J. Opt. Soc. Am. 38, 604 (1948).
  25. F. G. Back, J. Opt. Soc. Am. 43, 685 (1953).
  26. The experimental result that the power spectra of the various measuring apertures closely agree with the theoretically predicted spectra [via Eq. (1)] confirms the working validity of the power spectrum relationship given in Eq. (1).
  27. It is known that the visual ranking technique described is scale-dependent, i.e., all other factors being equal, a photograph with a low scale will have a higher rating than a photograph with a higher scale. On the other hand it can be shown that for physically reasonable forms of F(ν), the normalized low order moment of F(ν) is essentially scale-independent. In order for M to track properly the visual ranking therefore, M is made to be scale-dependent by the conversionM(cycles/m)2=1×106M(cycles/mm)2(Scale)2
  28. See Ref. 14, p. 320.
  29. J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
  30. A. S. Husain-Abidi, Pattern Recognition (Pergamon, New York, 1973), 5.3.
  31. J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Englewood Cliffs, N.J., 1966), p. 32.
  32. J. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 325.

1975

1974

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).

Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).

1973

N. Jensen, Photogramm. Eng. 39, 1321 (1973).

1972

E. M. Granger, K. N. Cupery, Photogr. Sci. Eng. 16, 221 (1972) [substituting f−1df for d(Log f) in the author’s second equation and using the relation (input power spectrum) |τ(f,θ)|2 = output power spectrum, where input power spectrum = f−m, m a constant].

1970

G. G. Lendaris, G. L. Stanley, IEEE Proc. 58, 198 (1970).

1969

1968

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C, 82 (1968).

D. Halford, IEEE Proc. 56, 251 (1968).

1963

At present there is no standard terminology in this area. The term power spectrum will be used in this paper, although other terms used in the literature are spectral power density, spectral density function, energy spectrum, Wiener-Khinchin spectrum, and Wiener spectrum. A good start at standardization is in the distinctly separate definitions given to power spectrum and Wiener spectrum by Thiry, J. Photogr. Sci. 11, 69, 1963.

1962

1960

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

The film’s light amplitude transmittance is actually a complex function, e.g., t(x,y) = h(x,y) exp[jθ(x,y)], where h(x,y) is the positive square root of the light intensity transmittance, and θ(x,y) describes a phase function caused by the emulsion relief image (signal-dependent), as well as film base thickness and index of refraction variations (signal-independent) [A. L. Ingalls, Photogr. Sci. Eng. 4, 135 (1960); H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972)]. The use of an index of refraction matching solution surrounding the film would minimize the first two phase components mentioned. However, this is an impractical technique to use for film that often must remain of archival quality. The emulsion relief image can be considered as a semisignal source since it nonlinearly tracks h(x,y). Experimental results demonstrate that the other two phase components have a relatively minimal deleterious effect upon the measured power spectra. For these reasons, liquid immersion of the film was not used.

1957

1953

1948

Arm, M.

Arsac, J.

J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Englewood Cliffs, N.J., 1966), p. 32.

Back, F. G.

Bennett, W. R.

Bracewell, R.

By evaluating the following Fourier transform relation [given in R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 245](2πju)m(2πjv)nP(u,v)↔F.T.(∂∂x)m(∂∂y)np(x,y)at x = y = 0, we have the relation for two-dimensional rectangular coordinate joint moments of order m + n:∫-∞∞∫-∞∞umvnP(u,v)dudv=1(2πj)m+n(∂∂x)m (∂∂y)np(0,0).Solving this moment equation for (m = 0, n = 2) and (m = 2, n = 0) and combining results give∫-∞∞∫-∞∞(u2+v2)P(u,v)dudv=-14π2[∂2p(x,y)dx2+∂2p(x,y)∂y2]x=y0,which is often called the moment of inertia and is the two-dimensional analog of Eq. (6).

Choens, R. C.

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).

Cox, D. R.

D. R. Cox, P. A. W. Lewis, Statistical Analysis of Series of Events (Wiley, New York, 1966).

Cupery, K. N.

E. M. Granger, K. N. Cupery, Photogr. Sci. Eng. 16, 221 (1972) [substituting f−1df for d(Log f) in the author’s second equation and using the relation (input power spectrum) |τ(f,θ)|2 = output power spectrum, where input power spectrum = f−m, m a constant].

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

DePalma, J. J.

DimiTriadis, B.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Granger, E. M.

E. M. Granger, K. N. Cupery, Photogr. Sci. Eng. 16, 221 (1972) [substituting f−1df for d(Log f) in the author’s second equation and using the relation (input power spectrum) |τ(f,θ)|2 = output power spectrum, where input power spectrum = f−m, m a constant].

Halford, D.

