Abstract

We demonstrate how the symmetric convolution-multiplication property of the discrete trigonometric transforms can be applied to problems in image reconstruction. This property allows for linear filtering of degraded images by means of point-by-point multiplication in the transform domain of trigonometric transforms. Specifically, in the transform domain of a type II discrete cosine transform, there is an asymptotically optimum energy compaction near d.c. for highly correlated images, which has advantages in reconstructing images with high-frequency noise. The symmetric convolution-multiplication property allows for scalar representations in the transform-domain space of discrete trigonometric transforms for linear reconstruction filters such as the Wiener filter. An analysis of the scalar Wiener filter’s performance in the trigonometric transform domain is given.

© 1998 Optical Society of America

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  1. N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
    [CrossRef]
  2. A. K. Jain, “A sinusoidal family of unitary transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-1, 356–365 (1979).
    [CrossRef]
  3. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  4. K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic, San Diego, Calif., 1990).
  5. S. A. Martucci, “Symmetric convolution and the discrete sine and cosine transforms,” IEEE Trans. Signal Process. 42, 1038–1051 (1994).
    [CrossRef]
  6. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  7. B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. AU-19, 285–288 (1971).
    [CrossRef]
  8. R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).
  9. B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Computers C-22, 805–812 (1973).
    [CrossRef]
  10. A. Graham, Kronecker Products and Matrix Calculus with Applications (Wiley, New York, 1981).
  11. S. A. Martucci, “Digital filtering of images using the discrete sine and cosine transforms,” Opt. Eng. 35, 119–127 (1996).
    [CrossRef]
  12. Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 803–816 (1984).
    [CrossRef]
  13. T. M. Foltz, B. M. Welsh, “Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms,” IEEE Trans. Image Process. (to be published).
  14. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).
  15. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).
  16. C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).
  17. S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  18. W. K. Pratt, “Generalized Wiener filter computation techniques,” IEEE Trans. Computers C-21, 636–641 (1972).
    [CrossRef]
  19. B. R. Hunt, T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).
  20. M. R. Whiteley, B. M. Welsh, M. C. Roggemann, “Limitations of Gaussian assumptions for the irradiance distribution in digital imagery: nonstationary image ensemble considerations,” J. Opt. Soc. Am. A 15, 802–810 (1998).
    [CrossRef]
  21. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Processing 42, 156–163 (1994).
    [CrossRef]
  22. J. W. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory IT-18, 232–240 (1972).
    [CrossRef]
  23. J. A. Stuller, B. Kurz, “Two-dimensional Markov representations of sampled images,” IEEE Trans. Commun. COM-24, 1148–1152 (1976).
    [CrossRef]
  24. T. M. Foltz, “Trigonometric transforms for image reconstruction,” Ph.D dissertation (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1998).
  25. M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  26. W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
    [CrossRef]

1998 (1)

1996 (1)

S. A. Martucci, “Digital filtering of images using the discrete sine and cosine transforms,” Opt. Eng. 35, 119–127 (1996).
[CrossRef]

1994 (2)

S. A. Martucci, “Symmetric convolution and the discrete sine and cosine transforms,” IEEE Trans. Signal Process. 42, 1038–1051 (1994).
[CrossRef]

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Processing 42, 156–163 (1994).
[CrossRef]

1984 (1)

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

1979 (1)

A. K. Jain, “A sinusoidal family of unitary transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-1, 356–365 (1979).
[CrossRef]

1977 (1)

W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
[CrossRef]

1976 (2)

J. A. Stuller, B. Kurz, “Two-dimensional Markov representations of sampled images,” IEEE Trans. Commun. COM-24, 1148–1152 (1976).
[CrossRef]

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

1974 (1)

N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
[CrossRef]

1973 (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Computers C-22, 805–812 (1973).
[CrossRef]

1972 (2)

W. K. Pratt, “Generalized Wiener filter computation techniques,” IEEE Trans. Computers C-21, 636–641 (1972).
[CrossRef]

J. W. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory IT-18, 232–240 (1972).
[CrossRef]

1971 (1)

B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. AU-19, 285–288 (1971).
[CrossRef]

Ahmed, N.

N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
[CrossRef]

Cannon, T. M.

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

Chen, W. H.

W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
[CrossRef]

Foltz, T. M.

