1. Introduction

As a novel far-infrared inspection technique, terahertz (THz) imaging has been successfully applied in noninvasive detection [1], substance identification [2], biological imaging [3], and so on. However, the requirement of slow mechanical scanning greatly hampers the advancement of the imaging technique. In 2008, W. L. Chan et. al firstly introduced a compressed sensing (CS) method into THz imaging to achieve a THz single-pixel imaging technique for enhancing the measurement efficiency. It only requires fewer measurements than the total number of image pixels and can recover a high-quality THz image [4]. Since then, single-pixel THz imaging based on CS has attracted growing attentions. Printed-circuit broads [4], metamaterials [5], and photoinduced dynamic patterns [6] have been utilized as measurement masks. Qualities of recovered images have been also compared with various measurement matrices, such as Bernoulli, Hadamard, and Gaussian masks [7]. Furthermore, it has been demonstrated that the amplitude, phase, and near-field information of a THz image can be accurately restored by CS [4,6]. However, the complex field information of a THz image recovered by CS has been not adequately exploited for further improving the image quality in previous reports.

In this work, an image reconstruction scheme of combining the CS and inverse Fresnel diffraction (IFD) algorithms is proposed for a single-pixel THz imaging system based on photoinduced dynamic masks. By taking advantage of the image reconstruction scheme, the THz temporal images are exactly recovered and the diffraction effect in the THz field is effectively eliminated to acquire a clear THz spectral image. The validity of the scheme is appropriately proved on the simulation and experiment.

2. Theory

Figure 1(a) presents the concept of the proposed investigation scheme. The incident THz beam illuminates a sample and the transmitted THz field carries the pattern of the sample. Passing through a certain propagation distance d, the image quality of the THz field is deteriorated due to the influence of the diffraction. A high-resistance silicon wafer is inserted on the path of the THz beam as the sampling medium. On the other hand, a near infrared pump beam is firstly spatially modulated by a spatial light modulator (SLM) to load the pattern of the measurement mask. The image of the pump beam is projected by a lens on the surface of the silicon wafer. On the exposed region, the photoinduced localized carriers are excited to enhance the absorption and reflection of the silicon wafer to the THz field [8]. Therefore, the photoexcited silicon wafer can be viewed as a THz SLM to realize multiplex sampling to the THz image [6]. Then, the combination of a THz lens and a single-pixel THz detector is utilized to acquire the integration of all sampling THz signals. Herein, the Hadamard encoding masks are chosen as the measurement matrix Φ in CS for effectively suppressing the system noise [6]. Φ is a M × N² matrix and its each row is applied to form a measurement mask which is loaded onto the SLM. The THz temporal image X can be considered as a N × N matrix at each time delay. Consequently, the original measurement signal Y with M dimensions is the multiplication between X and Φ, namely Y = ΦX. The procedure of the image reconstruction scheme is given in Fig. 1(b). By utilizing a log-barrier algorithm, the optimization problem $\min {\Vert X\Vert }_{l_1}$is solved and the THz temporal image is recovered from Y at each time delay, where ${\Vert X\Vert }_{l_1}$is the l₁-norm of the THz diffraction field represented in the Hadamard basis. The function l1qc_logbarrier of the l₁-magic software package is applied to solve the inverse imaging problem [9]. Then, the Fourier transformation is implemented on a series of recovered THz temporal images to obtain the THz spectral images. Finally, an IFD integral is operated on a THz spectral image to remove the influence of the diffraction and restore the sharp image of the sample. The reconstructed THz image can be expressed as [10]

 

Fig. 1 (a) Schematic illustration of single-pixel THz imaging based on photoinduced dynamic masks. (b) Procedure of the proposed image reconstruction scheme. (c)-(h) give the simulation results, including (c) sample, (d) original THz temporal image, (e) THz diffraction image at temporal peak position, (f) THz temporal image recovered by CS, (g) THz spectral image at 0.3 THz, (h) THz image reconstructed by the IFD algorithm.