D. Halford, IEEE Proc. 56, 251 (1968).

Helstrom, C.

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), p. 4.

Husain-Abidi, A. S.

A. S. Husain-Abidi, Pattern Recognition (Pergamon, New York, 1973), 5.3.

Ingalls, A. L.

The film’s light amplitude transmittance is actually a complex function, e.g., t(x,y) = h(x,y) exp[jθ(x,y)], where h(x,y) is the positive square root of the light intensity transmittance, and θ(x,y) describes a phase function caused by the emulsion relief image (signal-dependent), as well as film base thickness and index of refraction variations (signal-independent) [A. L. Ingalls, Photogr. Sci. Eng. 4, 135 (1960); H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972)]. The use of an index of refraction matching solution surrounding the film would minimize the first two phase components mentioned. However, this is an impractical technique to use for film that often must remain of archival quality. The emulsion relief image can be considered as a semisignal source since it nonlinearly tracks h(x,y). Experimental results demonstrate that the other two phase components have a relatively minimal deleterious effect upon the measured power spectra. For these reasons, liquid immersion of the film was not used.

Itakura, Y.

Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).

Jensen, N.

N. Jensen, Photogramm. Eng. 39, 1321 (1973).

Katz, A. H.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

Lendaris, G. G.

G. G. Lendaris, G. L. Stanley, IEEE Proc. 58, 198 (1970).

Lewis, P. A. W.

D. R. Cox, P. A. W. Lewis, Statistical Analysis of Series of Events (Wiley, New York, 1966).

Linfoot, E. H.

Lowry, E. M.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 140.

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

Smith, W.

W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 311.

Stanley, G. L.

G. G. Lendaris, G. L. Stanley, IEEE Proc. 58, 198 (1970).

Stark, H.

Stone, J.

J. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 325.

Stumpf, K.

K. Stumpf, R. Warren, “Wiener Spectra Flight Test Program,” Itek Corp. Technical Report, Lexington, Mass. (March1974), Vols. 1, 2.

Takagi, T.

Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C, 82 (1968).

Thiry,

At present there is no standard terminology in this area. The term power spectrum will be used in this paper, although other terms used in the literature are spectral power density, spectral density function, energy spectrum, Wiener-Khinchin spectrum, and Wiener spectrum. A good start at standardization is in the distinctly separate definitions given to power spectrum and Wiener spectrum by Thiry, J. Photogr. Sci. 11, 69, 1963.

Tsutsumi, S.

Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).

Tutumi, S.

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C, 82 (1968).

Vander Lugt, A.

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).

Walkup, J. F.

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).

Warren, R.

K. Stumpf, R. Warren, “Wiener Spectra Flight Test Program,” Itek Corp. Technical Report, Lexington, Mass. (March1974), Vols. 1, 2.

Appl. Opt.

IEEE Proc.

D. Halford, IEEE Proc. 56, 251 (1968).

G. G. Lendaris, G. L. Stanley, IEEE Proc. 58, 198 (1970).

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).

Infrared Phys.

Y. Itakura, S. Tsutsumi, T. Takagi, Infrared Phys. 14, 17 (1974).

IRE Trans. Inf. Theory

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Inf. Theory IT-6, 386 (1960).

J. Electr. Commun. Jpn.

T. Takagi, S. Tutumi, J. Electr. Commun. Jpn. 51-C, 82 (1968).

J. Opt. Soc. Am.

J. Photogr. Sci.

At present there is no standard terminology in this area. The term power spectrum will be used in this paper, although other terms used in the literature are spectral power density, spectral density function, energy spectrum, Wiener-Khinchin spectrum, and Wiener spectrum. A good start at standardization is in the distinctly separate definitions given to power spectrum and Wiener spectrum by Thiry, J. Photogr. Sci. 11, 69, 1963.

Opt. Eng.

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).

Photogr. Sci. Eng.

E. M. Granger, K. N. Cupery, Photogr. Sci. Eng. 16, 221 (1972) [substituting f−1df for d(Log f) in the author’s second equation and using the relation (input power spectrum) |τ(f,θ)|2 = output power spectrum, where input power spectrum = f−m, m a constant].

The film’s light amplitude transmittance is actually a complex function, e.g., t(x,y) = h(x,y) exp[jθ(x,y)], where h(x,y) is the positive square root of the light intensity transmittance, and θ(x,y) describes a phase function caused by the emulsion relief image (signal-dependent), as well as film base thickness and index of refraction variations (signal-independent) [A. L. Ingalls, Photogr. Sci. Eng. 4, 135 (1960); H. M. Smith, J. Opt. Soc. Am. 62, 802 (1972)]. The use of an index of refraction matching solution surrounding the film would minimize the first two phase components mentioned. However, this is an impractical technique to use for film that often must remain of archival quality. The emulsion relief image can be considered as a semisignal source since it nonlinearly tracks h(x,y). Experimental results demonstrate that the other two phase components have a relatively minimal deleterious effect upon the measured power spectra. For these reasons, liquid immersion of the film was not used.

