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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 2 — Jan. 18, 2010
  • pp: 1177–1190
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Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy

Yang Chen, Shengyang Huang, and Emma Pickwell-MacPherson  »View Author Affiliations


Optics Express, Vol. 18, Issue 2, pp. 1177-1190 (2010)
http://dx.doi.org/10.1364/OE.18.001177


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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

1. Introduction

Terahertz (1012 Hz, THz) technology is continuing to advance and an increasing number of applications are being investigated in areas ranging from terahertz characterization of breast cancer [1

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]

] to terahertz spectroscopy of explosives [2

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).

]. Different applications impose different requirements on the technology for instance in terms of the characteristics of terahertz devices, data acquisition rates and geometry restrictions for sample access. TeraView Ltd, Cambridge UK, has developed a prototype hand-held terahertz probe which is designed to be suitable for in vivo imaging of breast cancer (in reflection geometry) and we have been working with them to improve its capabilities. Addressing the requirements for fast data acquisition and freedom of movement of the probe (the user can wand the probe over the patient) has compromised the quality of the terahertz signal reaching the sample: both the bandwidth and signal to noise have been reduced. Thus, when we performed our usual deconvolution (inverse filtering coupled with a band pass double Gaussian filter) on data from the palm of the hand acquired using the terahertz probe, we were no longer able to resolve time domain features in the impulse response function that we had seen in data from our flat-bed imaging system (the TPI Imaga 1000, TeraView Ltd) processed in this way [3

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

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).

]. Adjusting the band bass filter to remove sufficient noise resulted in loss of too much signal. Therefore we have been investigating alternative processing methods with a view to being able to improve the resolution of the impulse response function extracted from the probe data.

In time series analysis, DWT often suffer from a lack of translation invariance. This problem can be solved by using the un-decimated stationary wavelet transform (SWT). The SWT is similar to the DWT in that the high-pass and low-pass filters are applied to the input signal at each level, but the output signal is never decimated. Instead, the filters are upsampled at each level [12

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

]. The translation-invariant property of the stationary wavelet will ensure that at each level in the SWT the lengths and relative positions of the decomposed approximation and detail coefficients are kept the same as in the time domain, such that for each level the threshold can be estimated from the preset noise intervals along the temporal axis. Thus we employ SWT in our algorithm to further improve on other existing approaches.

2. Theory

2.1 Baseline calculation

2.2 Double Gaussian filtered Inverse Filtering (DGIF)

Here, we outline two deconvolution approaches: direct inverse filtering and inverse filtering coupled with a double Gaussian filter, which are referred to as IF deconvolution and DGIF deconvolution for brevity. The IF deconvolution approach is based on the equation:
(gr)=f(hr)
(1)
where f is the target impulse response function, and g, h and r are the measured sample pulse, reference pulse and baseline pulse, respectively. Using the IF deconvolution, we obtain f by:

FFT(f)=FFT(gr)FFT(hr)  f=FFT-1(FFT(gr)FFT(hr))
(2)

To suppress the amplified noise effects caused by the division operation in Eq. (2), a band pass filter can be coupled into the IF deconvolution. Hitherto a double Gaussian filter has been our filter of choice:
filter=1HFet2HF21LFet2LF2
(3)
f=FFT-1(FFT(filter)FFT(gr)FFT(hr))
(4)
where t represents the time axis with zero in the middle. The parameters HF (relating to the high frequency cut off) and LF (relating to the low frequency cut off) in the filter function need to be set manually. Though suppressing noise to some extent, this DGIF deconvolution tends to over-smooth the resulting deconvolved time-domain pulses.

2.3 Hybrid Fourier-Wavelet Domain Deconvolution (FWDD)

The proposed Frequency-Wavelet Domain Deconvolution (FWDD) approach performs deconvolutions of terahertz pulses by first employing frequency domain Wiener filtering and then attenuating the leaked noise through stationary wavelet shrinkage. We use SWT with Daubechies wavelets because they have similar shape and length to our typical impulse function and produce good results in our experiment. If a function is s-times continuously differentiable, then it is said to have regularity s. The regularity or smoothness of our FWDD result is dependent on the regularity of the wavelet transform. Regularity of the wavelet transform is determined by the length of the filter used in the wavelet transform; the higher the regularity, the longer the filter length. Generally, the higher the differentiable order of the target signal, the higher the regularity of the wavelet representing the signal needs to be. In this work we choose the Daubechies4 (db4) wavelet as it has an appropriate order of differentiability to represent our terahertz signal.