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Firstly, the feasibility of the image reconstruction scheme is demonstrated on the simulation. A hollow letter “T” with a size of 10 mm × 10 mm and a slit width of 2 mm is picked up as the sample, as shown in Fig. 1(c). Then, the sample is impinged by a THz Gaussian beam which possesses a central wavelength of 1 mm, a pulse duration of 1 ps, and a full width at half maximum (FWHM) of 20 mm. On the output facet of the sample, the transmitted THz image at the temporal peak position is presented in Fig. 1(d). The pixel number of the image is 32 × 32. The time window and time resolution are 5 ps and 0.1 ps for the THz temporal signal. After passing through a propagation distance of 10 mm, the THz diffraction fields with different spectral components are obtained by using the Fourier transformation and the Fresnel diffraction integral [10]. Further, the THz diffraction field in the time domain is acquired by using the inverse Fourier transformation and the THz temporal image at the peak position is given in Fig. 1(e). It can be obviously seen that the THz image becomes ambiguous due to the diffraction. Now, the image reconstruction scheme is operated to restore a high-quality image. It should be noted that the full-sampled measurement is implemented in the simulation for ensuring the reconstruction effect, namely M = N². Therefore, 1024 Hadamard masks with 32 × 32 pixels are adopted to realize multiplex sampling. By utilizing the CS technique, the THz temporal image is recovered at each time delay and the recovered image at the peak position is exhibited in Fig. 1(f). Clearly, Fig. 1(f) is in good agreement with Fig. 1(e), which manifests that the main characteristics information of the THz temporal image is acquired by CS. By operating the Fourier transformation to a series of recovered THz temporal images, the THz spectral images are obtained and the amplitude pattern at 0.3 THz are presented in Fig. 1(g). Then, by using Eq. (1), the diffraction phenomenon of the THz field is effectively removed by means of the IFD algorithm and the sharp image of the letter “T” is obtained, as shown in Fig. 1(h). It should be explained that some slight noises are originated from the limited aperture of the sampling plane in the reconstructed image. The practicability of the image reconstruction method is testified on the theory.

As well known, the main information of an image can be accurately recovered by CS with an under-sampled measurement [4], namely M<N². Here, the reconstruction effects of the letter “T” with d = 10 mm are checked when random 100%, 75%, 50%, 20% of the total measurements are selected out in the simulation. Figure 2(a) exhibits the recovered THz temporal images with 100%, 75%, 50%, 20% of all measurements at the peak position. It can be seen that the loss of elaborate structures becomes gradually apparent in a recovered THz temporal image with reducing the measurement number. Figures 2(b) and 2(c) give the THz spectral images at 0.3 THz and the reconstructed images with different measurement numbers. The simulation results show that the sharpness of a reconstructed image almost keeps unchanging when a measurement subset varies from 100% to 50%. When the measurement subset is adjusted to 20%, the reconstructed image manifests more artifacts due to excessive under-sampling. Even so, the structure of the letter “T” can be still basically discerned. By calculating ∑_i[X_r(i)-X₀(i)]²/∑_iX₀(i)², the normalized mean squared error (MSE) of the reconstructed image is obtained and shown in Fig. 2(d), where X_r is the reconstructed image and X₀ is the original image acquired from Fig. 1(c). The curve of the MSE presents a typical variance tendency as a previous report [4], which rapidly drops from 5% to 20% measurements and then smoothly decays afterwards. The phenomenon describes that 20% measurements are adequate to acquire a discernible reconstructed image based on the sparsity of the sample in the Hadamard basis. The simulation results demonstrate that the image reconstruction scheme can works well in the case of under-sampling.

 

Fig. 2 Reconstruction effect with under-sampling in the simulation. (a) give the recovered THz temporal images of the letter “T” with 100%, 75%, 50%, 20% of all measurements at the peak position. (b) and (c) present the corresponding THz spectral images at 0.3 THz and reconstructed images with different measurement numbers, respectively. (d) shows the normalized mean squared error (MSE) between the reconstructed and original images with varying the measurement number.

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Furthermore, according to the simulation results, the effect of the CS technique on shortening the experimental time can be quantitatively evaluated. If the conventional mechanical scanning imaging method is applied to build a THz image, the time consumption should be the same as that of the CS technique with 100% measurements. Generally, it needs several hours. Under ensuring the same spatial resolution, the CS technique only needs half of the total data collection time to recover a THz image. When the requirement to the spatial resolution is not very rigorous, the CS technique only needs 20% of the total measurement time to acquire a THz image. It is no doubt that the advantage of CS will powerfully strengthen the practicability of THz imaging.

