Abstract

In rapid scan Fourier transform spectrometry, we show that the noise in the wavelet coefficients resulting from the filter bank decomposition of the complex insertion loss function is linearly related to the noise power in the sample interferogram by a noise amplification factor. By maximizing an objective function composed of the power of the wavelet coefficients divided by the noise amplification factor, optimal feature extraction in the wavelet domain is performed. The performance of a classifier based on the output of a filter bank is shown to be considerably better than that of an Euclidean distance classifier in the original spectral domain. An optimization procedure results in a further improvement of the wavelet classifier. The procedure is suitable for enhancing the contrast or classifying spectra acquired by either continuous wave or THz transient spectrometers as well as for increasing the dynamic range of THz imaging systems.

© 2003 Optical Society of America

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References

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Analyst

H.C. Goicoechea and A.C. Olivieri, ???Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS)???, Analyst, 124, 725-731 (1999).
[CrossRef]

Appl. Phys. B

D.M. Mittleman G. Gupta, R. Neelamani, R.G. Baraniuk J.V. Rudd and M. Koch ???Recent advances in terahertz imaging??? Appl. Phys. B 68, 1085-1094 (1999).
[CrossRef]

Appl. Phys. B.

D.M. Mittleman R. H. Jacobsen, R. Neelamani, R. G. Baraniuk and M. C. Nuss, ???Gas sensing using terahertz time-domain spectroscopy,??? Appl. Phys. B. 67, 379-390, (1998).
[CrossRef]

Computer

J. A. Nelder and R. Mead, ???Simplex method for function minimization,??? Computer, 7, 308???313 (1965).

Fluctuation and Noise Lett.

B. Ferguson and D. Abbott, ???Wavelet de-noising of optical terahertz imaging data,??? Fluctuation and Noise Lett., 1, L65-L70, (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D.M. Mittleman, R.H. Jacobsen, and M.C. Nuss, ???T-Ray Imaging,??? IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. Hadjiloucas, L.S. Karatzas and J.W. Bowen, ???Measurements of Leaf Water Content Using Terahertz Radiation,??? IEEE Trans. Microwave Theory Tech. MTT. 47, 142-149 (1999).
[CrossRef]

IEEE Trans. Signal Processing

B.G. Sherlock and D. M. Monro, ???On the space of orthonormal wavelets,??? IEEE Trans. Signal Processing, 46, 1716-1720, 1998.
[CrossRef]

J. Opt. Soc. Am A

S. Hadjiloucas, R.K.H. Galvao and J.W. Bowen, ???Analysis of spectroscopic measurements of leaf water content at THz frequencies using linear transforms,??? J. Opt. Soc. Am A 19, 2495-2509, (2002).
[CrossRef]

Microelectron. J.

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

Opt. Express

Opt. Lett.

Phys. Med. Biol.

P. Knobloch, C. Schildknecht, T. Kleine-Ostmann, M. Koch, S. Hoffmann, M. Hofmann, E. Rehberg, M. Sperling, K. Donhuijsen, G. Hein, and K. Pierz, ???Medical THz imaging: an investigation of histo-pathological samples,??? Phys. Med. Biol. 47, 3875-3884 (2002).
[CrossRef] [PubMed]

R.M. Woodward, B.E. Cole, V.P Wallace, R.J. Pye, D.D. Arnone, E.H. Linfield and M. Pepper ???Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue,??? Phys. Med. Biol. 47, 3853-3864 (2002).
[CrossRef] [PubMed]

S.W Smye, J.M. Chamberlain, A.J. Fitzgerald and E. Berry ???The interaction between Terahertz radiation and biological tissue,??? Phys. Med. Biol. 46 No 9 R101-R112 (2001).
[CrossRef] [PubMed]

A.J. Fitzgerald, E Berry, N.N. Zinovev, G.C. Walker, M.A. Smith and J.M. Chamberlain, ???An introduction to medical imaging with coherent terahertz frequency radiation,??? Phys. Med. Biol. 47 No 7 R67-R84 (2002).
[CrossRef] [PubMed]

P. Haring-Bolivar, M. Brucherseifer, M. Nagel, H. Kurz, A. Bosserhoff and R. Büttner "Label-free probing of genes by time-domain terahertz sensing,??? Phys. Med. Biol. 47, 3815-3822 (2002).
[CrossRef] [PubMed]

J. W. Handley, A.J. Fitzgerald, E. Berry and R.D. Boyle ???Wavelet compression in medical terahertz pulsed imaging,??? Phys. Med. Biol. 47, 3885-3892 (2002).
[CrossRef] [PubMed]

Other

G. Strang and T. Nguyen, Wavelets and Filter Banks, (Wellesley-Cambridge Press, Wellesley, 1996).

I. Daubechies, Ten lectures on wavelets, (Society for Industrial and Applied Mathematics, SIAM, Philadelphia, 1992).
[CrossRef]

P. P. Vaidyanathan, Multirate Systems and Filter Banks, (Prentice-Hall, Englewood Cliffs, 1993).

A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, (Prentice-Hall, Englewood Cliffs, 1989).