T. M. Foltz, “Trigonometric transforms for image reconstruction,” Ph.D dissertation (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1998).

T. M. Foltz, B. M. Welsh, “Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms,” IEEE Trans. Image Process. (to be published).

Fralick, S. C.

W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Graham, A.

A. Graham, Kronecker Products and Matrix Calculus with Applications (Wiley, New York, 1981).

Horn, R. A.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).

Hunt, B. R.

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Computers C-22, 805–812 (1973).
[CrossRef]

B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. AU-19, 285–288 (1971).
[CrossRef]

Jain, A. K.

A. K. Jain, “A sinusoidal family of unitary transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-1, 356–365 (1979).
[CrossRef]

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Johnson, C. R.

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Kurz, B.

J. A. Stuller, B. Kurz, “Two-dimensional Markov representations of sampled images,” IEEE Trans. Commun. COM-24, 1148–1152 (1976).
[CrossRef]

Martucci, S. A.

S. A. Martucci, “Digital filtering of images using the discrete sine and cosine transforms,” Opt. Eng. 35, 119–127 (1996).
[CrossRef]

S. A. Martucci, “Symmetric convolution and the discrete sine and cosine transforms,” IEEE Trans. Signal Process. 42, 1038–1051 (1994).
[CrossRef]

Matson, C. L.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Processing 42, 156–163 (1994).
[CrossRef]

Natarajan, T.

N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Pratt, W. K.

W. K. Pratt, “Generalized Wiener filter computation techniques,” IEEE Trans. Computers C-21, 636–641 (1972).
[CrossRef]

Rao, K. R.

N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
[CrossRef]

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic, San Diego, Calif., 1990).

Roggemann, M. C.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

Smith, C. H.

W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
[CrossRef]

Stuller, J. A.

J. A. Stuller, B. Kurz, “Two-dimensional Markov representations of sampled images,” IEEE Trans. Commun. COM-24, 1148–1152 (1976).
[CrossRef]

Therrien, C. W.

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

Wang, Z.

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

Welsh, B. M.

M. R. Whiteley, B. M. Welsh, M. C. Roggemann, “Limitations of Gaussian assumptions for the irradiance distribution in digital imagery: nonstationary image ensemble considerations,” J. Opt. Soc. Am. A 15, 802–810 (1998).
[CrossRef]

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

T. M. Foltz, B. M. Welsh, “Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms,” IEEE Trans. Image Process. (to be published).

Whiteley, M. R.

Woods, J. W.

J. W. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory IT-18, 232–240 (1972).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

Yip, P.

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic, San Diego, Calif., 1990).

IEEE Trans. Acoust., Speech, Signal Process. (1)

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Audio Electroacoust. AU-19, 285–288 (1971).
[CrossRef]

IEEE Trans. Commun. (2)

J. A. Stuller, B. Kurz, “Two-dimensional Markov representations of sampled images,” IEEE Trans. Commun. COM-24, 1148–1152 (1976).
[CrossRef]

W. H. Chen, C. H. Smith, S. C. Fralick, “A fast computational algorithm for the discrete cosine transform,” IEEE Trans. Commun. COM-25, 1004–1009 (1977).
[CrossRef]

IEEE Trans. Computers (3)

N. Ahmed, T. Natarajan, K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers C-23, 90–93 (1974).
[CrossRef]

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Computers C-22, 805–812 (1973).
[CrossRef]

W. K. Pratt, “Generalized Wiener filter computation techniques,” IEEE Trans. Computers C-21, 636–641 (1972).
[CrossRef]

IEEE Trans. Inf. Theory (1)

J. W. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory IT-18, 232–240 (1972).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

A. K. Jain, “A sinusoidal family of unitary transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-1, 356–365 (1979).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. A. Martucci, “Symmetric convolution and the discrete sine and cosine transforms,” IEEE Trans. Signal Process. 42, 1038–1051 (1994).
[CrossRef]

IEEE Trans. Signal Processing (1)

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Processing 42, 156–163 (1994).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

B. R. Hunt, T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. Syst. Man Cybern. SMC-6, 876–882 (1976).

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

Opt. Eng. (1)

S. A. Martucci, “Digital filtering of images using the discrete sine and cosine transforms,” Opt. Eng. 35, 119–127 (1996).
[CrossRef]

Other (12)

T. M. Foltz, “Trigonometric transforms for image reconstruction,” Ph.D dissertation (Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1998).

