THz imaging technology is widely used in nondestructive testing (NDT) [1
], security screening [4
], and medical scanning [11
]. One typical THz imaging mode is based on the time-domain spectroscopic (TDS) approach, which is predicated on simultaneous determination of the field amplitude, phase and polarization [15
]. It has been widely used, and its performance has been highly enhanced by researchers over the years [16
]. In order to improve the spatial resolution to that beyond the realm of the diffraction limit, THz near-field technology with detecting probes of subwavelength aperture size should be considered [15
]. However it may not be practical in some situations, such as in human screening scenarios, where, a detector array would be a workable alternative. In particular, charge-coupled devices (CCDs) in the THz range (5–13 THz) [20
], and bolometers [21
] with much higher responsivities in the THz frequency range are available and can be configured to suit specific imaging applications. Reading out online will be allowed in the form of single-shot recording at a pixel size of a few micrometers. Thus it will be preferred for real-time systems. However, the budget and complexity of a large-sized system are invariably determining factors in some applications, e.g., in a human THz imager with a size of 1 m × 0.5 m, such a detector array might not be practical. Another mode pertaining to conventional optics and microwaves [22
] is readily deployable without drastic modifications to realize image reconstruction: an antenna array could be utilized whose elements can be reduced substantially by the employment of the multi-input/multi-output (MIMO) technology [23
], or the sparse array technology [24
]. Moreover, as the frequency band we considered in the experiments is in the millimeter wave region (W-band, 75–110 GHz) of the electromagnetic spectrum, which is in the low THz region of frequency, the method of three-dimensional electromagnetic imaging is available as well.
The problem of electromagnetic imaging is essentially an inverse problem of electromagnetic field scattering [28
], governed by the scalar Helmholtz equation, subject to the proper initial and boundary conditions. However, the nonlinearity of this equation makes it difficult to obtain an analytical solution. Fortunately for most practical problems a numerical solution is sufficient. However, the process of iterations always tends to kill the potential of real-time image reconstruction. Therefore a variety of approximations were proposed in which Born approximation [29
] is the most widely used in image reconstruction technology. Under such approximations the scalar Helmholtz equations become linear, which can be solved in conventional of ways.
In this study, image reconstruction is realized by digital signals only. Thus the objective is to transform complex signals into images, two-dimensional (2D) or three-dimensional (3D). A direct method is to form an equation through samples from each channel according to linearized Helmholtz equations, which is always an ill-posed problem, so regulation of the inverse problem is an important tool for this kind of processing [31
]. For example, the simplest regulation method is Tikhonov regulation [32
], which is still widely used to date for its efficiency. However, it is obvious that Tikhonov regulation sacrifices precision for robustness. This may negatively impact the quality of the resulting images. Another primary type of regulation applies a recursive process, which may be more precise but more computationally intensive. What perhaps is more troublesome is the data volume involved therein. In order to achieve a high quality image, the parameter matrix is always densely meshed into a grid, and an incredible amount of RAM is essential, especially in 3D cases. Interestingly compressive sampling technology [33
] is famous for its low sampling rate, compared with that required by the Nyquist sampling rate, which eases the burden of sampling. This feature is quite suitable for some special occasions, such as those acquired with synthetic aperture radar by high speed aircrafts [35
]. Another advantage of compressive sensing in image reconstruction is that super resolution can be realized through it, which means resolution beyond the Rayleigh criterion, and resultant better focused images [36
Although methods of regulation and compressive sensing feature in some critical points, matched filtering still plays a major role in imaging applications. The primary reasons may be its robustness and relatively low computational burden. Firstly, matched filtering was deduced from the principle of maximizing signal to noise ratio, which seems to perform well against noise. Secondly, as long as the scenario conformed, no other parameters are required. Therefore it is broadly applicable. Thirdly, matched filtering is free of any searching iterations, consequently it outperforms all the other algorithms in efficiency, especially when Fast Fourier Transformation (FFT) is available.