E. R. Malinowski, Factor analysis in chemistry, (Wiley, New York, 1991).

X.-C. Zhang, ???Next Rays? T. Ray! ???, Plenary session, 26th International Conference on Infrared and millimeter waves, Toulouse France, September 2001.

D.D. Arnone, C. Ciesla, and M. Pepper, ???Terahertz imaging comes into view,??? in Issue April 2000 of Physics World, (Institute of Physics and IOP Publishing Limited 2000), pp. 35-40.

R.M. Woodward, B. Cole, V.P. Wallace, D.D. Arnone, R. Pye, E.H. Linfield, M. Pepper and A.G. Davies, ???Terahertz pulse imaging of in-vitro basal cell carcinoma samples,??? in OSA Trends in Optics and Photonics (TOPS) 56, Conference on Lasers and Electro-Optics (CLEO 2001), Technical Digest, Postconference Edition (Optical Society of America, Washington D.C., 2001), 329-330.

Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1.

Diagram of the tree algorithm used for the generation of the wavelet approximation (blue) and detail (red) coefficients. The frequency response of each filter combination is depicted.

Fig. 2.
Fig. 2.

Procedure for parameterizing wavelet filter banks by Q angles. z-1 represents the unit time delay operator.

Fig. 3.
Fig. 3.

Graphical illustration of the transformations involved in the optimization process.

Fig. 4.
Fig. 4.

Right: Interferograms xa (t) after mean-centering, time-centering and apodization: (a) Leather, (b) Lycra, (c) Background, (d) Difference. Left: (a) Spectra Xa (ω) of leather (green), lycra (brown) and background (orange), (b) spectrum ΔXa (ω) and (c) complex insertion loss ΔL(ω) of the difference between leather and lycra (graphs are in absolute values).

Fig. 5.
Fig. 5.

Contour plots of the real part (left) and imaginary part (right) of the db4 continuous wavelet transform for (a) leather, (b) lycra and (c) difference between them.

Fig. 6.
Fig. 6.

(a) Energy of each coefficient resulting from the db4 filter bank with one decomposition level. (b) Noise amplification factor for each coefficient. (c) Value of the objective function for each coefficient (ratio between the energy of the coefficient and its noise amplification factor). In this graph, the first 500 coefficients correspond to the 1st level approximation (blue region), whereas the remaining 500 correspond to the 1st level detail (red region).

Fig. 7.
Fig. 7.

Comparison between the original db4 low-pass filter weights (circles) and the low-pass filter weights resulting from the optimization (squares) for the wavelet coefficient that had the largest Ξ value. Eight points are depicted since the dbQ family has filters with 2Q taps. The inset shows the Ξ values before (black bars) and after (orange bars) the optimization for the 5 wavelet coefficients used in the classification model.

Fig. 8.
Fig. 8.

Classification errors (%) as a function of noise level in the interferograms. Non-optimized db4 wavelet (green), optimized wavelet (red) and Euclidean distance (blue) classifiers. The inset shows a leather interferogram with (a) no artificially added noise and noise with standard deviation of (b) 0.1 and (c) 0.5.

Equations (30)