M. C. Roggemann, B. M. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, Fla., 1996).

A. Graham, Kronecker Products and Matrix Calculus with Applications (Wiley, New York, 1981).

T. M. Foltz, B. M. Welsh, “Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms,” IEEE Trans. Image Process. (to be published).

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992).

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Mass., 1991).

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1992).

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

R. A. Horn, C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, New York, 1991).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications (Academic, San Diego, Calif., 1990).

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Figures (7)

Fig. 1
Fig. 1

Satellite object.

Fig. 2
Fig. 2

Inverse trigonometric filtering example using the Hanning-windowed Gaussian degradation filter with 1/e width of 4 pixels. (a) Blurred object, (b) object recovered with inverse filter, (c) noisy blurred object [(SNR)=20 dB], (d) noisy object unrecoverable with inverse filter.

Fig. 3
Fig. 3

Scalar Wiener filtering example using the same degradation filter as in the inverse filtering example. (a) Object estimate using the Fourier Scalar Wiener filter, (b) object estimate using the trigonometric scalar Wiener filter.

Fig. 4
Fig. 4

Normalized mse versus filter dimension.

Fig. 5
Fig. 5

Normalized mse versus correlation coefficient.

Fig. 6
Fig. 6

Normalized mse versus SNR.

Fig. 7
Fig. 7

Normalized mse versus PSF 1/e width.

Equations (135)

