Pre- and post-processing for tomographic reconstruction of terahertz time-domain spectroscopy

Hye-Jin Hong; Jinho Park; Hochong Park; Joo-Hiuk Son; Chang-Beom Ahn

doi:10.1364/OE.21.019943

1. Introduction

Terahertz ray (T-ray) is an electromagnetic wave between infra-red and micro waves. It has applications in many areas such as security, nondestructive test, and biomedical imaging [1–4]. T-ray tomographic imaging using the delay time of the time-domain spectroscopy has emerged as a new imaging method because it can be a non-invasive technique to probe bio-terahertz interaction [5–8].

The delay time of the time-domain spectroscopy provides depth information in the reflection-type T-ray tomography. The refractive index is a ratio of the velocity of the T-ray in free space divided by that inside a sample. If the refractive index is greater than 1, velocity inside the sample is less than that in free space.

There have been several terahertz tomography approaches. Among them are: time-of-flight tomography using the delay time of pulsed T-ray [5,6,9,10], terahertz computed tomography similar to the x-ray computed tomography acquiring projection data from various rotational angles [11,12], and terahertz diffraction tomography [13]. The terahertz computed tomography and the diffraction tomography may be applicable more generally, since they use a more general formulation of terahertz signal and employ mechanisms for rotational and translational motions of source and detector pairs to record position-dependent terahertz signal. In time-of-flight tomography, since it acquires depth information solely from the delay time of terahertz pulse, it has been applied only to simple planar objects composed of well defined planes [5,9] or measurements of radar cross section from surface [10]. If the technique is applied to the tomography of non-planar bio-samples, it is necessary to measure the travel time of T-ray in free space before the sample and that inside the sample to compensate for the different velocities of the T-ray for each scan point.

To illustrate the tomographic reconstruction with a two-dimensional scanning of the time-domain spectroscopy, the wave fronts inside a phantom composed of several layers with homogeneous refractive index are shown in Fig. 1.

Fig. 1 Illustration of a phantom (a), recorded wave front as a function of time in the vertical direction (refractive index, n = 2) (b), and a reconstructed plane on a time index (c). The location of the reconstructed plane and the time index are shown in (a) and (b) with red dotted lines, respectively.

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In Fig. 1, a conceptual phantom is shown in (a), which was piled up by squares with equal thickness. We assumed that the refractive indices of the squares are 2. The Terahertz beam is assumed to be incident from above (z). The vertical and horizontal axes in (a) are arbitrarily chosen as z and x, and the axis normal to the sheet is assumed as y. A corresponding wave front recorded is shown in (b) as a function of time in the vertical direction. Due to a lower velocity of the T-ray inside the sample, it takes a longer time for the T-ray to pass the sample. Therefore, a plane (c) is reconstructed on the time index shown in (b) with a dotted line. As one can see, (c) is not a proper cross sectional plane of the phantom at the location shown in (a) with a dotted line. To acquire distortion-free tomographic planes, the different velocities of the T-ray in free space and inside a sample should be properly compensated for. In other words, the data recorded as a function of time should be converted to the data as a function of depth.

In this paper, we report on a simple method to measure the refractive index of the sample and the velocity compensation method necessary for the tomographic reconstruction of the T-ray time domain spectroscopic image of non-planar bio samples.

2. Measurement of refractive index

A schematic diagram to measure the refractive index of the sample using one-dimensional measured data (time-domain spectroscopy) is shown in Fig. 2.

Fig. 2 Schematic diagram to measure the refractive index of a sample is shown.

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Aluminum tape was placed on the table to serve as a highly-reflective reference surface beneath the sample. One terahertz measurement was taken with the sample in place and another in free space without the sample. The T-ray passing through the sample shows highly reflected signals at the surface ( $t_{1}$ ) and at the boundary between the sample and the table ( $t_{2}$ ) due to the large impedance differences between free space and the sample.

On the other hand, the T-ray passing through free space is reflected highly from the table ( $t_{3}$ ). If the sample is homogeneous with an equal refractive index, the refractive index of the sample may be expressed as

n = \frac{v_{a i r}}{v_{s a m p l e}} = \frac{\frac{d_{a i r}}{t_{a i r}}}{\frac{d_{s a m p l e}}{t_{s a m p l e}}}

where

v_{a i r}

and

v_{s a m p l e}

are velocities of the T-ray in free space and inside the sample;

d_{a i r}

and

d_{s a m p l e}

are distances from the sample surface to the floor and the sample as shown in Fig. 2;

t_{a i r}

and

t_{s a m p l e}

are transit times for the T-rays in free space and inside the sample for the distances

d_{a i r}

and

d_{s a m p l e}

, respectively.

