## Optimal discrimination and classification of THz spectra in the wavelet domain

Optics Express, Vol. 11, Issue 12, pp. 1462-1473 (2003)

http://dx.doi.org/10.1364/OE.11.001462

Acrobat PDF (405 KB)

### 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

## 1. Introduction

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

5. Z. Jiang and X.-C. Zhang, “2D measurement and spatio-temporal coupling of few-cycle THz pulses,” Opt. Express **5**, 243–248 (1999), http://www.opticsexpress.org/oearchive/source/13775.htm. [CrossRef] [PubMed]

6. T. Löffler, T. Bauer, K.J. Siebert, H.G. Roskos, A. Fitzgerald, and S. Czasch, “Terahertz dark-field imaging of biomedical tissue,” Opt. Express **9**, 616–621 (2001), http://www.opticsexpress.org/oearchive/source/37294.htm. [CrossRef] [PubMed]

15. 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]

16. 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]

17. R.K.H. Galvão, S. Hadjiloucas, and J.W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. **27**, 643–645 (2002). [CrossRef]

18. 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]

21. 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]

## 2. Propagation of noise in the wavelet domain

*b*(

*t*) is negligible when compared to the noise in the sample interferograms

*x*(

*t*). A vector

*x*(

*t*) of length 2

*J*, (

*t*=0, 1, …, 2

*J*-1), can be assumed to be a stochastic process composed of a signal term

*x*(

_{m}*t*) and a zero mean noise term

*n*(

*t*) so that:

*x*(

_{m}*t*)=

*E*[

*x*(

*t*)],

*E*[

*n*(

*t*)]=0, ∀

*t*,

*E*[

*n*(

*t*

_{1})

*n*(

*t*

_{2})]=0, ∀

*t*

_{1}≠

*t*

_{2}and

*E*[

*n*

^{2}(

*t*)]=

*σ*

^{2}, ∀

*t*with

*E*denoting the expectation operator and

*σ*the noise standard deviation. For an apodization window

*w*(

*t*), the apodized sample interferogram

*x*(

_{a}*t*) is

*x*(

_{a}*t*)=

*x*(

*t*)

*w*(

*t*) which can also be divided into signal and noise terms as:

*X*(

_{a}*ω*) of the apodized interferogram can be written as:

*z*(

*t*)=

*w*

^{2}(

*t*), it follows that:

*Z*(

*ω*) is the DFT of

*z*(

*t*). Let

*L*(

*ω*) be the complex insertion loss of the sample, that is,

*B*(

_{a}*ω*) is the DFT of the apodized background interferogram. From Eq. (3) it follows that:

*N*(

_{ab}*ω*) is the apodized noise in the background. Under the hypothesis that this noise in the background is negligible compared to the background signal, 8a becomes:

*L*(

*ω*)=

*L*(

_{m}*ω*)+

*M*(

*ω*). The noise term

*M*(

*ω*) has the following properties:

## 3. Wavelet transform of the complex insertion loss function

**L**=[

*L*(

*ω*

_{0})

*L*(

_{ω}

*1*) …

*L*(

*ω*

_{J-1})], obtained after eliminating the right half of the DFT result, can be written as:

*ω*is the

_{n}*n*

^{th}frequency point in the DFT and the basis elements

*ψ*(wavelets) are built from a mother wavelet function

_{a,b}*ψ*as:

*a*∊R* is the “Scale” (or “Dilation”) which determines the width of the wavelet and subscript

*b*∊R defines the position of the wavelet with respect to signal

**L**. Equations (11) and (12) define a continuous wavelet transform, in the sense that the real-valued parameters

*a*and

*b*are allowed to vary continuously. The discrete wavelet transform is obtained by taking parameters

*a*and

*b*from a dyadic grid defined in the following manner:

*s*and

*r*are integer numbers. In this case, the wavelet transform results in a set of coefficients indexed by

*s*(scale level) and

*r*(translation index). These coefficients can be obtained in a computationally efficient manner by using the tree algorithm [22,23

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

**c**is the result of convolving

_{0}**L**with a “scaling function”, which is the low-pass counterpart of the mother wavelet. This can be regarded as a low-pass pre-filtering procedure. The wavelet transform coefficients at scale level

*s*are stored in sequence

**d**

_{s}.