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

HC=WMHF,CWM-1,
[WM-1]mn=exp-j 2πMmn,m, n=0, 1,, M-1,
[WM]mn=1Mexp+j 2πMmn,
m, n=0, 1,, M-1,
WM-1d=(WM-1h)  (WM-1θ),
AB
=[A]00B[A]01B[A]0(N1-1)B[A]10B[A]11B[A]1(N1-1)B[A](N1-1)0B[A](N1-1)1B[A](N1-1)(N1-1)B,
ϑF=(WN1-1WN2-1)θ.
HBC=(WN1WN2)HF,BC(WN1-1WN2-1),
(WN1-1WN2-1)d=(WN1-1WN2-1)h  (WN1-1WN2-1)θ,
da=S2e,N-1[S1e,Nhar  C2e,Nθ],
da=-C2e,N-1[S1e,Nhar  S2e,Nθ],
ds=C2e,N-1[C1e,Nhsr  C2e,Nθ],
ds=S2e,N-1[C1e,Nhsr  S2e,Nθ],
da=S2e,N-1HT,aC2e,Nθ=HSC,aθ,
da=-C2e,N-1HT,aS2e,Nθ=HSC,aθ,
ds=C2e,N-1HT,sC2e,Nθ=HSC,sθ,
ds=S2e,N-1HT,sS2e,Nθ=HSC,sθ,
h(n1, n2)=haa(n1, n2)+has(n1, n2)+hsa(n1, n2)+hss(n1, n2),
ϑT,ss=(C2e,N1C2e,N2)θ,
hT,aa=(S1e,N1S1e,N2)haarr,
hT,as=(C1e,N1S1e,N2)hasrr,
hT,sa=(S1e,N1C1e,N2)hsarr,
hT,ss=(C1e,N1C1e,N2)hssrr.
d=(S2e,N1-1S2e,N2-1)[hT,aa  ϑT,ss]+(C2e,N1-1S2e,N2-1)[hT,as  ϑT,ss]+(S2e,N1-1C2e,N2-1)[hT,sa  ϑT,ss]+(C2e,N1-1C2e,N2-1)[hT,ss  ϑT,ss].
HT,aa=diag{hT,aa},HT,as=diag{hT,as},
HT,sa=diag{hT,sa},HT,ss=diag{hT,ss},
HBSC,aa=(S2e,N1-1S2e,N2-1)HT,aa(C2e,N1C2e,N2),
HBSC,as=(C2e,N1-1S2e,N2-1)HT,as(C2e,N1C2e,N2),
HBSC,sa=(S2e,N1-1C2e,N2-1)HT,sa(C2e,N1C2e,N2),
HBSC,ss=(C2e,N1-1C2e,N2-1)HT,ss(C2e,N1C2e,N2),
HBSC,sa=(S2e,N1-1C2e,N2-1)HT,sa(C2e,N1C2e,N2),
HBSC,sa=(S2e,N1-1S2e,N2-1)HT,sa(C2e,N1S2e,N2),
HBSC,sa=-(C2e,N1-1C2e,N2-1)HT,sa(S2e,N1C2e,N2),
HBSC,sa=-(C2e,N1-1S2e,N2-1)HT,sa(S2e,N1S2e,N2),
(WN1WN2)FF,BC(WN1-1WN2-1)
=[(WN1WN2)HF,BC(WN1-1WN2-1)]-1,
FF,BC(k1, k2)=1HF,BC(k1, k2),
θˆ=(Fa+Fs)d=(Fa+Fs)(Ha+Hs)θ.
θˆ=[(-C2e,N-1FaS2e,N)(S2e,N-1HaC2e,N)+(-C2e,N-1FaS2e,N)(S2e,N-1HsS2e,N)+(C2e,N-1FsC2e,N)(-C2e,N-1HaS2e,N)+(C2e,N-1FsC2e,N)(C2e,N-1HsC2e,N)]θ,
FsHs-FaHa=I,
FaHs+FsHa=0,
-Ha(k)Hs(k)Hs(k)Ha(k)Fa(k)Fs(k)=10,
Fa(k)=-Ha(k)Ha2(k)+Hs2(k),
Fs(k)=Hs(k)Ha2(k)+Hs2(k),
Fs(k)=1Hs(k).
θˆ=(Faa+Fas+Fsa+Fss)d=(Faa+Fas+Fsa+Fss)(Haa+Has+Hsa+Hss)θ.
θˆ=(C1-1C2-1){Faa(S1S2)[(S1-1S2-1)×Haa(C1C2)+(S1-1S2-1)Has(S1C2)+(S1-1S2-1)Hsa(C1S2)+(S1-1S2-1)×Hss(S1S2)]+Fas(C1S2)[(C1-1S2-1)×Haa(S1C2)-(C1-1S2-1)Has(C1C2)+(C1-1S2-1)Hsa(S1S2)-(C1-1S2-1)×Hss(C1S2)]+Fsa(S1C2)[(S1-1C2-1)×Haa(C1S2)+(S1-1C2-1)Has(S1S2)-(S1-1C2-1)Hsa(C1C2)-(S1-1C2-1)×Hss(S1C2)]+Fss(C1C2)[(C1-1C2-1)×Haa(S1S2)-(C1-1C2-1)Has(C1S2)-(C1-1C2-1)Hsa(S1C2)+(C1-1C2-1)×Hss(C1C2)]}θ,
