Abstract

In terahertz reflection imaging, a deconvolution process is often employed to extract the impulse function of the sample of interest. A band-pass filter such as a double Gaussian filter is typically incorporated into the inverse filtering to suppress the noise, but this can result in over-smoothing due to the loss of useful information. In this paper, with a view to improving the calculation of terahertz impulse response functions for systems with a low signal to noise ratio, we propose a hybrid Frequency-Wavelet Domain Deconvolution (FWDD) for terahertz reflection imaging. Our approach works well; it retrieves more accurate impulse response functions than existing approaches and these impulse functions can then also be used to better extract the terahertz spectroscopic properties of the sample.

© 2010 OSA

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  1. P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, “Terahertz pulsed spectroscopy of freshly excised human breast cancer,” Opt. Express 17(15), 12444–12454 (2009).
    [CrossRef] [PubMed]
  2. M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).
  3. E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
    [CrossRef] [PubMed]
  4. B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).
  5. D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
    [CrossRef]
  6. B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32(12), 943–953 (2001).
    [CrossRef]
  7. D. L. Donoho, “De-noising by soft-thresholding,” IEE Trans. Inf. Theory 41(3), 613–627 (1995).
    [CrossRef]
  8. K. R. Castleman, “Digital Image Processing,” Prentice-Hall, Englewood Cliffs, NJ (1996).
  9. S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).
  10. R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
    [CrossRef]
  11. J. Pearce, H. H. Choi, D. M. Mittleman, J. White, and D. Zimdars, “Terahertz wide aperture reflection tomography,” Opt. Lett. 30(13), 1653–1655 (2005).
    [CrossRef] [PubMed]
  12. J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).
  13. S. Huang, P. C. Ashworth, K. W. C. Kan, Y. Chen, V. P. Wallace, Y. T. Zhang, and E. Pickwell-MacPherson, “Improved sample characterization in terahertz reflection imaging and spectroscopy,” Opt. Express 17(5), 3848–3854 (2009).
    [CrossRef] [PubMed]
  14. S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
    [CrossRef]
  15. G. Nason and B. Silverman, “The Stationary Wavelet Transform and some Statistical Applications,” Wavelets and Statistics, Springer Lecture Notes in Statistics 13, 281–300 (1995).
  16. P. C. Ashworth, P. O'Kelly, A. D. Purushotham, S. E. Pinder, M. Kontos, M. Pepper, and V. P. Wallace, “An intra-operative THz probe for use during the surgical removal of breast tumors,” 2008 33rd International Conference on Infrared, Millimeter and Terahertz Waves, Vols 1 and 2, 767–769 (2008).

2009 (3)

2005 (1)

2004 (2)

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
[CrossRef]

2003 (1)

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

2001 (2)

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32(12), 943–953 (2001).
[CrossRef]

1997 (1)

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).

1996 (2)

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).

1995 (2)

G. Nason and B. Silverman, “The Stationary Wavelet Transform and some Statistical Applications,” Wavelets and Statistics, Springer Lecture Notes in Statistics 13, 281–300 (1995).

D. L. Donoho, “De-noising by soft-thresholding,” IEE Trans. Inf. Theory 41(3), 613–627 (1995).
[CrossRef]

Abbott, D.

B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32(12), 943–953 (2001).
[CrossRef]

Ahuja, A. T.

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

Arnone, D. D.

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Ashworth, P. C.

Baraniuk, R.

R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
[CrossRef]

Baraniuk, R. G.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).

Carfantan, H.

J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).

Chen, Y.

Choi, H.

R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
[CrossRef]

Choi, H. H.

Cluff, J. A.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Cole, B. E.

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Crawley, D.

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Donoho, D. L.

D. L. Donoho, “De-noising by soft-thresholding,” IEE Trans. Inf. Theory 41(3), 613–627 (1995).
[CrossRef]

Ferguson, B.

B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32(12), 943–953 (2001).
[CrossRef]

Fitzgerald, A. J.

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Ghael, S. P.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).

Huang, S.

Huang, S. Y.

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

Kan, K. W. C.

Kemp, M. C.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Krim, H.

J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).