In the simulation, the influence of the propagation distance d to the reconstruction effect is also checked. A resolution chart is selected as the testing sample, as shown in Fig. 3(a). It possesses the grid lines of four periods, including p = 1.34 mm, 2.68 mm, 4.02 mm and 5.36 mm. The duty cycles of the grid lines are 0.5. To more completely cover all of the grid lines, the FWHM of the incident THz beam is adjusted as 30 mm. Figure 3(b) exhibits the transmitted THz temporal image at the peak position on the output facet of the resolution chart. The diffraction distance d is set as 10 mm, 15 mm, 26 mm, 32 mm, respectively. The reconstructed images with different d at 0.3 THz are obtained by applying the image reconstruction scheme, as shown in Figs. 3(c)-3(f). Obviously, the spatial resolution of the reconstructed image gradually deteriorates with increasing the diffraction distance. With d = 10 mm, all of the grid lines can be easily discerned. When d is equal to 15 mm, 26 mm, 32 mm in sequence, the images of the grid lines with p = 1.34 mm, 2.68 mm, 4.02 mm successively become ambiguous. To more intuitively analyze the variance of the spatial resolution, the normalized THz amplitude profile curves on the grid lines with p = 4.02 mm are extracted from Figs. 3(b)-3(f), as shown in Fig. 3(g). The position of these curves is marked by the white dashed line in Figs. 3(b). It can be observed that the images of the grid lines gradually mix each other with increasing d. Until d = 32 mm, the grid lines with p = 4.02 mm cannot be clearly identified. The phenomenon is easy to understand. According to the angular spectrum theory [10], a spatial spectrum component with a higher frequency has a larger deflection angle with respect to the optical axis, so higher frequency spatial components may be lost with increasing d due to the limited aperture of the sampling plane. Therefore, the loss of the image detail information is more seriously with a larger d.

 

Fig. 3 Influence of the diffraction distance d to the reconstruction effect in the simulation. (a) gives the photo of a resolution chart with four periods, including p = 1.34 mm, 2.68 mm, 4.02 mm and 5.36 mm. (b) presents the original THz temporal image on the output facet of the resolution chart. (c)-(f) show the reconstructed images with d = 10 mm, 15 mm, 26 mm 32 mm at 0.3 THz, respectively. (g) gives the normalized amplitude curves extracted from (b)-(f). The position of these curves is marked by the white dashed line in (b).

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3. Experiment

Now, the image reconstruction scheme is carried out on the experiment. A single-pixel THz imaging system is built and its schematic diagram is given in Fig. 4(a). A Spectra-Physics femto-second laser amplifier is chosen as the light source, which possesses a central wavelength of 800 nm, a pulse duration of 50 fs, a repetition ratio of 1 kHz, and an average power of 750 mW. The incident laser is split into three beams, including the pump, probe and control beams. In the path of the pump beam, a <110> ZnTe crystal with a thickness of 1 mm is irradiated by the pump beam expanded by a concave lens (L1) with a focal length of 30 mm. A linearly polarized THz radiation is excited and an off-axis metallic parabolic mirror (PM1) is applied to collimate the THz beam. The FWHM of the THz beam approximately reaches 26 mm. Then, a sample and a silicon wafer (Si1) with a thickness of 0.5 mm are inserted into the path of the collimated THz beam. A glass coated by indium tin oxide (ITO) and another parabolic mirror (PM2) are used to reflect and focus the transmitted THz beam for the single-pixel detection. In the path of the probe beam, the laser is guided to focus on the sensor crystal (another <110> ZnTe with a thickness of 1 mm) through a motorized linear stage (stage1) and a convex lens (L2). Here, another silicon wafer (Si2) are used to achieve the superposition between the THz and probe beams. In the sensor crystal, the birefringence of the crystal is varied by the THz field and further the polarization of the probe beam is modulated to carry the THz signal in the time domain. The modulated probe beam is measured by a combination of a convex lens (L3), a quarter wave plate (QWP), a Wollaston prism (PW), and a balanced detector. The THz temporal signal is extracted by using the conventional electric-optic sampling technique [11]. In the path of the control beam, another motorized linear stage (stage2) is used to adjust the optical path difference with respect to the THz beam. A lens combination (a concave lens L4 and a convex lens L5) are adopted to expand the control beam and the FWHM of the control beam is about 21 mm. A visible-light SLM (Holoeye, PLUTO-NIR-011) is applied to load a series of measurement masks onto the control beam and the image of the control beam is projected on the Si1 by a convex lens (L6). In addition, before and after the SLM, a pair of polarizers (P1 and P2) with orthogonal directions are inserted into the path of the control beam to filter the background intensity. A mechanical chopper is mounted on the path of the control beam and the THz temporal signal modulated by photoinduced carriers is exclusively measured by using a lock-in technique. Figure 4(b) shows the variance of the THz temporal peak signal with adjusting the optical path difference between the control and THz beams, when the control beam with a homogeneous spatial profile is projected on the Si1. The positive time region indicates that the control beam arrives at the Si1 prior to the THz beam. In the measurement, the time delay between the control and THz beams is adjusted as 4 ps for maximizing the modulation depth and avoiding the diffusion of photoinduced carriers as far as possible. Figure 4(c) gives a modulated THz temporal signal with the time delay of 4 ps. Besides, it should be explained that the experimental measurement matrices only include values of 1 and 0 corresponding to the exposed and unexposed regions of the Si1, so two sets of measurement matrices with positive and inverse masks are separately used in the experiment and their subtraction is implemented to realize the desired Hadamard matrices with [-1, 1] [6].