Photogramm. Eng.

N. Jensen, Photogramm. Eng. 39, 1321 (1973).

Other

See Ref. 7, Eq. (5).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 140.

By evaluating the following Fourier transform relation [given in R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 245](2πju)m(2πjv)nP(u,v)↔F.T.(∂∂x)m(∂∂y)np(x,y)at x = y = 0, we have the relation for two-dimensional rectangular coordinate joint moments of order m + n:∫-∞∞∫-∞∞umvnP(u,v)dudv=1(2πj)m+n(∂∂x)m (∂∂y)np(0,0).Solving this moment equation for (m = 0, n = 2) and (m = 2, n = 0) and combining results give∫-∞∞∫-∞∞(u2+v2)P(u,v)dudv=-14π2[∂2p(x,y)dx2+∂2p(x,y)∂y2]x=y0,which is often called the moment of inertia and is the two-dimensional analog of Eq. (6).

W. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 311.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. R. Cox, P. A. W. Lewis, Statistical Analysis of Series of Events (Wiley, New York, 1966).

See Ref. 16 and also note the following. If the scene spectrum falls off only as ν−2, the theoretical existence of∫0∞ν3P(ν)dνat first appears to be less assured (see Bracewell, Ref. 12, p. 141). However, for this case P(ν) will actually fall off faster than ν−2 because of the effect of the sensor system’s spatial frequency response. In any case, over the finite limits actually used, the moment will exist in practice.

K. Stumpf, R. Warren, “Wiener Spectra Flight Test Program,” Itek Corp. Technical Report, Lexington, Mass. (March1974), Vols. 1, 2.

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), p. 4.

A. S. Husain-Abidi, Pattern Recognition (Pergamon, New York, 1973), 5.3.

J. Arsac, Fourier Transforms and the Theory of Distributions (Prentice-Hall, Englewood Cliffs, N.J., 1966), p. 32.

J. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 325.

The experimental result that the power spectra of the various measuring apertures closely agree with the theoretically predicted spectra [via Eq. (1)] confirms the working validity of the power spectrum relationship given in Eq. (1).

It is known that the visual ranking technique described is scale-dependent, i.e., all other factors being equal, a photograph with a low scale will have a higher rating than a photograph with a higher scale. On the other hand it can be shown that for physically reasonable forms of F(ν), the normalized low order moment of F(ν) is essentially scale-independent. In order for M to track properly the visual ranking therefore, M is made to be scale-dependent by the conversionM(cycles/m)2=1×106M(cycles/mm)2(Scale)2

See Ref. 14, p. 320.

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Figures (6)

Fig. 1
Fig. 1

Basic coherent optical power spectrum analyzer.

Fig. 2
Fig. 2

Image quality sensitivity range of power spectra. The scene image corresponding to the solid curve was judged by experienced viewers to be of higher quality than the scene image corresponding to the dashed curve. Note the inversion in the high frequency, low SNR region (measurements made on instrument shown in Fig. 3, 0.4-mm diam aperture).

Fig. 3
Fig. 3

Coherent optical power spectrum analyzer used in experimentation. The annular apertures in the power spectrum plane are sequentially introduced by the on-line computer control system, each centering on the dc. All the energy passed through a given annular aperture is then detected by the photomultiplier tube and recorded on punched paper tape.

Fig. 4
Fig. 4

Power spectrum merit factor M vs observed image quality. Power spectra measured with various aperture sizes from fifty-four frames of film imagery, converted to the merit factor M [see Eq. (10)].

Fig. 5
Fig. 5

Power spectrum merit factor M ^ vs observed image quality. The same film imagery spectra as used in Fig. 4, but without any aperture compensation; thus M ^ equals the quantity given in Eq. (13).

Fig. 6
Fig. 6

The power spectrum merit factor M (20–100-cycles/mm range) from film imagery acquired under various degrees of defocus in units of optical path difference (relative to a reference position of OPD = 0). Seven independent spectra per frame were measured with frame sample sizes (left to right in figure) of 1,3,4,4,1.