Mittleman, D. M.

J. Pearce, H. H. Choi, D. M. Mittleman, J. White, and D. Zimdars, “Terahertz wide aperture reflection tomography,” Opt. Lett. 30(13), 1653–1655 (2005).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

Nason, G.

G. Nason and B. Silverman, “The Stationary Wavelet Transform and some Statistical Applications,” Wavelets and Statistics, Springer Lecture Notes in Statistics 13, 281–300 (1995).

Neelamani, R.

R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
[CrossRef]

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

Pearce, J.

Pepper, M.

P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, “Terahertz pulsed spectroscopy of freshly excised human breast cancer,” Opt. Express 17(15), 12444–12454 (2009).
[CrossRef] [PubMed]

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Pesquet, J. C.

J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).

Pickwell, E.

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

Pickwell-MacPherson, E.

Pinder, S. E.

Provenzano, E.

Purushotham, A. D.

Sayeed, A. M.

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).

Silverman, B.

G. Nason and B. Silverman, “The Stationary Wavelet Transform and some Statistical Applications,” Wavelets and Statistics, Springer Lecture Notes in Statistics 13, 281–300 (1995).

Taday, P. F.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Tribe, W. R.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Wallace, V. P.

P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, “Terahertz pulsed spectroscopy of freshly excised human breast cancer,” Opt. Express 17(15), 12444–12454 (2009).
[CrossRef] [PubMed]

S. Huang, P. C. Ashworth, K. W. C. Kan, Y. Chen, V. P. Wallace, Y. T. Zhang, and E. Pickwell-MacPherson, “Improved sample characterization in terahertz reflection imaging and spectroscopy,” Opt. Express 17(5), 3848–3854 (2009).
[CrossRef] [PubMed]

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Wang, Y. X. J.

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

White, J.

Woodward, R.

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Yeung, D. K.

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

Zhang, Y. T.

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

S. Huang, P. C. Ashworth, K. W. C. Kan, Y. Chen, V. P. Wallace, Y. T. Zhang, and E. Pickwell-MacPherson, “Improved sample characterization in terahertz reflection imaging and spectroscopy,” Opt. Express 17(5), 3848–3854 (2009).
[CrossRef] [PubMed]

Zimdars, D.

IEE Trans. Inf. Theory (1)

D. L. Donoho, “De-noising by soft-thresholding,” IEE Trans. Inf. Theory 41(3), 613–627 (1995).
[CrossRef]

IEEE Sel. Top. Quantum Electron. (1)

D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE Sel. Top. Quantum Electron. 2(3), 679–692 (1996).
[CrossRef]

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

J. C. Pesquet, H. Krim, and H. Carfantan, “Time-invariant orthonormal wavelet representations,” IEEE Trans. Acoust., Speech, Signal Process. 44, 1964–1970 (1996).

IEEE Trans. Signal Process. (1)

R. Neelamani, H. Choi, and R. Baraniuk, “Forward: Fourier-Wavelet regularized deconvolution for Ill-conditioned systems,” IEEE Trans. Signal Process. 52(2), 418–433 (2004).
[CrossRef]

Microelectron. J. (1)

B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32(12), 943–953 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (2)

S. Y. Huang, Y. X. J. Wang, D. K. Yeung, A. T. Ahuja, Y. T. Zhang, and E. Pickwell-Macpherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54(1), 149–160 (2009).
[CrossRef]

E. Pickwell, B. E. Cole, A. J. Fitzgerald, M. Pepper, and V. P. Wallace, “In vivo study of human skin using pulsed terahertz radiation,” Phys. Med. Biol. 49(9), 1595–1607 (2004).
[CrossRef] [PubMed]

Proc. Soc. Photo Opt. Instrum. Eng. (1)

B. E. Cole, R. Woodward, D. Crawley, V. P. Wallace, D. D. Arnone, and M. Pepper, “Terahertz imaging and spectroscopy of human skin, in-vivo,” Proc. Soc. Photo Opt. Instrum. Eng. 4276, 1–10 (2001).

Terahertz for Military and Security Applications (1)

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Terahertz for Military and Security Applications 5070, 44–52 (2003).

