Abstract

Locally narrow-band images can be modeled as two-dimensional (2D) spatial AM–FM signals with several applications in image texture analysis and computer vision. We formulate an image-demodulation problem and present a solution based on the multidimensional energy operator Φ(f) = ||∇f||2f2f. This nonlinear operator is a multidimensional extension of the one-dimensional (1D) energy-tracking operator Ψ(f) = (f′)2ff″, which has been found useful for demodulating 1D AM–FM and speech signals. We discuss some interesting properties of the multidimensional operator and develop a multidimensional energy-separation algorithm to estimate the amplitude envelope and instantaneous frequencies of 2D spatially varying AM–FM signals. Experiments are also presented on applying this 2D energy-demodulation algorithm to estimate the instantaneous amplitude contrast and spatial frequencies of image textures bandpass filtered by means of Gabor filters. The attractive features of the multidimensional energy operator and the 2D energy-separation algorithm are their simplicity, efficiency, and ability to track instantaneously varying spatial-modulation patterns.

© 1995 Optical Society of America

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References

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  1. A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
    [CrossRef]
  2. J. P. Havlicek, A. C. Bovik, P. Maragos, “Modulation models for image processing and wavelet-based image demodulation,” presented at the 26th Annual Asilomar Conference on Signals, Systems and Computers, Monterey, California, October 1992.
  3. J. E. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2, 1160–1169 (1985).
    [CrossRef] [PubMed]
  4. P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1818, 177–186 (1993).
    [CrossRef]
  5. J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N. Mex., April 1990 (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 381–384.
  6. J. F. Kaiser, “On Teager’s energy algorithm and its generalization to continuous signals,” in Proceedings of the IEEE Digital Signal Processing Workshop, Mohonk (New Paltz), N. Y., September 1990(Institute of Electrical and Electronics Engineers, New York, 1990).
  7. P. Maragos, J. F. Kaiser, T. F. Quatieri, “On separating amplitude from frequency modulations using energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, Calif., March 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. II: 1–4.
  8. P. Maragos, T. F. Quatieri, J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 421–424.
    [CrossRef]
  9. P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
    [CrossRef]
  10. P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
    [CrossRef]
  11. T.-H. Yu, S. K. Mitra, J. F. Kaiser, “A novel nonlinear filter for image enhancement,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1452, 303–309 (1991).
    [CrossRef]
  12. S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
    [CrossRef]
  13. H. M. Teager, S. M. Teager, “Evidence for nonlinear production mechanisms in the vocal tract,” NATO Advanced Study Institute on Speech Production and Speech Modelling, Bonas, France, July 1989 (Kluwer, Boston, Mass., 1990), pp. 241–261.
  14. A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
    [CrossRef]

1993 (3)

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
[CrossRef]

1992 (1)

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

1985 (1)

Bovik, A. C.

A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
[CrossRef]

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

J. P. Havlicek, A. C. Bovik, P. Maragos, “Modulation models for image processing and wavelet-based image demodulation,” presented at the 26th Annual Asilomar Conference on Signals, Systems and Computers, Monterey, California, October 1992.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1818, 177–186 (1993).
[CrossRef]

Daugman, J. E.

Emmoth, T.

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

Gopal, N.

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

Havlicek, J. P.

J. P. Havlicek, A. C. Bovik, P. Maragos, “Modulation models for image processing and wavelet-based image demodulation,” presented at the 26th Annual Asilomar Conference on Signals, Systems and Computers, Monterey, California, October 1992.

Kaiser, J. F.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, T. F. Quatieri, J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 421–424.
[CrossRef]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N. Mex., April 1990 (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 381–384.

J. F. Kaiser, “On Teager’s energy algorithm and its generalization to continuous signals,” in Proceedings of the IEEE Digital Signal Processing Workshop, Mohonk (New Paltz), N. Y., September 1990(Institute of Electrical and Electronics Engineers, New York, 1990).

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On separating amplitude from frequency modulations using energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, Calif., March 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. II: 1–4.

T.-H. Yu, S. K. Mitra, J. F. Kaiser, “A novel nonlinear filter for image enhancement,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1452, 303–309 (1991).
[CrossRef]

Li, H.

S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
[CrossRef]

Lin, I.-S.

S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
[CrossRef]

Maragos, P.

A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1818, 177–186 (1993).
[CrossRef]

P. Maragos, T. F. Quatieri, J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 421–424.
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On separating amplitude from frequency modulations using energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, Calif., March 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. II: 1–4.