Equations (32)

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P ( u , v ) = k 2 λ 2 F 2 | - - a ( x , y ) t ( x , y ) × exp [ - j 2 π ( x u + v y ) ] d x d y | 2 .
P ( u , v ) = F ( u , v ) * H ( u , v ) ,
H ( u , v ) = - - [ a ( x , y ) * a ( - x , - y ) ] exp [ - j 2 π ( u x + v y ) ] d x d y , F ( u , v ) = - - [ t ( x , y ) * t ( - x , - y ) ] exp [ - j 2 π ( u x + v y ) ] d x d y ,
t ( x , y ) * t ( - x , - y ) = lim X Y 1 4 X Y - Y Y - X X t ( τ , η ) t ( τ + x , η + y ) d τ d η ,
p ( r ) = 0 ν P ( ν ) J 0 ( ν r ) d ν ,
0 ν 2 n + 1 P ( ν ) d ν = p ( 2 n ) ( 0 ) [ ( n ! ) 2 2 2 n ( - 1 ) n ( 2 n ) ! ] for n = 0 , 1 , 2 , 3 , .
0 ν 3 P ( ν ) d ν = - 2 p ( 0 ) .
0 2 π 0 ν P ( ν ) d ν d θ = 2 π 0 ν P ( ν ) d ν ,
0 ν 3 P ( ν ) d ν 0 ν P ( ν ) d ν = - 2 p ( 0 ) p ( 0 ) .
0 ν 3 [ F ( ν ) * H ( ν ) ] d ν 0 ν [ F ( ν ) * H ( ν ) ] d ν = - 2 ( f h ) f h | r = 0 ,
- 2 ( f h ) f h | r = 0 = - 2 ( f h f h + f h f h + 2 f h f h ) r = 0 = - 2 ( f f + h h ) r = 0 .
M = 0 ν 3 F ( ν ) d ν 0 ν F ( ν ) d ν = 0 ν 3 P ( ν ) d ν 0 ν P ( ν ) d ν - 0 ν 3 H ( ν ) d ν 0 ν H ( ν ) d ν .
ν max = 4.2 ( 1 / R 1 + 1 / R 2 + ) - 3 / 5 ,
c = γ η ν c 2 ν c - 5 ν c + 5 P ( ν ) ν d ν ÷ 0 200 P ( ν ) ν d ν ,
ν γ ν η ν 3 P ( ν ) d ν ÷ 0 200 ν P ( ν ) d ν .
ν γ ν η ν 2 P ( ν ) d ν
ν γ ν η ν 3 P ( ν ) d ν
P ( u , v ) = | - - a ( x , y ) t ( x , y ) exp [ - j 2 π ( x u + y v ) ] d x d y | 2 = - - a ( x , y ) t ( x , y ) exp [ - j 2 π ( x u + y v ) ] d x d y × - - a ( x , y ) t ( x , y ) exp [ j 2 π ( x u + y v ) ] d x d y .
P ( u , v ) = lim R S 1 4 R S - S S - R R T ( r , s ) A ( u - r , v - s ) d r d s × lim W Z 1 4 W Z - W W - Z Z T ( w , z ) A ( u - w , v - z ) d z d w .
P ( u , v ) = lim R S 1 4 R S - S S - R R × T ( r , s ) T ( r , s ) A ( u - r , v - s ) A ( u - r , v - s ) d r d s = lim R S 1 4 R S - S S - R R T ( r , s ) 2 A ( u - r , v - s ) 2 d r d s .
E [ P ( u , v ) ] = - - E [ T ( r , s ) 2 ] A ( u - r , v - s ) 2 d r d s ,
P ( u , v ) = - - F ( r , s ) H ( u - r , v - s ) d r d s = F ( u , v ) * H ( u , v ) .
p ( r ) = 0 ν P ( ν ) J 0 ( ν r ) d ν ,
p ( r ) = n = 0 ( - 1 ) n m 2 n + 1 ( n ! ) 2 2 2 n r 2 n ,
m 2 n + 1 = 0 ν 2 n + 1 P ( ν ) d ν n = 0 , 1 , 2 , 3 , .
p ( r ) = k = 0 p ( k ) ( 0 ) r k k ! k = 0 , 1 , 2 , 3 , ,
p ( k ) ( 0 ) = d ( k ) p ( r ) d r k | r = 0 ,
k = 0 p ( k ) ( 0 ) r k k ! = n = 0 ( - 1 ) n m 2 n + 1 ( n ! ) 2 2 2 n r 2 n .
p ( 2 n ) ( 0 ) r 2 n ( 2 n ) ! = ( - 1 ) n m 2 n + 1 ( n ! ) 2 2 2 n r 2 n .
m 2 n + 1 = 0 ν 2 n + 1 P ( ν ) d ( ν ) = p ( 2 n ) ( 0 ) ( n ! ) 2 2 2 n ( - 1 ) n ( 2 n ) ! .
0ν3P(ν)dν
M(cycles/m)2=1×106M(cycles/mm)2(Scale)2

Metrics