Wavelet Applications in Signal and Image Processing V (1)

S. P. Ghael, A. M. Sayeed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Wavelet Applications in Signal and Image Processing V 3169, 389–399 (1997).

Wavelets and Statistics, Springer Lecture Notes in Statistics (1)

G. Nason and B. Silverman, “The Stationary Wavelet Transform and some Statistical Applications,” Wavelets and Statistics, Springer Lecture Notes in Statistics 13, 281–300 (1995).

Other (2)

P. C. Ashworth, P. O'Kelly, A. D. Purushotham, S. E. Pinder, M. Kontos, M. Pepper, and V. P. Wallace, “An intra-operative THz probe for use during the surgical removal of breast tumors,” 2008 33rd International Conference on Infrared, Millimeter and Terahertz Waves, Vols 1 and 2, 767–769 (2008).

K. R. Castleman, “Digital Image Processing,” Prentice-Hall, Englewood Cliffs, NJ (1996).

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Figures (8)

Fig. 1
Fig. 1

The schematic decomposition of SWT

Fig. 2
Fig. 2

A flow chart to highlight the extra processes involved in our proposed approach compared to our original method.

Fig. 3
Fig. 3

The refractive index (a) and absorption coefficient (b) of isopropanol calculated using our newly proposed FWDD approach and compared to our previous methods and transmission spectroscopy data.

Fig. 4
Fig. 4

(a) A photograph to illustrate the measurement of the palm using the terahertz probe; (b) A schematic diagram of the skin layers contributing to the reflections observed in our terahertz measurements.

Fig. 5
Fig. 5

The resulting response functions calculated using palm data from a) the flat-bed imaging system, processed using a double Gaussian filter; b) the probe, processed using the same double Gaussian as in (a); c) the probe, processed with various double Gaussian filters; and d) the probe, processed using the new (FWDD) approach.

Fig. 6
Fig. 6

(a) The impulse functions along the palm using the FWDD approach with the inset photograph indicating the position of the measurements on the palm. (b) A b-scan image showing the variation in stratum corneum thickness with palm position. The color bar represents E(n)/Emax. The white dashed line indicates the interface between the quartz window and the stratum corneum; and the black dashed line indicates the interface between the stratum corneum and the epidermis.

Fig. 7
Fig. 7

Pulse separation when SNR is 32 dB by (a) DGIF, LF = 2048 and HF = 20; (b) FWDD, β = 0.001; and (c) ForWaRD approaches.

Fig. 8
Fig. 8

Pulse separation when SNR is 22 dB by (a) DGIF, LF = 2048 and HF = 30; (b) FWDD, β = 0.005; and (c) ForWaRD approaches.

Equations (13)

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

( g r ) = f ( h r )
FFT ( f ) = FFT ( g r ) FFT ( h r )     f = FFT -1 ( FFT ( g r ) FFT ( h r ) )
f i l t e r = 1 HF e t 2 HF 2 1 LF e t 2 LF 2
f = FFT -1 ( FFT ( f i l t e r ) FFT ( g r ) FFT ( h r ) )
f W i e n e r = FFT 1 ( FFT ( g r ) FFT ( h r ) ( | FFT ( h r ) | 2 | FFT ( h r ) | 2 + β N σ 2 S ) )
S = ( g - r ) - mean ( g - r ) 2 2 N σ 2 ( h - r ) 2 2
mean ( g r ) = t ( g ( t ) r ( t ) ) N
[ c A , c D ] = SWT( f W i e n e r )
c D ^ k , = s i g n ( c D k , ) ( | c D k , | T _ c D k ) +
f F W D D = ISWT ( [ c A , c D ^ ] )
k ˜ s a m p l e cos θ s a m = ( 1 M ) k ˜ q u a r t z 2 θ q u a r t z 2 + ( 1 + M ) k ˜ q u a r t z θ q u a r t z k ˜ a i r cos θ a i r ( 1 M ) k ˜ a i r cos θ a i r + ( 1 + M ) k ˜ q u a r t z θ q u a r t z
M = FFT ( g r ) FFT ( h r )
M = FFT ( f F W D D )

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