 

Fig. 4 (a) Optical configuration of a single-pixel THz imaging system in which a photoexcited high-resistance silicon wafer is adopted to implement multiplex sampling to the THz wave front. (b) Variance of the THz temporal peak signal with adjusting the time delay between the control and THz beams. (c) THz temporal signal modulated by photoinduced carriers.

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Three metallic hollow letters “T”, “H”, and “Z” are chosen as the samples and their photos are given in Fig. 5(a). Their sizes are 10 mm × 10 mm and the slit widths are 2 mm. The distance between the sample and the Si1 is set as d = 10 mm. On the surface of the Si1, the total area of the exposed region is 21 mm × 21 mm and 1024 Hadamard masks with 32 × 32 pixels are adopted to achieve multiplex sampling. With the full-sampled measurement, the THz diffraction field in the time domain is measured and recovered by CS. Figure 5(b) exhibits the recovered THz temporal images of three letters at the peak position, which vividly show that their image quality is seriously deteriorated due to the diffraction. By operating the Fourier transformation, the THz spectral images of three letters at 0.3 THz are extracted and presented in Fig. 5(c), which more evidently manifest the influence of the diffraction to the sharpness of these images. By utilizing Eq. (1), the reconstructed images of three letters at 0.3 THz are obtained, as shown in Fig. 5(d). It is easy to find that the diffraction phenomenon of the THz field is pronouncedly weakened and the image quality is significantly improved. The experimental result adequately demonstrates the feasibility of the proposed scheme on the experiment, which implies that both amplitude and phase information of the THz field can be effectively restored by CS and further digital image processing methods can be applied on the recovered THz image.

 

Fig. 5 (a) Photos of the samples, including three metallic hollow letters “T”, “H”, and “Z”. (b) THz temporal images recovered by CS at the peak position. (c) THz spectral images at 0.3 THz by operating the Fourier transformation. (d) Reconstructed THz images by the IFD algorithm.

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Here, the reconstruction effect with under-sampling is also checked on the experiment. Figure 6(a) gives the recovered THz temporal images of the letter “T” with 100%, 75%, 50%, 29% measurements at the peak position, which manifest the same phenomenon as Fig. 2(a). With reducing the sampling rate, the recovered image exhibits more and more information loss. Figures 6(b) and 6(c) show the corresponding THz spectral images at 0.3 THz and the reconstructed images, respectively. From the experimental results, it can be seen that the image quality basically remains unchanged at sampling rates from 100% to 50%. Until 29% measurements, some notable artifacts appear in the reconstructed image. However, the outline of the letter “T” can be still roughly identified. Here, the MSE between the reconstructed and reference images is also calculated with varying the sampling rate, as shown in Fig. 6(d). It should be noted that the reference image is a 32 × 32 grayscale picture acquired from the photo of the sample. The curve of the MSE shows the similar tendency as Fig. 2(d), which manifests that 29% measurements are enough to obtain a discernible reconstructed image. The slight divergence between the experimental and simulation results is originated from the laser intensity fluctuation and unexpected measurement errors in the experiment. Even so, the experimental phenomena confirm that the image reconstruction scheme is available with an under-sampled measurement.