J. P. Havlicek, A. C. Bovik, P. Maragos, “Modulation models for image processing and wavelet-based image demodulation,” presented at the 26th Annual Asilomar Conference on Signals, Systems and Computers, Monterey, California, October 1992.

Mitra, S. K.

T.-H. Yu, S. K. Mitra, J. F. Kaiser, “A novel nonlinear filter for image enhancement,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1452, 303–309 (1991).
[CrossRef]

S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
[CrossRef]

Quatieri, T. F.

A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1818, 177–186 (1993).
[CrossRef]

P. Maragos, T. F. Quatieri, J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 421–424.
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On separating amplitude from frequency modulations using energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, Calif., March 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. II: 1–4.

Restrepo, A.

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

Teager, H. M.

H. M. Teager, S. M. Teager, “Evidence for nonlinear production mechanisms in the vocal tract,” NATO Advanced Study Institute on Speech Production and Speech Modelling, Bonas, France, July 1989 (Kluwer, Boston, Mass., 1990), pp. 241–261.

Teager, S. M.

H. M. Teager, S. M. Teager, “Evidence for nonlinear production mechanisms in the vocal tract,” NATO Advanced Study Institute on Speech Production and Speech Modelling, Bonas, France, July 1989 (Kluwer, Boston, Mass., 1990), pp. 241–261.

Yu, T.-H.

S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
[CrossRef]

T.-H. Yu, S. K. Mitra, J. F. Kaiser, “A novel nonlinear filter for image enhancement,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1452, 303–309 (1991).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. C. Bovik, N. Gopal, T. Emmoth, A. Restrepo, “Localized measurement of emergent image frequencies by Gabor wavelets,” IEEE Trans. Inf. Theory 38, 691–712 (1992).
[CrossRef]

IEEE Trans. Signal Process. (3)

A. C. Bovik, P. Maragos, T. F. Quatieri, “AM–FM energy detection and separation in noise using multiband energy operators,” IEEE Trans. Signal Process. 41, 3245–3265 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (9)

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1818, 177–186 (1993).
[CrossRef]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N. Mex., April 1990 (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 381–384.

J. F. Kaiser, “On Teager’s energy algorithm and its generalization to continuous signals,” in Proceedings of the IEEE Digital Signal Processing Workshop, Mohonk (New Paltz), N. Y., September 1990(Institute of Electrical and Electronics Engineers, New York, 1990).

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On separating amplitude from frequency modulations using energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, San Francisco, Calif., March 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. II: 1–4.

P. Maragos, T. F. Quatieri, J. F. Kaiser, “Speech nonlinearities, modulations, and energy operators,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 421–424.
[CrossRef]

T.-H. Yu, S. K. Mitra, J. F. Kaiser, “A novel nonlinear filter for image enhancement,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1452, 303–309 (1991).
[CrossRef]

S. K. Mitra, H. Li, I.-S. Lin, T.-H. Yu, “A new class of nonlinear filters for image enhancement,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Toronto, Canada, May 1991 (Institute of Electrical and Electronics Engineers, New York, 1991), pp. 2525–2528.
[CrossRef]

H. M. Teager, S. M. Teager, “Evidence for nonlinear production mechanisms in the vocal tract,” NATO Advanced Study Institute on Speech Production and Speech Modelling, Bonas, France, July 1989 (Kluwer, Boston, Mass., 1990), pp. 241–261.

J. P. Havlicek, A. C. Bovik, P. Maragos, “Modulation models for image processing and wavelet-based image demodulation,” presented at the 26th Annual Asilomar Conference on Signals, Systems and Computers, Monterey, California, October 1992.

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Figures (4)

Fig. 1
Fig. 1

(a) Perspective plot of original 2D AM–FM signal

Fig. 2
Fig. 2

Frequency responses (represented as intensities) of the 2D Gabor filters used in the filter bank. There are 40 filters arranged in a polar waveletlike tesselation on eight rays, with five filters per ray, plus one filter centered at (Ω1, Ω2) = (0, 0). Each of the 41 filter responses in the figure has been independently scaled for maximum dynamic range in the available gray levels (from Ref. 1).