FaaHaa-FasHas-FsaHsa+FssHss=I,
FaaHas+FasHaa-FsaHss-FssHsa=0,
FaaHsa-FasHss+FsaHaa-FssHas=0,
FaaHss+FasHsa+FsaHas+FssHaa=0,
Haa(k1, k2)-Has(k1, k2)-Hsa(k1, k2)Hss(k1, k2)Has(k1, k2)Haa(k1, k2)-Hss(k1, k2)-Hsa(k1, k2)Hsa(k1, k2)-Hss(k1, k2)Haa(k1, k2)-Has(k1, k2)Hss(k1, k2)Hsa(k1, k2)Has(k1, k2)Haa(k1, k2)Faa(k1, k2)Fas(k1, k2)Fsa(k1, k2)Fss(k1, k2)=1000,
Faa(k1, k2)=1A(k1, k2)Haa3(k1, k2)+Haa(k1, k2)Has2(k1, k2)+Haa(k1, k2)Hsa2(k1, k2)-Haa(k1, k2)Hss2(k1, k2)+2Has(k1, k2)Hsa(k1, k2)Hss(k1, k2),
Fas(k1, k2)=-1A(k1, k2)Has3(k1, k2)+Has(k1, k2)Haa2(k1, k2)-Has(k1, k2)Hsa2(k1, k2)+Has(k1, k2)Hss2(k1, k2)+2Haa(k1, k2)Hsa(k1, k2)Hss(k1, k2),
Fsa(k1, k2)=-1A(k1, k2)Hsa3(k1, k2)+Hsa(k1, k2)Haa2(k1, k2)-Hsa(k1, k2)Has2(k1, k2)+Hsa(k1, k2)Hss2(k1, k2)+2Haa(k1, k2)Has(k1, k2)Hss(k1, k2),
Fss(k1, k2)=1A(k1, k2)Hss3(k1, k2)-Hss(k1, k2)Haa2(k1, k2)+Hss(k1, k2)Has2(k1, k2)+Hss(k1, k2)Hsa2(k1, k2)+2Haa(k1, k2)Has(k1, k2)Hsa(k1, k2),
A(k1, k2)={[Haa(k1, k2)+Hss(k1, k2)]2+[Has(k1, k2)-Hsa(k1, k2)]2}×{[Haa(k1, k2)-Hss(k1, k2)]2+[Has(k1, k2)+Hsa(k1, k2)]2}.
Fss(k1, k2)=1Hss(k1, k2).
θˆ=μθ+CθθHBCT(HBCCθθHBCT+Cww)-1(d-HBCμθ).
θˆ=RθθHBCT(HBCRθθHBCT+Rww)-1d,
FBC=RθθHBCT[HBCRθθHBCT+Rww]-1,
FF,BC(k1, k2)=HF,BC*(k1, k2)|HF,BC(k1, k2)|2+WF2(k1, k2)¯ϴF2(k1, k2)¯,
(Fa+Fs)[(Ha+Hs)Rθθ(Ha+Hs)T+Rww]
=Rθθ(Ha+Hs)T,
(-C2e,N-1FaS2e,N)(S2e,N-1HaC2e,N)Rθθ(S2e,N-1HaC2e,N)T
+(-C2e,N-1FaS2e,N)(S2e,N-1HaC2e,N)
×Rθθ(C2e,N-1HsC2e,N)T+(S2e,N-1FaC2e,N)
×(C2e,N-1HsC2e,N)Rθθ(S2e,N-1HaC2e,N)T
+(S2e,N-1FaC2e,N)(C2e,N-1HsC2e,N)
×Rθθ(C2e,N-1HsC2e,N)T+(S2e,N-1FsS2e,N)
×(S2e,N-1HaC2e,N)Rθθ(S2e,N-1HaC2e,N)T
+(S2e,N-1FsS2e,N)(S2e,N-1HaC2e,N)
×Rθθ(C2e,N-1HsC2e,N)T+(C2e,N-1FsC2e,N)
×(C2e,N-1HsC2e,N)Rθθ(S2e,N-1HaC2e,N)T
+(C2e,N-1FsC2e,N)(C2e,N-1HsC2e,N)
×Rθθ(C2e,N-1HsC2e,N)T+(-C2e,N-1FaS2e,N)
×RwwS2e,NTS2e,N-T+(C2e,N-1FsC2e,N)
×RwwC2e,NTC2e,N-T
=C2e,N-1C2e,NRθθ(S2e,N-1HaC2e,N)T
+C2e,N-1C2e,NRθθ(C2e,N-1HsC2e,N)T,
C2e,N-1[-Fa(HaRϴsϴsHa+RWaWa)
+FsHsRϴsϴsHa]S2e,N-T+C2e,N-1[-FaHaRϴsϴsHs
+Fs(HsRϴsϴsHs+RWsWs)]C2e,N-T