Image reconstruction using matched filtering algorithms can generally be divided into two classes: time domain algorithms and frequency domain ones. In fact, the distinctions between them may be ascertained from how and when to perform spectral analysis, and how to make some approximations to reduce the cost of computation without impacting on the image quality too much. For example, the Range Doppler algorithm (RDA) [28
] can be relegated to the frequency domain, since compensations are made in frequency domain, whereas Back Projection algorithm (BPA) [38
] refers to a time-domain algorithm. As frequency domain algorithms take advantage of FFT, pulse compression processing can be implemented efficiently. Thus they are preferred in the cases of real-time systems. Range Migration algorithm (RMA) [28
] relies on fewer approximations than RDA, which makes it efficient and relatively more precise than RDA. However interpolation errors cannot be avoided, and cause defocusing in range direction. BPA applies no approximations, but consumes unacceptable amount of computational resources. In practice efficiency will always be in contradiction with precision, and some kind of compromise has to be struck. However, the phase-shift migration algorithm (PMA) [41
] can be an appropriate alternative which retains the high precision of BPA, while being free of interpolation errors. It has been applied in image reconstruction of targets under the illumination of THz Gaussian beams [43
], as such, it will be reasonable for quasi-optical systems. However when it comes to the scenario of the multistatic case, the angular divergence of the beam, which is 120 degrees in our work, will be much larger than 30 degrees considered previously [45
]. Thus the precision of Gaussian beam approximation will no longer obtain. In the present study, PMA is applied to THz imaging reconstruction under the monostatic as well as the spherical wave assumption. PMA is then extended to the multistatic case for the first time, and made available in the simulation. Finally, the performance of PMA is studied in comparison with BPA and RMA in detail in the simulations and experiments.
4. Experimental Results
In this section, we present results of the monostatic experiment. Figure 11
demonstrates the setup of the experiment and the target to be imaged. A model N5247A network analyzer (Agilent, Santa Clara, CA, USA) is used to transmit, receive and process the signals. The signal used is a step-frequency continuous wave, and will be received after down-conversion. The signals received are analyzed in frequency domain, where it will be convenient to process. Mechanical raster scanning is realized by a 2D scanning platform. The parameters applied in this experiment are listed in Table 3
The slice figure results in Figure 12
prove the poor positioning performance of RMA again. The slice position agrees well between Figure 12
a,c, while an error of 0.08 m happens in RMA. In Figure 13
, some notable cases happen in Figure 13
b. Firstly, some red artifacts appear around the image. It results from the errors introduced by interpolation. In fact, the poor positioning performance must be relevant to defocus in the range direction from this point of view.
Secondly, the color of the fan in Figure 13
b tends to be blue while the other two images appear red. This is also due to the appearance of artifacts. The plot function of the 3D reflectivity coefficients will be scaled to the maximum datum after noises-removing operation. The color in an image does not refer to an absolute range but a relative quantity. Therefore the three images cannot be contrasted directly in color. It simply means that the artifacts appear nearer to the scanning aperture than the target.
In this section, some phenomena in the simulations and experiments will be discussed further. It is known that RMA takes the advantage of FFT in all three dimensions and achieves the highest efficiency. However as the samples distribute unequally in the range direction, interpolation called “Stolt interpolation [40
]” should be applied to rearrange the spectrum. Thus a sinc interpolation is introduced and the error caused by the sinc interpolation is smaller than the resolution in the range direction in principle. However, as the interpolation samples are limited, slight defocus in the range direction will be introduced after all. And one of the ramifications may be its poor positioning performance. From another point of view, the reason may be explained in the following way: the sinc interpolation is conducted by moving data in wavenumber domain along the range direction with a sinc weighting function, thus the position tends to shift. Another problem is the artifacts appearing in Figure 13
b. However, it is noticed that in the results of the simulations (Figure 9
b) the artifacts do not appear. No noise is included in the simulations, while it will certainly appear in the experimental data. As interpolation causes defocus, the signals will be added incoherently in some voxels. Thus peaks may appear in some unexpected positions that are considered as artifacts.
The algorithm of BP is very classical. Its principle is simple and easy to understand. Recently some modified BPAs have been proposed, which aim to accelerate. However, they are not yet suitable for real-time systems. Whereas BPA processes the echoes directly without any approximation, it is supposed to be closest to ideality in resolution. Thus in this article the result of BPA is introduced as a criterion of image quality. Furthermore, it is preferable to mesh the grid more densely which will introduce more computation.
In fact, Phase-shift Migration Algorithm relates to RMA and BPA internally. It applies Equation (7) directly without carrying out interpolation. Thus it retains the performance of BPA, but operates a little slower than RMA. However, PMA can be implemented in parallel, and has the potential of real-time processing.