Fig. 3
Fig. 3

(a) Intensity image I of a 256 × 256 pixel texture (sea fan); (b) energy Φ(I) of the intensity image; (c) energy Φ(IĪ) of the zero-mean image; (d) bandpass-filtered image f = I * g, where g is the impulse response of a Gabor filter with the passband centered at horizontal and vertical frequencies of 27.2 and 27.2 cycles per image, respectively; (e) bandpass image energy Φ(f); (f) amplitude envelope of f estimated with the DESA; (g) instantaneous frequency Ω1 of f estimated with the DESA; (h) instantaneous frequency Ω2 of f estimated with the DESA; (i) frequency vectors (Ω1, Ω2), decimated and scaled, superimposed on the bandpass image. {Images in (f)–(h) have been filtered by means of a 3 × 3 median. All image plots are normalized so that intensities are in [0, 255].}

Fig. 4
Fig. 4

(a) Intensity image I of a 240 × 250 pixel texture (wood); (b) energy Φ(I) of the intensity image; (c) energy Φ(IĪ) of the zero-mean image; (d) bandpass-filtered image f = I * g where g is the impulse response of a Gabor filter with the passband centered at horizontal and vertical frequencies of 35.5 and 14.7 cycles per image, respectively; (e) bandpass image energy Φ(f); (f) amplitude envelope of f estimated with the DESA; (g) instantaneous frequency Ω1 of f estimated with the DESA; (h) instantaneous frequency Ω2 of f estimated with the DESA; (i) frequency vectors (Ω1, Ω2), decimated and scaled, superimposed upon the bandpass image. {Images in (f)–(h) have been filtered by means of a 3 × 3 median. All image plots are normalized so that intensities are in [0, 255].}

Equations (67)