In Eq. (1), $d_{a i r}$ may be expressed as

d_{a i r} = d_{s a m p l e} + d_{g a p} ≅ d_{s a m p l e}

where

d_{g a p}

, the distance between the bottom of the sample and the floor is assumed to be much smaller than

d_{s a m p l e}

, and is neglected. Thus, Eq. (1) may be approximately expressed as

n ≅ \frac{t_{s a m p l e}}{t_{a i r}} = \frac{t_{2} - t_{1}}{t_{3} - t_{1}}

where

t_{1}

is the reflected time of the T-ray at the surface of the sample;

t_{2}

is the reflected time at the boundary of the sample adjoining to the floor;

t_{3}

is the reflected time at the floor without passing through the sample.

3. Compensation method by the refractive index

The depth of the T-ray data is given by the integral length of the velocity of the T-ray multiplied by a unit time. Generally, a sample may be composed of many elements and structures having different refractive indices. However, we assumed a homogeneous sample with a constant refractive index. So only two media, free space and a sample, are considered.

The depth of the data at $(l_{1} + l_{2}) Δ t$ may be given by

\begin{array}{l} d = l_{1} Δ t ​ v_{a i r} + l_{2} Δ t v_{s a m p l e} \\ = l_{1} Δ t n v_{s a m p l e} + l_{2} Δ t v_{s a m p l e} \\ = l_{1} n Δ d + l_{2} Δ d \\ = (l_{1} n + l_{2}) Δ d \end{array}

Where

Δ d = Δ t v_{s a m p l e}

In Eq. (4),

l_{1}

is the number of time units for the T-ray to travel in free space before the surface of the sample;

l_{2}

is the number of time units for the T-ray to travel inside the sample to reach the location;

Δ t

is the unit time; and

Δ d

is the unit distance in the sample.

From Eq. (4), the data acquired at the time of $(l_{1} + l_{2}) Δ t$ is converted to the one with the depth of $(l_{1} n + l_{2}) Δ d$ . Disregarding the units for now, the difference between the distance and time indices is $(l_{1} n + l_{2}) - (l_{1} + l_{2}) = (n - 1) l_{1}$ . Thus the two-dimensional terahertz spectroscopy data as a function of time can be converted to the one as a function of depth by simply shifting the data by $(n - 1) l_{1}$ points and changing the time unit to the depth unit given by Eq. (5). The time index $l_{1}$ can be detected by the first peak signal reflected at the surface of the sample.

The recorded wave fronts shown in Fig. 1(b) are transformed as a function of depth using the velocity compensation method and are shown in Fig. 3. Since the velocity of the T-ray in free space is double the speed inside the sample (n = 2), compensation can be done by shifting the data by $l_{1}$ grids as previously described. The grid in Fig. 3(a) is the unit distance of the T-ray inside the phantom. Perfect reconstruction can be made as shown in Fig. 3(b).

Fig. 3 Restored wave front as a function of depth after the compensation (n = 2) (a) and a reconstructed plane (b) for the location shown in (a) with a dotted line.

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4. Experiment

The proposed pre-processing method is applied to two-dimensional multi slice tomography with a reflection-type T-ray time-domain spectroscopy system with a pulsed mode laser source [6]. The sample is a parched anchovy in 10 x 38 mm. The image matrix size is 56 x 170 pixels. Transverse resolution is 250μm x 250μm. The number of sampling points in the temporal direction is 512 with the sampling interval of 70.3fs [6]. Using Eq. (3), the average refractive index of the parched anchovy is 1.6. Thus, the depth resolution is 21μm in free space, and is 13μm inside the anchovy.

Photographs of the sample, dissection image, and positive peak image of the T-ray are shown in Fig. 4. A positive peak image is displayed in log scale for a better visualization.

Fig. 4 Photograph of the patched anchovy (a), dissection image (b) and positive peak image of the T-ray in log scale (c). Horizontal and vertical dotted lines are added in (c) for profile plot.