*H*and

*G*represent respectively low-pass and high-pass filters associated with the mother wavelet adopted. Symbol (↓2) denotes the down-sampling operation, which consists of eliminating every other coefficient of a sequence. Down-sampling ensures that the number of points remains the same after the wavelet transform. In each scale level s two sets of coefficients are generated:

**d**(which are also called “detail” coefficients in this context) and

_{s}**c**(“approximation” coefficients). The tree algorithm is iterative in the sense that coefficients

_{s}**c**

_{s}are further decomposed in order to generate

**c**and

_{s+1}**d**. The filtering and down-sampling operations can be summarized in the following equations, in which sequences

_{s+1}*h*(

*i*) and

*g*(

*i*), of 2

*Q*points each (

*i*=0, 1,…2

*Q*-1), are the impulse response functions of the low-pass and high-pass filters

*H*and

*G*respectively.

*i*denotes the index of elements in the filtering sequence, whilst 2

*Q*denotes the number of points in the sequence). It is worth noting that most works use the data vector itself as the input to the tree algorithm (instead of

**c**). This results in the

_{0}**d**coefficients obtained being actually approximations to the wavelet transform coefficients. This approximation improves as

_{s}*s*increases. However, in most cases, it may not be essential to obtain the actual wavelet transform coefficients, but just to generate coefficients in which the energy of the signal is compressed. In this case, both

**c**and

_{s}**d**coefficients can be regarded as the result of an energy compressing linear transform and can thus be used for filtering and feature extraction. In this filter bank, the low-pass filtering result undergoes successive filtering iterations with the number of iterations

_{s}*S*(

*S*<log

_{2}(

*J*)) chosen by the analyst. The final result of the decomposition of data vector

**L**is a vector

**p**resulting from the concatenation of row vectors

**c**

_{s}and

**d**

_{s}in the following manner:

**d**

_{S},

**d**

_{S-1},

**d**

_{S-2}) associated with broad features in the data vector, and coefficients in smaller scales (e.g.

**d**

_{1},

**d**

_{2}

**d**

_{3}…) associated with narrower features such as sharp peaks. In this manner, the double index (

*s,r*) of the filter bank result can be replaced with a single index

*k*indicating the position of the coefficient in vector

**p**.

*h*

_{0},

*h*

_{1}, …,

*h*

_{2Q-1}} and {

*g*

_{0},

*g*

_{1}, …,

*g*

_{2Q-1}} satisfy the following conditions:

*l*an arbitrary point in the length of the sequence, the structure in Fig. 1 is termed a quadrature-mirror filter (QMF) bank [22–24]. A QMF bank is said to enjoy a perfect reconstruction (PR) property, because vector

**L**can be reconstructed from vector

**p**which means that there is no loss of information in the decomposition process. Moreover if the convolution operation is performed in a circular manner [25], after data are periodized to minimize border effects, the transformation from

**L**to

**p**can be represented by a matrix operation

**p**

_{1×J}=

**L**

_{1×J}

**V**

_{J×J}where

**V**is an orthogonal matrix, i.e.

**V**

^{T}

**V**=

**I**. For a single decomposition level, for instance, the

**V**matrix is given as

**V**=[

*|*

**H***], where*

**G***and*

**H***are sub-matrices built from the low-pass and high-pass filter impulse responses as*

**G***p*(

*k*) be the

*k*

^{th}coefficient resulting from the decomposition of

**L**by the filter bank shown in Fig. 1, that is,

*v*(

_{k}*n*) is the

*n*

^{th}element of the

*k*

^{th}column of the transformation matrix

**V**associated with the filter bank. Since

*L*(

*ω*)=

_{n}*L*(

_{m}*ω*)+

_{n}*M*(

*ω*), it follows that:

_{n}**V**) are real-valued, the noise term

*f*(

*k*) has the following properties:

*x*(

*t*) is taken as the difference between two sample interferograms, the wavelet coefficients

*p*(

*k*),

*k*=0, 1, …,

*J*-1 can be used as discriminant values. In this case,

*Ξ*(

*k*) will reflect the signal-to-noise ratio in the

*k*

^{th}discriminant coefficient. Wavelet coefficients with low signal-to-noise ratio are discarded. The maximization of the objective function

*Ξ*(

*k*) circumvents the main difficulty of the wavelet transform which is how to choose the best mother wavelet for a particular application.