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

f ( x , y ) = a ( x , y ) cos [ ϕ ( x , y ) ] ,
Ψ ( f ) ( t ) [ f ( t ) ] 2 - f ( t ) f ( t ) ,
Ψ d ( f ) ( n ) f 2 ( n ) - f ( n - 1 ) f ( n + 1 )
f ( t ) = a ( t ) cos [ ϕ ( t ) ] ,
Ψ { a ( t ) cos [ ϕ ( t ) ] } a 2 ( t ) ω 2 ( t ) ,
Φ ( f ) ( x ) f ( x ) 2 - f ( x ) 2 f ( x ) ,
f = ( f x 1 , , f x d )
f 2 = ( f x 1 ) 2 + + ( f x d ) 2
2 f = k = 1 d 2 f x k 2
Φ ( f ) = k = 1 d ( f x k ) 2 - f ( 2 f x k 2 ) = k = 1 d Ψ k ( f ) ,
Ψ k ( f ) ( f x k ) 2 - f 2 f x k 2 .
( f g ) = g f + f g , 2 ( f g ) = g 2 f + f 2 g + 2 ( f ) · ( g )
Φ ( f g ) = f 2 Φ ( g ) + g 2 Φ ( f ) .
Φ ( f + g ) = Φ ( f ) + Φ ( g ) + 2 ( f ) · ( g ) - f 2 g - g 2 f .
Φ [ f ( x ) + c ] = Φ ( f ) ( x ) - c 2 f ( x ) .
Φ [ exp ( c · x ) ] = 0 ,
Φ [ A exp ( c · x ) f ( x ) ] = A 2 exp ( 2 c · x ) Φ [ f ( x ) ] .
f ( x ) = A cos ( ω c · x + θ )
ω c = ( ω c , 1 , , ω c , d )
Φ [ A cos ( ω c · x + θ ) ] = A 2 ( k = 1 d ω c , k 2 ) = A 2 ω c 2 .
f x k ( x ) = [ A cos ( ω c · x + θ ) ] x k = - A ω c , k sin ( ω c · x + θ )
Φ ( f x k ) ( x ) = ( A ω c , k ) 2 ω c 2
ω c , k = [ Φ ( f x k ) / Φ ( f ) ] 1 / 2 ,             k = 1 , 2 , , d ,
A = Φ ( f ) / [ k = 1 d Φ ( f x k ) ] 1 / 2 .
f ( x ) = a ( x ) cos [ ϕ ( x ) ] ,
ω ( x ) ϕ ( x ) = [ ω 1 ( x ) , , ω d ( x ) ]
ω k ( x ) ϕ x k ( x )
ω k ( x ) = ω c , k + ω m , k q k ( x ) ,
Φ [ a cos ( ϕ ) ] = a 2 ω 2 - ½ a 2 sin ( 2 ϕ ) 2 ϕ + cos 2 ( ϕ ) Φ ( a ) .
μ a = 1 ( 2 π ) d - ω a ω a - ω a ω a A ( u ) d u 1 d u d ,
a ( x ) a max μ a ,
| a x k | ω a μ a ,
| 2 a x k 2 | ω a 2 μ a ,
Φ ( a ) ( x ) 2 d ω a 2 μ a 2 ,
Φ [ a cos ( ϕ ) ] a 2 ω 2 ,
E ( x ) = Φ [ a cos ( ϕ ) ] - a 2 ω 2
E ( x ) ( 2 d ω a 2 + 1 2 k = 1 d ω m , k ω f , k μ q k ) μ a 2 ,
2 ϕ x k 2 = ω m , k q k x k .
ω a min k ω c , k ,             k = 1 d ω m , k ω f , k ω c 2
f x k = a x k cos ( ϕ ) - a ω k sin ( ϕ ) .
Φ ( f x k ) Φ [ a ω k sin ( ϕ ) ] a 2 ω k 2 ω 2
[ Φ ( f x k ) / Φ ( f ) ] 1 / 2 ω k ( x ) ,             k = 1 , 2 , , d ,
Φ ( f ) / [ k = 1 d Φ ( f x k ) ] 1 / 2 a ( x ) .
f ( t ) = [ f 1 ( t ) , f 2 ( t ) , , f n ( t ) ] ,
f = ( f 1 , f 2 , , f n )
Θ ( f ) ( t ) f ( t ) 2 - f ( t ) · f ( t ) .
Θ ( f ) = k = 1 n Ψ ( f k ) .
C ( f ) ( t ) f ( t ) 2 - Re [ f * ( t ) f ( t ) ] ,
C ( f ) = Θ { [ Re ( f ) , Im ( f ) ] } = Ψ [ Re ( f ) ] + Ψ [ Im ( f ) ] .
Φ d ( f ) ( m , n ) Ψ d , 1 ( f ) ( m , n ) + Ψ d , 2 ( f ) ( m , n ) ,
Φ d ( f ) ( m , n ) = 2 f 2 ( m , n ) - f ( m - 1 , n ) f ( m + 1 , n ) - f ( m , n - 1 ) f ( m , n + 1 ) ,
Ψ d , 1 ( f ) ( m , n ) f 2 ( m , n ) - f ( m - 1 , n ) f ( m + 1 , n )
Φ d [ A cos ( Ω 1 m + Ω 2 n + θ ) ] = A 2 [ sin 2 ( Ω 1 ) + sin 2 ( Ω 2 ) ] .
f ( m , n ) = a ( m , n ) cos [ ϕ ( m , n ) ] .
Ω 1 ( m , n ) ϕ m = Ω c , 1 + Ω m , 1 q 1 ( m , n )
Ω a min k Ω c , k ,             Ω f 1 ,             Ω m , k Ω c , k ,
Φ d { a ( m , n ) cos [ ϕ ( m , n ) ] } a 2 ( m , n ) { sin 2 [ Ω 1 ( m , n ) ] + sin 2 [ Ω 2 ( m , n ) ] } .
g 1 ( m , n ) = [ f ( m + 1 , n ) - f ( m - 1 , n ) ] / 2 ,
g 2 ( m , n ) = [ f ( m , n + 1 ) - f ( m , n - 1 ) ] / 2 ,
Φ d [ g 1 ] a 2 sin 2 [ Ω 1 ] ( sin 2 [ Ω 1 ] + sin 2 [ Ω 2 ] ) ,
Φ d [ g 2 ] a 2 sin 2 [ Ω 2 ] ( sin 2 [ Ω 1 ] + sin 2 [ Ω 2 ] ) ,
arcsin ( { Φ d [ f ( m + 1 , n ) - f ( m - 1 , n ) ] 4 Φ d [ f ( m , n ) ] } 1 / 2 ) Ω 1 ( m , n ) ,
arcsin ( { Φ d [ f ( m , n + 1 ) - f ( m , n - 1 ) ] 4 Φ d [ f ( m , n ) ] } 1 / 2 ) Ω 2 ( m , n ) ,
2 Φ d [ f ( m , n ) ] { Φ d [ f ( m + 1 , n ) - f ( m - 1 , n ) ] + Φ d [ f ( m , n + 1 ) - f ( m , n - 1 ) ] } 1 / 2 a ( m , n ) .
Φ ( I ) = Φ ( I - I ¯ ) - I ¯ 2 I .
Φ d ( I ) ( m , n ) = Φ d ( I - I ¯ ) ( m , n ) - I ¯ [ ( i , j ) B I ( m + i , n + j ) - 4 I ( m , n ) ] ,
f ( m , n ) = 0.5 [ 1 + 0.5 cos ( π 30 m + π 50 n ) ] cos [ π 3 m + π 5 n + 2 sin ( π 30 m ) + sin ( π 50 n + π 2 ) ] , m , n = 1 , , 100 ;

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