It is observed that our detected terahertz pulses have a non-zero signal before the reflection off the quartz/sample interface. Since the signal before the sample reflection should be zero we deduce that this must be due to system noise or fluctuations. Similarly, when measuring a semi-infinite sample layer, we have observed that there is also a non-zero signal after the reflection due to the sample, but if there were no noise then this should be zero too. We therefore treat this region as noise. Thus, by analyzing the two intervals before and after the main sample reflection, the magnitude and distribution properties of the system noise can be obtained. This is the basis of the threshold estimation for the wavelet shrinkage in our proposed FWDD approach. The new approach consists of the following steps:

2.3.1 Wiener filtering

Frequency domain Wiener filtering is performed through following Eqs. (5-7) to achieve a crude deconvolution but ensuring that there is no blurring of the main impulse function:
fWiener=FFT1(FFT(gr)FFT(hr)(|FFT(hr)|2|FFT(hr)|2+βNσ2S))
(5)
S=(g-r)-mean(g-r)22Nσ2(h-r)22
(6)
mean(gr)=t(g(t)r(t))N
(7)
where S is the mean power spectral density of subtracted measurements(gr). N is the total number of points in t time domain axis. The Lp norm operator is denoted by and is defined:Xp:=(i=1n|xi|p)1p. Nσ2 is the estimated noise power and the noise variance σ2 is estimated from the subtracted measurements (gr) using the median estimator on the finest level wavelet coefficients of (gr) [7

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

]. The regularization parameter β can be used to modulate the effect of Wiener filtering in the tradeoff between noise-suppressing and pulse-preserving. Manually setting β to be a small value between 0.001 and 0.05 avoids over smoothing and any noise remaining after the Wiener filtering can be well addressed by applying the following stationary wavelet shrinkage step.

2.3.2 Stationary wavelet shrinkage

The translation-invariant SWT is used to preserve the relative temporal positions in wavelet decomposition and reconstruction. Along the temporal axis at each level k, it transforms a 1-D signal fF(n) into the approximation coefficients vector cAk,l and detail coefficients vector cDk,l by convolving fF(n) with a low-pass filterΨand a high-pass filterΦ respectively. In contrast to the well-known general discrete wavelet transform, SWT does not down sample the signal and instead upsamples the filters at each decomposition level. Figure 1
Fig. 1 The schematic decomposition of SWT
shows the schematic decomposition of SWT:

Wavelet shrinkage is a signal denoising technique based on the idea of thresholding the wavelet coefficients. Wavelet coefficients having small absolute value are considered to encode mostly noise and very fine details of the signal. In contrast, the important information is encoded by the coefficients having large absolute values. Removing the small absolute value coefficients and then reconstructing the signal should produce a signal with less noise. After applying SWT on the result from Wiener filtering, we obtain the approximation and detail coefficients cAk, and cDk,, at each level k [15

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).

]. We used a maximum level of 5 for the wavelet decomposition as no significant improvement was observed for higher levels to justify the extra computational expense.

[cA,cD]=SWT(fWiener)
(8)

For the wavelet shrinkage at each level k, we need the threshold parameters T_cDk for the detail coefficients. We define T_cDk to be the maximum magnitudes of the detail coefficients within the indicated noise intervals at each level k. We note that at each level k in the SWT, the lengths and relative positions of the decomposed approximation coefficientscAk,and detail coefficients cDk, are kept the same as in the temporal axis of the original data forfWiener. The positions of the noise intervals (indicated by the red rectangles in Fig. 2
Fig. 2 A flow chart to highlight the extra processes involved in our proposed approach compared to our original method.
) are thus also kept unchanged for each level k. As a default, we choose the length of each red box to be one quarter of the length of the measured signal and position them to avoid the main pulse. For example, for a measurement with 512 data points, the box preceding the main pulse starts 10 points from the beginning of the signal and finishes at point 10 + 512/4 = 138, and the box following the main pulse starts at 512-138 = 374. Their separation in this case is therefore 374-138 = 236 data points. The separation of the boxes is not too crucial, but we sometimes need to change the position if they overlap with the main pulse, this occasionally happens for measurements with fewer data points (eg 256). By considering the noise after the main pulse as well as before it, we are able to improve the resulting impulse function than if only the noise interval before the pulse was used. This is because in many cases the magnitude of the noise after the main pulse is greater than that preceding the main pulse. With these acquired T_cDk at each level k we then perform shrinkage to obtain the estimated cD^k, by thresholding the coefficients cDk, along the whole temporal axis:

cD^k,=sign(cDk,)(|cDk,|T_cDk)+
(9)