*t*

_{1},

*t*

_{2},

*t*

_{3},

*t*

_{4}. Each interferogram can be regarded as a point in a four-dimensional space. After the Fourier transformation and upon elimination of the right half of the resulting vectors, the patterns become spectra with real and imaginary parts at two frequencies

*ω*

_{1}and

*ω*

_{2}. In Figs. 3a and 3b, the two patterns to be discriminated are represented as points in two complex planes, each plane associated with one of the frequencies. The larger circles represent the noise associated with the patterns, which is equal in all directions because the Fourier transform is orthogonal. The difference Δ

*X*is represented in each plane (Figs. 3(c) and (d)) as a vector joining the two points. Ratioing against the background at each frequency is equivalent to rotating and contracting the difference vectors, which then become differences in complex insertion loss Δ

_{a}*L*. Since the background intensity is different at each frequency, the noise level becomes different in Δ

*L*(

*ω*

_{1}) and Δ

*L*(

*ω*

_{2}). Since, the noise is no-longer equal in all directions, as shown in Figs. 3(e) and (f), the Euclidean distance classifier is no-longer optimal [16

16. 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]

*Ξ*(

*k*) performs a further rotation of the four axes trying to find a single direction for the

*k*

^{th}wavelet coefficient in which the projection OB of the difference is maximal with respect to the projection AC of the noise.

## 4. Classification of spectra in the wavelet domain

10. 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]

16. 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]

28. 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]

*Ξ*value, which means that they are basically associated with the noise. Moreover, by comparing Fig. 6a and 6c, it can be seen that large coefficients may not have a large

*Ξ*value, since their noise propagation factor may also be large. In this sense, a deterministic approach to the selection of wavelet coefficients, which only takes into account the magnitude of the coefficients (Fig. 6(a)) without considering the noise, might not yield good results. Function

*Ξ*(Fig. 6(c)) can be employed to select the wavelet coefficients to include in the classification model. Statistical criteria [29] could be used to weight the discriminative power of the model against its complexity, in order to determine the best number of wavelet coefficients to be used. Without loss of generality, the 5 wavelets with the largest

*Ξ*-value will be employed in a simulated study to illustrate the utility of the proposed optimization procedure.

*Ξ*-values before and after the optimization. As can be seen, a considerable improvement was obtained in the objective function for all coefficients. For illustration purposes, Fig. 7 depicts the effect of the optimization on the low-pass filter weights

*h*(

*i*) for the wavelet coefficient that had the largest

*Ξ*-value. The original db4 filter weights are represented as circles, whereas the filter weights resulting from the optimization are represented as squares. The thick vertical lines are used to help the visualization of the magnitude of the weights. This graph shows that the filters were significantly modified by the optimization algorithm.

## 5. Conclusion

## References and Links

1. | D.M. Mittleman, R.H. Jacobsen, and M.C. Nuss, “T-Ray Imaging,” IEEE J. Sel. Top. Quantum Electron. |

2. | X.-C. Zhang, ‘Next Rays? T. Ray!’, Plenary session, 26 |

3. | 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. |

4. | Z. Jiang and X.-C. Zhang, “Single-Shot Spatial-Temporal THz Field Imaging,” Opt. Lett. |

5. | Z. Jiang and X.-C. Zhang, “2D measurement and spatio-temporal coupling of few-cycle THz pulses,” Opt. Express |

6. | T. Löffler, T. Bauer, K.J. Siebert, H.G. Roskos, A. Fitzgerald, and S. Czasch, “Terahertz dark-field imaging of biomedical tissue,” Opt. Express |

7. | S.W Smye, J.M. Chamberlain, A.J. Fitzgerald, and E. Berry “The interaction between Terahertz radiation and biological tissue,” Phys. Med. Biol. |

8. | 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. |

9. | 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. |

10. | S. Hadjiloucas, L.S. Karatzas, and J.W. Bowen, “Measurements of Leaf Water Content Using Terahertz Radiation,” IEEE Trans. Microwave Theory Tech. MTT. |

11. | P.Y. Han, G.C. Cho, and X.-C. Zhang, “Time-domain transillumination of biomedical tissue with terahertz pulses,” Opt. Lett. |

12. | D.D. Arnone, C. Ciesla, and M. Pepper, “Terahertz imaging comes into view,” |

13. | 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 |

14. | 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. |

15. | 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. |

16. | 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 |

17. | R.K.H. Galvão, S. Hadjiloucas, and J.W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. |

