Cylindrical Three-Dimensional Millimeter-Wave Imaging via Compressive Sensing

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Introduction
Millimeter-wave (MMW) imaging techniques have been applied for the detection of concealed weapons and contraband carried by personnel at airports and other secure locations [1][2][3][4][5].Millimeter waves can penetrate common clothing barriers to form an image of a person as well as any concealed items with different reflectivity or emissivity.Also, millimeter waves are nonionizing and, therefore, pose no known health hazard at moderate power levels, not like X-ray systems.
However, a large number of antennas and switches (maybe from hundreds to ten thousands) are needed to construct one-or two-dimensional (1D or 2D) array to obtain threedimensional (3D) images, which increases the cost of the imaging system and then limits its wide application.In order to reduce the complexity and cost of the system, we propose a compressive sensing-(CS-) based method for 3D MMW imaging.The area of CS was initiated in 2006 by Candès et al. [6] and by Donoho [7].
CS has been widely explored in the domain of radar imaging [8][9][10][11], in which the image was usually converted into a vector.And the 1D optimization algorithm was used to recover the image.However, if we convert the 3D image into a vector, a huge sensing matrix will be introduced.Then the computational complexity will be largely increased.In this paper, we introduce an operator as the sensing matrix to represent the traditional imaging process based on wavenumber domain algorithm (WDA) [12].Thus, large-scale matrix computations are avoided.The image resolution of the WDA is determined by signal bandwidth, signal frequency, and the length of synthetic aperture.And the resolution of the CSbased imaging algorithm is determined mainly by the signalto-noise ratio and the matching degree of signal model to the requirements of CS, such as the restricted isometry property (RIP).The simulation and experiment results show that the CS-based method offers a much more accurate image than the traditional WDA just using a small subset of the full samples.
The rest of the paper is organized as follows.In Section 2, the CS imaging method based on the traditional WDA is introduced.The results demonstrating the efficiency and accuracy of the method are shown in Section 3. Section 4 summarizes the conclusions.

Raw data
Elevation FFT Figure 2: Scheme of the wavenumber domain algorithm.

CS-Based Method for 3D MMW Imaging
For the 3D imaging geometry is illustrated in Figure 1.An object formed by a distribution of independent scatterers of reflectivity (, , ) is located at the center of the imaging scene.The backscattered field (, ,   ) after demodulation is given by where  = 2/ is the spatial frequency,  represents the light speed,  ,  (, , ) = √ ( 0 cos  − ) 2 + ( 0 sin  − ) 2 + (  − ) 2 is the distance from each antenna element at ( 0 cos ,   ,  0 sin ) to each scattering center at (, , ), and  0 is the radius of the cylindrical aperture, as shown in Figure 1.
The scheme of the traditional WDA [12] for near-field cylindrical 3D imaging is illustrated in Figure 2. We represent the imaging process as an operator.Then, the inverse process can be represented as the inversion of the operator.Thus the imaging process is given by where the operator  denotes the imaging process as shown in Figure 2; that is, Then (, ,   ) can be expressed as the inversion of (2) as follows: where  −1 denotes the inverse process of (3).Due to the sparsity of the image scene (, , ), only a small subset of scattered data (, ,   ) is enough to be used to reconstruct the image exactly by solving the optimization problem [13]: where ‖ ⋅ ‖ 0 is simply the number of nonzero components of , and  represents a sparse transform, making  a sparse vector.However, solving (5) where ‖ ⋅ ‖ 1 and ‖ ⋅ ‖ 2 denote the ℓ 1 norm and ℓ 2 norm, respectively.In this paper, we use the total-variation (TV) as a sparse transform.The parameter  is used to balance the twin objectives of minimizing both error and sparsity.The operators  and  −1 correspond to the Fourier matrices if we express the signal model directly.Thus,  and  −1 satisfy the RIP required by CS, and both (5) and ( 6) have the same unique solution [6].There are many state-ofthe-art optimization algorithms designed to solve the convex optimization problem of (6) [14][15][16][17].Here we use a nonlinear conjugate gradient (CG) descent algorithm described in [18].The computational complexities of  −1 [(, , )] and [(, ,   )] both are an order of (NMP)⋅log 2 (NMP)+NMP⋅ , ,  and  are the sizes of the scattered data,  being the interpolator length.And the computational complexity of the key part of the CG algorithm during an iteration is an order of (NMP)log 2 (NMP) [18].

Results
The aforementioned algorithm is tested with numerical simulations and experimental data.The simulation parameters are shown in Table 1. Figure 3 shows the target model with 8 point-like scatterers with equal radar cross section.The 3D imaging result (isosurface of −13 dB relative to the maximum value) of WDA using the full sampling data and the result of the proposed CS-based method using only 25% of the full data are shown in Figures 4(a) and 4(b), respectively.Clearly, even with a small subset of the full data, the CSbased method can provide a more accurate imaging result with higher resolution than WDA.
The experiments on real measured data are performed.A stepped-frequency W-band radar with two circular polarized horn antennas is used for the measurement, which is shown in Figure 5. Three screws and a card are measured, respectively, as shown in Figure 6.The diameter of the cross section of the screw is about 1 cm.
The corresponding MMW imaging results of WDA using the full measured data are shown in Figures 7(a) and 8(a), respectively.And the results of the CS-based method using only 25% of the full data are demonstrated in Figures 7(b) and 8(b), respectively.Due to the fact that the card is a distributed target, the imaging result in Figure 8(b) is not so good as the point-like target in Figure 7(b).However, the main characteristics remained; for example, the chip inside the card can be easily seen in Figure 8

Conclusion
A CS-based method has been proposed for cylindrical near-field 3D MMW imaging.The imaging process and its inversion are represented by an operator and its inversion, respectively, to avoid large-scale matrix computation.Thus the CS approach can be applied to the cylindrical 3D imaging by using the operator as the sensing matrix.Simulation and experiment results show that the CS-based method can offer highly accurate and reliable results, even using only a small subset of the full samples compared with the traditional wavenumber domain algorithm.

Table 1 :
Simulation parameters for full set of samples.Parameters Values Center frequency (GHz) 91 Bandwidth (GHz) 10 Sampling interval along azimuth-range direction (m) 0.004 Number of samples along azimuth-range direction 64 Sampling interval along elevation-range direction (m) 0.004 Number of samples along elevation-range direction 64

Figure 4 :
Figure 4: 3D MMW images of the target model: (a) the result of WDA using the full data and (b) the result of the CS-based method using only 25% of the full data.

Figure 6 :Figure 7 :
Figure 6: Photo of three screws and a card.

Figure 8 :
Figure 8: MMW images of the card: (a) the result of WDA using the full data and (b) the result of the CS-based method using only 25% of the full data.