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

x ( t ) = x m ( t ) + n ( t )
x a ( t ) = [ x m ( t ) + n ( t ) ] w ( t ) = x m ( t ) w ( t ) + n ( t ) w ( t )
X a ( ω ) = DFT [ x a ( t ) ] = DFT [ x m ( t ) w ( t ) + n ( t ) w ( t ) ] = X ma ( ω ) + N a ( ω )
X ma ( ω ) = DFT [ x m ( t ) w ( t ) ]
N a ( ω ) = DFT [ n ( t ) w ( t ) ]
E [ N a ( ω ) ] = t E [ n ( t ) ] 0 w ( t ) e j ω t = 0
E [ N a ( ω 1 ) N a * ( ω 2 ) ] = E [ ( t n ( t ) w ( t ) e j ω 1 t ) ( t n ( t ) w ( t ) e + j ω 2 t ) ] =
= E [ ( t 1 t 2 n ( t 1 ) w ( t 1 ) e j ω 1 t 1 n ( t 2 ) w ( t 2 ) e + j ω 2 t 2 ) + ( t n 2 ( t ) w 2 ( t ) e j ( ω 1 ω 2 ) t ) ]
= ( t 1 t 2 E [ n ( t 1 ) n ( t 2 ) ] 0 w ( t 1 ) e j ω 1 t 1 w ( t 2 ) e + j ω 2 t 2 ) + ( t E [ n 2 ( t ) ] σ 2 w 2 ( t ) e j ( ω 1 ω 2 ) t )
E [ N a ( ω 1 ) N a * ( ω 2 ) ] = σ 2 t z ( t ) e j ( ω 1 ω 2 ) t = σ 2 Z ( ω 1 ω 2 )
L ( ω ) = X ma ( ω ) B a ( ω ) + X ma ( ω ) B a ( ω ) ( N a ( ω ) X ma ( ω ) ) 2 + ( N ab ( ω ) B a ( ω ) ) 2
L ( ω ) = X ma ( ω ) B a ( ω ) + N a ( ω ) B a ( ω )
E [ M ( ω ) ] = 1 B a ( ω ) E [ N a ( ω ) ] 0 = 0
E [ M ( ω 1 ) M * ( ω 2 ) ] = E [ N a ( ω 1 ) B a ( ω 1 ) N a * ( ω 2 ) B a * ( ω 2 ) ] = E [ N a ( ω 1 ) N a * ( ω 2 ) ] B a ( ω 1 ) B a * ( ω 2 ) = σ 2 Z ( ω 1 ω 2 ) B a ( ω 1 ) B a * ( ω 2 )
p ( a , b ) = n = 0 J 1 L ( ω n ) ψ a , b ( ω n )
ψ a , b ( ω ) = 1 a ψ ( ω b a )
a = 2 s , b = r 2 s
c s + 1 ( r ) = i = 0 2 Q 1 h ( i ) c s ( 2 r i + 1 )
d s + 1 ( r ) = i = 0 2 Q 1 g ( i ) c s ( 2 r i + 1 )
p = [ c S d S d S I d 1 ] .
i = 0 2 Q 1 2 l h i h i + 2 l = { 1 , l = 0 0 , 0 < l < Q
g i = ( 1 ) i + 1 h 2 Q 1 i , i = 0 , 1 , , 2 Q 1
V = 0 0 h 2 Q 4 h 2 Q 2 h 2 Q 1 0 h 2 Q 5 h 2 Q 3 h 2 Q 2 0 h 2 Q 6 h 2 Q 4 h 2 Q 3 h 2 Q 1 h 2 Q 7 h 2 Q 5 h 0 h 2 0 0 0 h 1 0 0 0 h 0 0 0 0 0 h 2 Q 2 0 0 0 h 2 Q 3 h 2 Q 1 0 0 g 2 Q 4 g 2 Q 2 g 2 Q 1 0 g 2 Q 5 g 2 Q 3 g 2 Q 2 0 g 2 Q 6 g 2 Q 4 g 2 Q 3 g 2 Q 1 g 2 Q 7 g 2 Q 5 g 0 g 2 0 0 0 g 1 0 0 0 g 0 0 0 0 0 g 2 Q 2 0 0 0 g 2 Q 3 g 2 Q 1
p ( k ) = n L ( ω n ) v k ( n )
p ( k ) = n [ L m ( ω n ) + M ( ω n ) ] v k ( n ) = n L m ( ω n ) v k ( n ) + n M ( ω n ) v k ( n )
E [ f ( k ) ] = n E [ M ( ω n ) ] 0 v k ( n ) = 0
E [ f ( k ) f * ( k ) ] = E [ ( n M ( ω n ) v k ( n ) ) ( n M * ( ω n ) v k ( n ) ) ]
= E [ n 1 n 2 M ( ω n 1 ) M * ( ω n 2 ) v k ( n 1 ) v k ( n 2 ) ] = n 1 n 2 E [ M ( ω n 1 ) M * ( ω n 2 ) ] v k ( n 1 ) v k ( n 2 )
= σ 2 n 1 n 2 Z ( ω n 1 ω n 2 ) B a ( ω n 1 ) B a * ( ω n 2 ) v k ( n 1 ) v k ( n 2 ) NAF ( k ) = σ 2 NAF ( k )
Ξ ( k ) = p m ( k ) p m * ( k ) NAF ( k )

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