18. | D.M. Mittleman, G. Gupta, R. Neelamani, R.G. Baraniuk, J.V. Rudd, and M. Koch “Recent advances in terahertz imaging” Appl. Phys. B |

19. | B. Ferguson and D. Abbott, “Wavelet de-noising of optical terahertz imaging data,” Fluctuation and Noise Lett. , |

20. | B. Ferguson and D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. , |

21. | J. W. Handley, A.J. Fitzgerald, E. Berry, and R.D. Boyle “Wavelet compression in medical terahertz pulsed imaging,” Phys. Med. Biol. |

22. | G. Strang and T. Nguyen, |

23. | I. Daubechies, |

24. | P. P. Vaidyanathan, |

25. | A.V. Oppenheim and R.W. Schafer, |

26. | B.G. Sherlock and D. M. Monro, “On the space of orthonormal wavelets,” IEEE Trans. Signal Processing , |

27. | J. A. Nelder and R. Mead, “Simplex method for function minimization,” Computer , |

28. | 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 , |

29. | E. R. Malinowski, |

**OCIS Codes**

(100.7410) Image processing : Wavelets

(170.1580) Medical optics and biotechnology : Chemometrics

(300.6270) Spectroscopy : Spectroscopy, far infrared

(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

(320.7100) Ultrafast optics : Ultrafast measurements

(330.4270) Vision, color, and visual optics : Vision system neurophysiology

(330.6180) Vision, color, and visual optics : Spectral discrimination

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 25, 2003

Revised Manuscript: June 2, 2003

Published: June 16, 2003

**Citation**

Roberto Galvão, Sillas Hadjiloucas, John Bowen, and Clarimar Coelho, "Optimal discrimination and classification of THz spectra in the wavelet domain," Opt. Express **11**, 1462-1473 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-12-1462

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### References

- D.M. Mittleman, R.H. Jacobsen, and M.C. Nuss, ???T-Ray Imaging,??? IEEE J. Sel. Top. Quantum Electron. 2, 679-692 (1996). [CrossRef]
- X.-C. Zhang, ???Next Rays? T. Ray! ???, Plenary session, 26th International Conference on Infrared and millimeter waves, Toulouse France, September 2001.
- 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]
- Z. Jiang and X.-C. Zhang, "Single-Shot Spatial-Temporal THz Field Imaging," Opt. Lett. 23, 1114-1116 (1998). [CrossRef]
- Z. Jiang and X.-C. Zhang, ???2D measurement and spatio-temporal coupling of few-cycle THz pulses,??? Opt. Express 5, 243-248 (1999), <a href="http://www.opticsexpress.org/oearchive/source/13775.htm">http://www.opticsexpress.org/oearchive/source/13775.htm.</a>. [CrossRef] [PubMed]
- T. Löffler, T. Bauer, K.J. Siebert, H.G. Roskos, A. Fitzgerald and S. Czasch, ???Terahertz dark-field imaging of biomedical tissue,??? Opt. Express 9, 616-621 (2001), <a href="http://www.opticsexpress.org/oearchive/source/37294.htm">http://www.opticsexpress.org/oearchive/source/37294.htm</a>. [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]
- 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]
- P.Y. Han, G.C. Cho and X.-C. Zhang, ???Time-domain transillumination of biomedical tissue with terahertz pulses,??? Opt. Lett. 25, 242-244 (2000). [CrossRef]
- 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.
- 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. 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]
- R.K.H. Galvão, S. Hadjiloucas and J.W. Bowen, ???Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,??? Opt. Lett. 27, 643-645 (2002). [CrossRef]
- 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]
- B. Ferguson and D. Abbott, ???Wavelet de-noising of optical terahertz imaging data,??? Fluctuation and Noise Lett., 1, L65-L70, (2001). [CrossRef]
- B. Ferguson and D. Abbott, ???De-noising techniques for terahertz responses of biological samples,??? Microelectron. J., 32, 943-953, (2001). [CrossRef]
- 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]
- 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).
- B.G. Sherlock and D. M. Monro, ???On the space of orthonormal wavelets,??? IEEE Trans. Signal Processing, 46, 1716-1720, 1998. [CrossRef]
- J. A. Nelder and R. Mead, ???Simplex method for function minimization,??? Computer, 7, 308???313 (1965).
- 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]
- E. R. Malinowski, Factor analysis in chemistry, (Wiley, New York, 1991).

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