> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Compressive Sensing Imaging of 3-D Object by a Holographic Algorithm Shiyong Li, Member, IEEE, Guoqiang Zhao, Houjun Sun, and Moeness Amin, Fellow, IEEE hazard at moderate power levels. MMW imaging systems can Abstract—Existing three-dimensional (3-D) compressive be classified into two types: passive imaging systems and active sensing-based millimeter-wave (MMW) imaging methods require imaging systems. Both types have their own offerings and a large-scale storage of the sensing matrix and immense challenges. This paper focuses on active MMW imaging computations owing to the high dimension matrix-vector model techniques, which typically implement large scale antenna employed in the optimization. To overcome this shortcoming, we arrays to illuminate the whole human body, leading to high propose an efficient compressive sensing (CS) method based on a holographic algorithm for near-field 3-D MMW imaging. An system cost. Compressive sensing (CS) and sparse interpolation-free holographic imaging algorithm is developed reconstruction techniques, on the other hand, have been used to and used as a sensing operator, in lieu of the nominal sensing reduce the number of array elements, thereby the system cost, matrix typically used in the CS iterative optimization procedure. without degradation of image quality [12]-[14]. CS has been In so doing, the problem induced by the large-scale sensing matrix extensively studied in radar imaging [15]-[19], microwave is avoided. With no interpolations required, both the imaging [20]-[23], array synthesis and diagnosis [24], [25], and computational speed and the image quality can be improved. direction-of-arrival estimation [26]-[28]. Simulation and experimental results are provided to demonstrate CS methods are typically based on the matrix-vector model, the performance of the proposed method in comparison with which employs a large-scale sensing matrix in 3-D MMW those of the K based CS and the traditional Fourier-based imaging for personnel inspection and security applications. imaging techniques. Such model was incorporated in [29] for a 3-D compressive phased array imaging. CS was applied to single-frequency 2-D Index Terms—Near-field, millimeter-wave (MMW) imaging, MMW holographic imaging in [30] and [31], where a compressive sensing (CS), holographic algorithm. Fourier-based imaging operator represented the sensing matrix. This replacement, in essence, avoided storing and processing of a large-scale sensing matrix which, in turn, simplified imaging I. INTRODUCTION and permitted its realization on an ordinary personal computer. ILLIMETER-WAVE (MMW) has attractive In [32], the 3-D K algorithm, referred to as range migration characteristics compared with waves in the microwave M algorithm, was used in combination with the CS principle for band or lower frequency bands. These include higher carrier image reconstruction. Nevertheless, this algorithm includes the frequency and wider usable frequency band that enables higher forward and inverse Stolt interpolations that entail require high target cross-range and down-range resolutions. Another computations and can lead to reduced image fidelity. An important feature of MMW is the design of small and light interpolation-free SAR imaging algorithm, named range systems and equipment. Accordingly, it is beneficial to adopt stacking, was proposed in [33] and extended to 3-D imaging in MMW for short-range broadband communications [1], [2], [34]. The range stacking reconstruction method forms the target high-resolution sensing [3], [4], and radio astronomy [5]. image at different range points by matched filtering the SAR MMW imaging techniques have been widely developed and signal in the spatial frequency domain. The result is integrated applied to non-destructive testing [6], biomedical diagnosis [7], over frequencies to yield the marginal Fourier transform of the and personnel security inspection [8]-[11]. target function. MMWs are capable of penetrating regular clothing to form In [35] and [36], we considered a single-frequency based an image of a person and concealed objects. Most importantly, auto-focus holographic imaging algorithm. The auto-focus was MMWs are non-ionizing and, therefore, pose no known health obtained by calculating the amplitude integral values of the images reconstructed at different focusing range bins. These Manuscript received February 16, 2018. This work was supported by the values draw to a minimum when the image is well focused. In National Natural Science Foundation of China under Grant 61771049. this paper, we extend the above algorithm to the wideband S. Li, G. Zhao and H. Sun are with the Beijing Key Laboratory of Millimeter Wave and Terahertz Technology, Beijing Institute of Technology, Beijing signal, and