Here, soft thresholding is used withsign(x)=x/|x|. (x)+=0 if x<0, and (x)+=x if x0. The maximum detail coefficients’ magnitudes from the main pulse are generally higher than those of the noise within the indicated noise intervals. Coefficients with larger magnitudes tend to represent useful information and smaller coefficients tend to represent noise, therefore in the shrinkage process, we only keep the coefficients which are higher than the highest coefficient from within the noise. Then we can obtain the deconvolved pulse fFWDD by performing an inverse stationary wavelet transform (ISWT, the inverse process of SWT) from the approximation coefficients cAk, and the thresholded detail coefficients cD^k, at each level k.

fFWDD=ISWT([cA,cD^])
(10)

2.4 Calculation of the complex refractive index

We use Fresnel theory to determine the complex refractive index of the sample from sample and reference measurements. The resulting equation (as explained in [9

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).

]) is:
k˜samplecosθsam=(1M)k˜quartz2θquartz2+(1+M)k˜quartzθquartzk˜aircosθair(1M)k˜aircosθair+(1+M)k˜quartzθquartz
(11)
where, k˜samplesinθsam=12 and k˜sample=njc2ωαwith n and α representing the refractive index and absorption coefficient respectively. In our original approach, M was the frequency ratio of the sample and reference, given as:

M=FFT(gr)FFT(hr)
(12)

In this new FWDD approach, we use the Fourier transform of the impulse function resulting from the inverse stationary wavelet transform to determine M such that:

M=FFT(fFWDD)
(13)

3. Experimental methods and equipment

We initially apply our theory to the same data previously used to illustrate the improvement due to our baseline calculation method [13

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]

]. In this way we can highlight the additional improvement due to our FWDD approach. These data were acquired from the reflection geometry flat-bed system, TPI Imaga 1000, TeraView Ltd. Details about the system and experimental procedure are given in reference [3

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]

] and [13

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]

] respectively.

To further test our theory we apply it to in vivo measurements of the palm of the hand acquired using the terahertz probe by TeraView Ltd. In contrast to the TPI Imaga 1000, which employs free space optics, the probe system utilizes optical fibers to guide the laser beam to the terahertz devices. Short 90 fs pulses (centered at 800 nm) from a Ti:Sapphire laser are guided along an optical fiber shielded with an electrical cable to the probe head where a photoconductive emitter and receiver are embedded for generating and detecting THz pulses. The bandwidth achieved by this system is 0.1-2 THz. More details are given in reference [16

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).

].

4. Results and discussion

4.1 Frequency domain validation

From the traces in Fig. 3 we can clearly see the improvements as our processing method becomes more refined. The most significant improvements are seen in the absorption coefficient. This is most likely to be because the calculation of the absorption coefficient in reflection geometry is predominantly dependent on the phase information, and this is very sensitive to the experimental set up. When the measured baseline was used, the absorption coefficient barely matched the transmission spectroscopy data up to 0.5 THz. The calculated baseline method extended this up to about 1 THz; and using the FWDD approach combined with the calculated baseline has extended this further to about 1.8 THz. Thus the combined approach has nearly quadrupled the useable bandwidth compared to the standard measured baseline and processing approach.

The reason for this improvement is that the FWDD deconvolution produces a more correct impulse function of isopropanol in the time domain. This is then used in the calculation of the complex refractive index and thus results in a very close match with the standard transmission data up to about 1.8 THz. To illustrate the improvements in the time domain impulse response function we use in vivo measurements of skin as this is a multi-layered structure and better shows the strength of our approach.

4.3 Time domain validation and in vivo skin measurements

To illustrate the effectiveness of the FWDD deconvolution in the time domain processing we took in vivo reflection data of the palm of a hand using the terahertz probe system. As depicted in 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.
, the skin of the palm of the hand has a layered structure: the epidermis lies beneath the stratum corneum and has a different refractive index. There are two troughs in the reflected waveform. This is because the incident terahertz light is reflected when it enters a medium of different refractive index. The first reflection is due to the interface between the quartz window and the stratum corneum, and the second reflection is due to the interface between the stratum corneum and the epidermis. The reflections have the opposite phase to the incident pulse because it has entered a medium of higher refractive index. The separation (optical delay) between the troughs is dependent on the refractive index and the thickness of the stratum corneum. Since the skin has a high water content which attenuates the signal, we are not able to see reflections off the epidermis/dermis interface. In some cases there is a narrow peak before the first trough; this is attributed to surface dryness of the stratum corneum [3

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]

].