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Cut views of the sample are shown in Fig. 5 along the dotted lines shown in Fig. 4(c). Two vertical axes in Fig. 5 represent the time of the delay (left) and the depth (right) of the first reflected signal. The first peak time of the T-ray is plotted with a blue solid line, and the corresponding depth of the surface of the sample is shown with a red solid line after the velocity compensation. Note that the depth profile is not obtainable by simply scaling of the time profile as demonstrated in Fig. 1.

Fig. 5 Cut views of the sample in horizontal (a) and vertical (b) directions.

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Four tomographic images of the parched anchovy are shown in Figs. 6 and 7 before and after the velocity compensation. The maximum intensity is projected (MIP) among 5 adjacent depth images with a depth resolution of 13μm, which makes the depth resolution or equivalent slice thickness 65μm. By comparing Figs. 6 and 7, the tomographic images after the velocity compensation (Fig. 7) have better defined boundaries and show the internal structures more clearly. Log scale is taken to make for severe attenuation inside the sample. The log scale reduces the large reflected signal at the surface, which enhances the small signal inside the sample. The velocity compensated images in linear scale are also shown in Fig. 8 for comparison.

Fig. 6 The Maximum intensity projection (MIP) images of the T-ray tomography are shown in log scale before the velocity compensation.

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Fig. 7 The MIP images of the T-ray tomography in log scale after the velocity compensation. The resolution in the transverse plane is 250μm x 250μm, and the depth resolution is 65μm.

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Fig. 8 The MIP images of the T-ray tomography in linear scale.

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Since the depth resolution (13μm) inside the sample is much higher compared to the transverse resolution (250μm) [9], some post-processing along the depth direction is desirable. For instance, T-ray tomography with an isotropic resolution in both the transverse plane and the depth direction (250μm x 250μm x 250μm) is reconstructed from 19 adjacent depth images after the velocity compensation. Reconstructed images by the maximum intensity projection, averaging, and the short-time Fourier transform along the depth direction are shown in Figs. 9-11 in log scale. The images shown in Fig. 9 are obtained by the maximum intensity projection. The technique is a nonlinear process, which may reduce the contribution from small signal. The images shown in Fig. 10 are done by averaging, which improves signal-to-noise ratio. The images shown in Fig. 11 are the average spectral responses of the terahertz wave in the range of 0.374THz~1.123THz obtained by the short-time Fourier transform. By using a narrow spectral width rather than the whole spectra of the terahertz wave, the responses have higher resolution. The technique also improves signal-to-noise ratio similar to the averaging method. Although the three post-processing methods do not make significant differences in the reconstructed images, they have distinct characters in improving the signal-to-noise ratio and resolution in the reconstructed images, which will be useful in the terahertz tomography depending on the sample and measurement conditions.

Fig. 9 Five MIP images of the T-ray tomography of the anchovy are shown with an isotropic resolution of 250 μm in both the transverse plane and the depth direction.

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Fig. 10 Five average images of T-ray tomography of the anchovy with an isotropic resolution of 250 μm.

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Fig. 11 Narrow band spectral responses (0.374THz ~1.123THz) of the T-ray tomographic images of the anchovy are shown with an isotropic resolution of 250 μm by the short-time Fourier transform.

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5. Conclusion

Terahertz tomography is obtained for a biological sample. Due to different velocities of the terahertz wave in free space and inside the sample, compensation is necessary to obtain tomographic images. A simple method to measure the refractive index of the sample is proposed. By the suggestive velocity compensation method, tomographic images are successfully obtained. To make isotropic resolution, several image processing methods such as the maximum intensity projection, averaging, and the short-time Fourier transform along the slice direction (the propagation direction of the terahertz wave) are suggested. Log scale display of the terahertz tomography is also suggested.

Acknowledgments

This research was conducted during the sabbatical year, 2011 supported by Kwangwoon University. This research was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0001291) and by a grant from the Korean Health Technology R&D Project of the Ministry for Health, Welfare & Family Affairs, Republic of Korea (A101954).

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Pre- and post-processing for tomographic reconstruction of terahertz time-domain spectroscopy

Abstract

1. Introduction

2. Measurement of refractive index

3. Compensation method by the refractive index

4. Experiment

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (5)

Optics Express