Near-Field Millimeter-Wave Imaging via Circular-Arc MIMO Arrays
11 Communication
Near-Field Millimeter-Wave Imaging via Circular-Arc MIMO Arrays
Shiyong Li, Shuoguang Wang, Guoqiang Zhao, and Houjun Sun
Abstract —Millimeter-wave (MMW) imaging has a wideprospect in application of concealed weapons detection. Wepropose a circular-arc multiple-input multiple-output (MIMO)array scheme with uniformly spaced transmit and receiveantennas along the horizontal-arc direction, while scanningalong the vertical direction. The antenna beams of the circular-arc MIMO array can provide more uniform coverage of theimaging scene than those of the linear or planar MIMO arrays.Further, a near-field three-dimensional (3-D) imaging algorithm,based on the spatial frequency domain processing, is presentedwith analysis of sampling criteria and resolutions. Numericalsimulations, as well as comparisons with the back-projection (BP)algorithm, are provided to show the efficacy of the proposedapproach.
Index Terms —Millimeter-wave (MMW) imaging, near-field,circular-arc multiple-input multiple-output (MIMO) array,spatial frequency domain processing, back-projection (BP)algorithm.
I. I
NTRODUCTION
Millimeter-wave (MMW) imaging can provide highresolutions of the target under test. Thus, it is of interest ina wide applications, such as remote sensing [1], biomedicaldiagnosis [2], and indoor target tracking [3]. In recent years,there is an increasing demand for MMW to detect concealedweapons and contraband carried by personnel [4]–[7], due tothe fact that MMW can penetrate regular clothing to form animage of a person as well as the concealed objects with nohealth hazard at moderate power levels.Usually, the antenna apertures are formed by 1-D monostaticarrays accompanied by mechanical scanning. The monostaticarrays need to satisfy the Nyquist sampling criterion, whichleads to high number of antennas and switches. State-of-the-art MMW imaging systems employ multiple-input multiple-output (MIMO) arrays to reduce the number of elements[6]–[8]. Another merit of using MIMO lies in the factthat it can reduce the reconstructed artifacts induced bymultipath reflections, which are prevalent in the monostaticarray systems [5], [9], [10].Imaging systems combining 1-D linear MIMO arrays andsynthetic aperture radar (SAR) were discussed in [11]–[15]for concealed weapons detection. The two-dimensional (2-D) MIMO arrays with different topologies were studiedin [7], [16]–[20] for near-field imaging. In so doing, themechanical scanning is eliminated, which acheives real-timedata acquisition. However, the cost and complexity of 2-D
The work was supported by the National Natural Science Foundation ofChina under Grant 61771049.The authors are with the Beijing Key Laboratory of Millimeter Wave andTerahertz Technology, Beijing Institute of Technology, Beijing 100081, China.(e-mail: lisy [email protected]).
MIMO array systems are much higher than those constructedby a 1-D array with mechanical scanning perpendicular to thearray dimension.The 1-D ultrawideband (UWB) MIMO array was designedaccording to the effective aperture approach in [11] associatedwith a time-domain imaging algorithm. In [12], the 1-D linearMIMO arrays were optimized for short-range applicationsalso on the basis of effective aperture approach. In [13],imaging algorithms based on the modified Kirchhoff methodwere derived for linear MIMO arrays with receivers evenlydistributed. A spatial frequency domain imaging algorithm wasdeveloped in [14] for a linear MIMO array system associatedwith cylindrical scanning. The extended phase shift migrationalgorithm was studied in [15] for a MIMO-SAR systemworking at Terahertz band.The aforementioned MIMO arrays are all configured withantennas placed on a straight line or a plane. Usually, twoseparate dense transmit subarrays were set at both ends ofthe undersampled receive array for the 1-D MIMO arraydesign. This scheme can hardly provide an even illuminationof a large imaging area along the array direction. A single-frequency MIMO-arc array based azimuth imaging methodwas presented in [21], based on the geometry transformationfrom the arc array to the equivalent linear array [22].In this communication, we propose a circular-arc MIMOarray based three-dimensional (3-D) imaging scheme. Thetransmit and receive antennas are evenly placed alongthe horizontal-arc direction. And the array is mechanicallyscanned along its perpendicular direction to obtain acylindrical aperture, as illustrated in Fig. 1. The antennabeams of the circular-arc MIMO array cover the imagingarea more evenly than those of the linear or planar MIMOarrays. Furthermore, we present a near-field 3-D imagingalgorithm based on the spatial frequency domain processing,to fully utilize the fast Fourier transforms (FFTs), for thecircular-arc MIMO array system. The key to obtain theimaging algorithm is to solve the convolution of the Green’sfunctions in the Fourier domain, which is, however, hard tobe resolved. We solve it through using an approximation ofthe radial distances between antennas and target pixels in thecylindrical coordinates. Then, the spatial frequency domainimaging algorithm can be acquired based on the decompositionof the cylindrical waves into superposition of plane waves.The rest of the communication is organized as follows: Inthe next section, we formulate the imaging algorithm based onsolving the convolution of the Green’s functions in the spatialfrequency domain. The sampling criteria and resolutions areoutlined. Numerical results are shown in Section III. Andconcluding remarks follow at the end. a r X i v : . [ ee ss . SP ] F e b ( x , y , z )( x T , y T , z ’ ) ( x R , y R , z ’ ) yz Target x Rx:TR: Scanning
Fig. 1. Topology of the circular-arc MIMO array based MMW imaging.