To highlight the improvement in resolution that our FWDD algorithm has over DGIF and ForWaRD approaches, we simulate the reflections at normal incidence due to a material of refractive index 1.5 with thicknesses ranging from 200 µm down to 20 µm at 20 µm intervals and process the data using the three approaches. In Fig. 7 we perform the simulations for a system with a high signal to noise ratio (SNR). The same DGIF parameters used in the flat-bed system analysis are used for the higher SNR simulation (LF = 2048 and HF = 20). Gaussian white noise is added to the incident terahertz pulse resulting in a SNR of 32 dB. We see that both the FWDD and ForWaRD approaches are able to resolve down to 40 µm (0.4 ps of optical delay) with similar sharpness, but that the DGIF can only resolve down to 80 µm (0.8 ps of optical delay). When the SNR of the system is reduced to 22 dB the DGIF parameters are adjusted to LF = 2048 and HF = 30 so that the resulting reflections are still smooth. As illustrated in 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.
, the DGIF can now only resolve down to 120 µm (1.2 ps of optical delay) and the FWDD and ForWaRD down to 80 µm (0.8 ps). However the FWDD approach results in clearer separation of the reflections than ForWaRD – this is most noticeable for 100 µm and 80 µm. Since our terahertz probe has a lower SNR than the flat-bed system this sharper resolution of the FWDD approach is very important. One reason for this is that ForWaRD’s automatic parameter settings limit its performance in high noise conditions whereas by modulating β the FWDD approach is still able to perform well. Thus our FWDD algorithm copes well with the probe data as β can be adjusted to deal with the noisier signal.

The computation costs for the IFDG, ForWaRD and proposed FWDD method for the 1-D signals calculated in terms of CPU times are 0.0018 seconds, 0.87 seconds, and 0.80 respectively. The simulation was performed in Matlab 7.1 using an Intel CoreTM 2 Quad CPU, 4096 Mb RAM. It is noted that although the FWDD method uses SWT and not DWT, it is slightly computationally less expensive than the ForWaRD algorithm; this is likely to be due to the automatic parameter finding in ForWaRD. The upsampling involved in the SWT process is far more (450 times more in this case) computationally expensive than the DGIF as it creates a much larger sample set and this makes real-time processing less feasible, but still tolerable for 1-D cases.

5. Conclusions

We have shown that by using the FWDD approach combined with our baseline calculation method we are able to nearly quadruple the bandwidth of reflection spectroscopy measurements as compared to standard methods. The FWDD approach also results in vast improvements in the time domain analysis as compared to DGIF such that impulse functions of samples can be retrieved with much less noise and without loss of useful high frequency information. This enables the reflections off layered structures to be much more clearly resolved. By using the FWDD approach on reflection data acquired by the terahertz probe we are able to improve the impulse response function so much so that we are able to compensate for the reduced signal to noise performance of the probe due to its fast acquisition times and fiber coupled devices.

In conclusion, we have devised a robust data processing algorithm that is able to obtain sharper and clearer deconvolved impulse functions and more accurate complex refractive indices.

Acknowledgments

The authors would like to thank Dr David Seery, Department of Applied Mathematics 
and Theoretical Physics, Cambridge University, UK, for useful discussions. We gratefully acknowledge partial financial support for this work from the Research Grants Council of the Hong Kong Government and the Shun Hing Institute of Advanced Engineering, Hong Kong.

References and links

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).

OCIS Codes
(000.1430) General : Biology and medicine
(120.4825) Instrumentation, measurement, and metrology : Optical time domain reflectometry
(300.6495) Spectroscopy : Spectroscopy, teraherz
(110.6795) Imaging systems : Terahertz imaging

ToC Category:
Imaging Systems

History
Original Manuscript: November 16, 2009
Revised Manuscript: December 23, 2009
Manuscript Accepted: January 4, 2010
Published: January 8, 2010

Citation
Yang Chen, Shengyang Huang, and Emma Pickwell-MacPherson, "Frequency-wavelet domain deconvolution for terahertz reflection imaging and spectroscopy," Opt. Express 18, 1177-1190 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-1177


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References

  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).

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