+S2e,N-1[FaHsRϴsϴsHa+FsHaRϴsϴsHa]S2e,N-T
+S2e,N-1[FaHsRϴsϴsHs+FsHaRϴsϴsHs]C2e,N-T
=C2e,N-1RϴsϴsHaS2e,N-T+C2e,N-1RϴsϴsHsC2e,N-T,
-Fa(HaRϴsϴsHa+RWaWa)+FsHsRϴsϴsHa
=RϴsϴsHa,
-FaHaRϴsϴsHs+Fs(HsRϴsϴsHs+RWsWs)
=RϴsϴsHs,
FaHsRϴsϴsHa+FsHaRϴsϴsHa=0,
FaHsRϴsϴsHs+FsHaRϴsϴsHs=0.
Ha2(k)+Wa2(k)¯ϴs2(k)¯-Ha(k)Hs(k)-Ha(k)Hs(k)Hs2(k)+Ws2(k)¯ϴs2(k)¯Ha(k)Hs(k)Ha2(k)Hs2(k)Ha(k)Hs(k)F(k)Fs(k)
=-Ha(k)Hs(k)00,
Ha2+Hs2+Wa2¯ϴs2¯00Ha2+Hs2+Ws2¯ϴs2¯0000FaFs
=-HaHsHaHsHa2+Hs2+Wa2¯ϴs2¯+-HaHsHa2+Hs2+Ws2¯ϴs2¯0,
Ha2(k)+Hs2(k)+Wa2(k)¯ϴs2(k)¯00Ha2(k)+Hs2(k)+Ws2(k)¯ϴs2(k)¯Fa(k)Fs(k)=-Ha(k)Hs(k),
Ha(k)Hs(k)Ha2(k)+Hs2(k)+Wa2(k)¯ϴs2(k)¯
=Ha(k)Hs(k)Ha2(k)+Hs2(k)+Ws2(k)¯ϴs2(k)¯.
Wa2(k)¯=0,k=02Nσ2,k=1, 2,, N-14Nσ2,k=N,
Ws2(k)¯=4Nσ2,k=02Nσ2,k=1, 2,, N-10,k=N,,
Fa(k)=-Ha(k)Ha2(k)+Hs2(k)+Wa2(k)¯ϴs2(k)¯
Fs(k)=Hs(k)Ha2(k)+Hs2(k)+Ws2(k)¯ϴs2(k)¯,
Fs(k)=Hs(k)Hs2(k)+Ws2(k)¯ϴs2(k)¯.
(Faa+Fas+Fsa+Fss)[(Haa+Has+Hsa+Hss)
×Rθθ(Haa+Has+Hsa+Hss)T+Rww]
=Rθθ(Haa+Has+Hsa+Hss)T,
Haa2+Waa2¯ϴss2¯-HaaHas-HaaHsa-HaaHss-HaaHasHas2+Was2¯ϴss2¯HasHsa-HasHss-HaaHsaHasHsaHsa2+Wsa2¯ϴss2¯-HsaHssHaaHss-HasHss-HsaHssHss2+Wss2¯ϴss2¯HasHaa-Hss-HsaHsa-HssHaa-HasHssHsaHasHaa FaaFasFsaFssHaa-Has-HsaHss000
Fss(k1, k2)=Hss(k1, k2)Hss2(k1, k2)+Wss2(k1, k2)¯ϴss2(k1, k2)¯.
Faa(k1, k2)=0,
Fas(k1, k2)=0,
Fsa(k1, k2)
=-Hsa(k1, k2)Hsa2(k1, k2)+Hss2(k1, k2)+Wsa2(k1, k2)¯ϴss2(k1, k2)¯,
Fss(k1, k2)
=Hss(k1, k2)Hsa2(k1, k2)+Hss2(k1, k2)+Wss2(k1, k2)¯ϴss2(k1, k2)¯.
Faa(k1, k2)=0,
Fas(k1, k2)
=-Has(k1, k2)Has2(k1, k2)+Hss2(k1, k2)+Was2(k1, k2)¯ϴss2(k1, k2)¯,
Fsa(k1, k2)=0,
Fss(k1, k2)
=Hss(k1, k2)Has2(k1, k2)+Hss2(k1, k2)+Wss2(k1, k2)¯ϴss2(k1, k2)¯.
Faa(k1, k2)=Haa(k1, k2)Haa2(k1, k2)+Hsa2(k1, k2)+Waa2(k1, k2)¯ϴss2(k1, k2)¯,
Fas(k1, k2)=0,
Fsa(k1, k2)
=-Hsa(k1, k2)Haa2(k1, k2)+Hsa2(k1, k2)+Wsa2(k1, k2)¯ϴss2(k1, k2)¯,
Fss(k1, k2)=0.
Faa(k1, k2)
=Haa(k1, k2)Haa2(k1, k2)+Has2(k1, k2)+Waa2(k1, k2)¯ϴss2(k1, k2)¯,
Fas(k1, k2)
=-Has(k1, k2)Haa2(k1, k2)+Has2(k1, k2)+Was2(k1, k2)¯ϴss2(k1, k2)¯,
Fsa(k1, k2)=0,
Fss(k1, k2)=0.
22¯=tr{Rθθ-2FHRθθ+F(HRθθHT+Rww)FT},
T22¯=tr{Rϴϴ-2FHRϴϴ+F(HRϴϴHT+RWW)FT},

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