change the integral variables of the imaging model 100081, China. S. Li is also with the Center for Advanced Communications, to eliminate the Stolt interpolation. This changing is similar to Villanova University, Villanova, PA 19085 USA. (e-mail: the work in [33], [34]. However, unlike these references, we [email protected]). M. Amin is with the Center for Advanced Communications, Villanova first apply the inverse Fourier transform over the azimuth- and University, Villanova, PA 19085 USA (e-mail: [email protected]). elevation-frequencies with respect to the matched filtered data. > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 2 The results are then integrated over the fast-time frequencies, Antennas y which can be considered as a coherent summation of the x', y ', Z single-frequency holographic imaging results. These steps Target amount to a linear relationship between the scene and the measurements, defined by a sensing operator, and as such, the x CS problem can be readily formulated. Due to the fact that interpolations are avoided in the iterations underlying the optimization algorithm, the computational speed and o optimization solution can both be improved in comparison with z the 3-D -based CS method [32]. In order to deal with compressed data, which corresponds to selecting few antennas, we utilize a uniform-random spatial Fig. 1. Geometrical configuration of the imaging system. undersampling scheme, in lieu of the totally random undersampling scheme. The benefits of the former over the Dechirp-on-receive is used to demodulate the received signal, latter sampling strategy can be made evident by analyzing the and yields the intermediate frequency signal, mutual coherence measure of the sensing operator. In imaging, this measure, in essence, represents the maximum sidelobe sIF x', y ', f x , y , z exp j 2π f f 0 d , (3) value of the point spread function (PSF) in Fourier-based where f Kt is considered as the fast-time frequency. We imaging. Accordingly, it can be used to quantify the effect of assume the residual video phase (RVP) has been removed. The different undersampling schemes on performance. Also, we time-delay is given by, compare the PSFs of the proposed sensing operator and that of 2 2 2 the -based sensing operator, and show the superior 2x x ' y y ' z Z . performance of the former. d c The remainder of this paper is organized as follows. Section Due to the target located in the near-field, which is II presents the formulation of the 3-D interpolation-free illuminated by spherical waves, the square root of the above holographic imaging algorithm, which is used to construct the time-delay expression cannot be simplified, as in the case of operator considered as the sensing matrix of CS. In Section III, far-field. For a volume target, extending in all three we provide the CS imaging method and present the dimensions, uniform-random under-sampling scheme. The relationship 2 2 2 s x', y ', f x , y , z ej2 k x x ' y y ' z Z dxdydz between the mutual coherence and the point spread function is IF , analyzed. Numerical simulations and experimental results are (4) shown in Section IV. Finally, concluding remarks are presented where k2π f f / c is considered as the wavenumber. In in Section V. 0 the above equation,K we have ignored the propagation loss of II. 3-D INTERPOLATION-FREE HOLOGRAPHIC spherical waves which is characterized by 2 2 2 IMAGING ALGORITHM 1/x ' x y ' y Z x for the round-trip A. Formulation of the interpolation-free holographic propagation. The exponential term in (4) represents a spherical imaging algorithm wave emanating from . This term can be decomposed The configuration of the imaging system is shown in Fig. 1. into a superposition of plane wave components [8], as For a frequency modulated continuous wave (FMCW) 2 2 2 j2 k x x ' y y ' z Z jk x x' jk y y ' jk z Z transceiver, the transmitted signal can be expressed as, e exz' ey' e dk dk xy'' 1 2 (5) sT0 t exp j 2π f t Kt , (1) 2 where kx ' and ky ' are the Fourier transform variables where f0 is the center frequency, t is the fast-time variable corresponding to x ' and y ' , respectively. The spatial within one pulse repetition interval, and K is the frequency frequencies and range from 2k to 2k . Substituting slope of the transmitted signal. The backscattered signal from a (5) into (4), and using the Fourier transform, we obtain [8], point target is given by, jkzz Zjkx x jky y jk z xyz,,,, SIF kkkex y e e e dkdkdk x y z , (6) sRT x', y ', t x , y , z s t d , (2) where x,, y z represents the backscattering coefficient of 222 where kz4 k k x k y , SIF kx, k y , k FFT x , y s IF x , y , f the point target at location x,, y z , and is the round-trip d and FFTxy, indicates the 2-D fast Fourier transform (FFT) time-delay defined by the propagation speed of the over xy, .