 

Fig. 6 Reconstruction effect with an under-sampled measurement in the experiment. (a) gives the recovered THz images of the letter “T” with 100%, 75%, 50%, 29% measurements at the temporal peak position. (b) and (c) exhibit the corresponding THz spectral images at 0.3 THz and reconstructed images. (d) shows the MSE between the reconstructed and reference images with varying the measurement number. The reference image is obtained from a photo of the sample.

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Besides, it still should be pointed out that both the time consumptions of performing conventional mechanical scanning imaging and CS with 50% measurements are approximately four hours in the current experiment. The reason is that two sets of measurement matrices with positive and inverse masks need to be adopted for forming Hadamard encoding, so the data collection time is doubled in CS. According to a previous report [5], it is expected to solve the problem by applying a THz metamaterial spatial light modulator. Additionally, on the aspect of data processing time, it only needs tens of seconds to fulfill the image reconstruction scheme by using a personal computer (CPU: Intel(R) Core(TM) i7-4790, 3.6 GHz; RAM memory: 8.0 GB), which can be negligible. Therefore, it can be said that the CS technique has more room for development in THz imaging.

Finally, the influence of the propagation distance d to the reconstruction effect is also checked on the experiment. The distance between the sample and the Si1 is adjusted as 10 mm, 15 mm, and 20 mm, respectively. The metallic hollow letter “T” is used as the sample. Figures 7(a) and 7(b) exhibit the THz temporal images recovered by CS and the spectral images at 0.3 THz obtained by the Fourier transformation with d = 10 mm, 15 mm, and 20 mm. From these spectral images, it is easy to observe that the diffraction effect is more and more striking with increasing d. Notably, the shape of “T” cannot be identified from the spectral image with d = 20 mm. By using the IFD algorithm, the sharp images of “T” are obtained with different d, as shown in Fig. 7(c). It can be seen that the qualities of the reconstructed images with d = 10 mm and 15 mm are almost the same, but the reconstructed image with d = 20 mm presents a poorer image sharpness. The phenomenon is consistent with Fig. 3, which describes that increasing d causes the degradation of the spatial resolution in a reconstructed image.

 

Fig. 7 Influence of the diffraction distance d to the reconstruction effect in the experiment. (a) gives THz temporal images recovered by CS with d = 10 mm, 15 mm, and 20 mm for a metallic hollow letter “T”. (b) and (c) present the THz spectral images at 0.3 THz and reconstructed images with d = 10 mm, 15 mm, and 20 mm, respectively.

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4. Discussion

Undoubtedly, the introduction of CS is an important impetus to develop single-pixel THz imaging due to shortening the experimental time and avoiding the mechanical scanning system [4]. Furthermore, the application of photoinduced carriers of semiconductors is crucial for enhancing the practicability of the CS technique [6]. It allows to arbitrarily alter the measurement mask and realize rapid sampling to the THz wave front. Actually, it has been demonstrated that the amplitude and phase information of a THz field can be simultaneously extracted by CS in a previous report [4]. However, to our best knowledge, the recovered THz complex field information has been not sufficiently utilized for further image processing. Currently, almost all of THz imaging systems based on CS are a transmission type. Generally, the measurement masks are mounted close to the sample for weakening the diffraction effect [6,12]. The achievement of this work implies that conventional digital image processing techniques can be directly applied to a THz image recovered by CS, which can effectively prompt the advancement of single-pixel THz imaging. Particularly, a single-pixel THz imaging system with a reflection type can be realized based on this work. By taking advantage of the CS technique and the IFD algorithm, tomographic images with a high-quality can be acquired in a reflective single-pixel THz imaging system.

5. Conclusion

In conclusion, the IFD algorithm is introduced into THz images recovered by CS to enhance the image quality. The feasibility of the image reconstruction scheme is adequately demonstrated on the simulation and experiment. A single-pixel THz imaging system is built based on photoinduced dynamic masks, which utilizes the transient localized carriers on a silicon wafer to implement multiplex sampling to a THz field. THz temporal images are recovered by CS and a sharp THz spectral image is reconstructed by using the Fourier transformation and the IFD algorithm. In addition, the influences of an under-sampled measurement as well as the diffraction distance of a THz field to the reconstruction effect are also analyzed in detail. This work lays the foundation for the practical application of single-pixel THz imaging.