II. C
IRCULAR -A RC MIMO A
RRAY B ASED I MAGING
A. Imaging Algorithm
The circular-arc MIMO array based imaging geometry isshown in Fig. 1, where the transmit and receive antennas areuniformly spaced along a circular arc, meanwhile, scanningalong the vertical direction. The demodulated scattered wavesare given by, s ( k, θ T , θ R , z (cid:48) ) = (cid:90) (cid:90) (cid:90) g ( x, y, z ) e − j k ( R T + R R ) d x d y d z, (1)where k = πfc denotes the wavenumber, f is the workingfrequency, c is the speed of light, g ( x, y, z ) represents thescattering coefficient of the target located at ( x, y, z ) , R T and R R are the distances from the transmit antenna to the targetand from the target to the receive antenna, respectively, whichcan be expressed as, R T = (cid:113) ρ T + ( z − z (cid:48) ) ,R R = (cid:113) ρ R + ( z − z (cid:48) ) , where ρ T = (cid:112) ( x − R sin θ T ) + ( y + R cos θ T ) , (2) ρ R = (cid:112) ( x − R sin θ R ) + ( y + R cos θ R ) . (3)The transmit and receive positions are denoted by ( R , θ T , z (cid:48) ) and ( R , θ R , z (cid:48) ) , respectively, in the cylindrical coordinates,where R represents the radius of the circular array.Next, we derive a spatial frequency domain algorithm fromthis imaging scheme. Performing the Fourier transform onboth sides of (1) with respect to z (cid:48) and using the propertyof convolution in the Fourier domain, we have s ( k, θ T , θ R , k z (cid:48) ) = (cid:90) (cid:90) (cid:90) g ( x, y, z ) · (4) F z (cid:48) [ e − j kR T ] (cid:126) k z (cid:48) F z (cid:48) [ e − j kR R ]d x d y d z. The exponential terms e − j kR T and e − j kR R are referred to asthe free space Green’s functions, whose Fourier transformswith respect to z (cid:48) can be expressed as [23], F z (cid:48) [ e − j kR T ] = e − j √ k − k z (cid:48) ρ T e − j k z (cid:48) z , (5) F z (cid:48) [ e − j kR R ] = e − j √ k − k z (cid:48) ρ R e − j k z (cid:48) z . (6) Target
Array apertureTx Rx( x , y ) 𝜌 𝑇 𝜌 𝑅 𝜃 𝑇 𝜃 𝑅 𝑅 O 𝑥 𝑦 Fig. 2. Geometry of a simple circular-arc MIMO array and target.
Substituting (5) and (6) in (4), we obtain, s ( k, θ T , θ R , k z (cid:48) ) = (cid:90) (cid:90) (cid:90) g ( x, y, z ) · (7) [ e − j √ k − k z (cid:48) ρ T e − j k z (cid:48) z (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ R e − j k z (cid:48) z ]d x d y d z. According to the form of convolution, (7) is rewritten as, s ( k, θ T , θ R , k z (cid:48) ) = (cid:90) (cid:90) (cid:90) g ( x, y, z ) e − j k z (cid:48) z · (8) [ e − j √ k − k z (cid:48) ρ T (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ R ]d x d y d z. However, it is hard to find the analytical solution of theconvolution in the square brackets. In order to solve it,we approximate ρ T ≈ ρ R (denoted by ρ ) temporarily,according to the geometry information between the imagingscene (mainly located around the axis of a cylindrical surfaceconstructed by the mechanical scanned array) and the circular-arc MIMO array. Then, the convolution in the square bracketsin (8) can be expressed as, e − j √ k − k z (cid:48) ρ T (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ R (9) ≈ e − j √ k − k z (cid:48) ρ (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ = (cid:90) e − j √ k − ζ ρ e − j √ k − ( k z (cid:48) − ζ ) ρ d ζ. This integral can be calculated using the principle of stationaryphase (POSP) [24]. Assuming ϑ ( ζ ) = − (cid:112) k − ζ ρ − (cid:112) k − ( k z (cid:48) − ζ ) ρ , (10)then, using d ϑ/ d ζ = 0 , we obtain ζ = k z (cid:48) / . Thus, theconvolution in (9) is given by, e − j √ k − k z (cid:48) ρ T (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ R ≈ e jϑ ( ζ = k z (cid:48) / e − jπ/ (11) ≈ e − j (cid:114) k − k z (cid:48) ρ T e − j (cid:114) k − k z (cid:48) ρ R e − jπ/ . Note that we replace ρ by the original ρ T and ρ R afterthe convolution. Here, we omit the envelope and the constantterms.To evaluate the accuracy of (11), we show a numericalcomparison between e − j √ k − k z (cid:48) ρ T (cid:126) k z (cid:48) e − j √ k − k z (cid:48) ρ R , whichis calculated by the MATLAB function ‘conv’ as the accuratevalues, and e − j √ k − k z (cid:48) / ρ T e − j √ k − k z (cid:48) / ρ R e − jπ/ , under aconfiguration illustrated in Fig. 2 with parameters shown inTable I. The results are shown in Fig. 3. Clearly, the errorsinduced by the approximation ρ T ≈ ρ R are really small in theprocess of solving the convolution. TABLE IP
ARAMETERS FOR E VALUATING THE C ONVOLUTION
Parameters ValuesRadius of the circular array ( R ) θ T − ◦ Angle of receive antenna θ R ◦ Scanning step along z direction ∆ z ( x, y ) (0 . , (Unit: meter)(a)(b)Fig. 3. Comparison between the true convolution and the approximated one,(a) the real part, and (b) the imaginary part. Substituting (11) in (8) yields, s ( k, θ T , θ R , k z (cid:48) ) = (cid:90) (cid:90) (cid:90) g ( x, y, z ) e − jk z (cid:48) z · (12) e − jk ρ ρ T e − jk ρ ρ R d x d y d z, where k ρ = (cid:113) k − k z (cid:48) / , (13)and e − jπ/ is omitted. Based on (2) and (3), we decomposethe cylindrical waves into a superposition of plane waves, e − j k ρ ρ T = (cid:90) e − j k xT ( x − x T ) e − j k yT ( y − y T ) d k x T , (14) e − j k ρ ρ R = (cid:90) e − j k xR ( x − x R ) e − j k yR ( y − y R ) d k x R , (15)where x T/R = R sin θ T/R , y T/R = − R cos θ T/R , and k x T/R + k y T/R = k ρ T/R with k ρ T/R = k ρ . Here, the subscript‘ T /R ’ denotes the transmit or receive parts for conciseness.In order to achieve independent data with respect to k ρ T and k ρ R , we perform dimension increase by dividing k intotwo parts using the relation k = k T + k R , with k ρ T = (cid:113) k T − k z (cid:48) / , (16a) k ρ R = (cid:113) k R − k z (cid:48) / . (16b) 𝑠(𝑘 (𝑁+1)/2 ) 𝑠(𝑘 )𝑠(𝑘 )𝑠(𝑘 ) 𝑠(𝑘 (𝑁+3)/2 )𝑠(𝑘 )𝑠(𝑘 )𝑠(𝑘 )𝑠(𝑘 (𝑁+1)/2 ) 𝑠(𝑘 (𝑁+3)/2 ) 𝑠(𝑘 (𝑁+5)/2 ) 𝑠(𝑘 𝑁 ) Fig. 4. Dimension increasing from s ( k, · · · ) to s ( k T , k R , · · · ) . To achieve this goal, we reformulate the data s ( k, · · · ) according to the illustration in Fig. 4. In so doing, we geta data matrix corresponding to the independent polar spatialfrequency support ( k T , θ T ) and ( k R , θ R ) , respectively.Then, substituting (14) and (15) in (12), and rearranging theintegral sequence, we obtain, s ( k T , k R , θ T , θ R , k z (cid:48) ) = (cid:90) (cid:90) (cid:90) (cid:90) (cid:90) g ( x, y, z ) · (17) e − j( k xT + k xR ) x e − j( k yT + k yR ) y e − j k z (cid:48) z · e j k xT x T e j k yT y T e j k xR x R e j k yR y R d x d y d z d k x T d k x R . Clearly, the inner integrals over x , y , and z can beexpressed as a 3-D Fourier transform of g ( x, y, z ) (denotedby G ( k x , k y , k z ) ). Then, we have s ( k T , k R ,θ T , θ R , k z ) = (cid:90) (cid:90) G ( k x , k y , k z ) · (18) e j k xT x T e j k yT y T e j k xR x R e j k yR y R d k x T d k x R , where k x = k x T + k x R , (19a) k y = k y T + k y R , (19b) k z = k z (cid:48) . (19c)The exponential terms associated with the integrals inthe right side of (18) can be canceled in the cylindricalcoordinates. The spatial frequency relations between theCartesian and the cylindrical coordinates are given by, k x T/R = k ρ T/R sin φ T/R , (20a) k y T/R = − k ρ T/R cos φ T/R . (20b)The differential elements in (18) can then be approximated by d k x T d k x R ≈ k ρ T k ρ R cos φ T cos φ R d φ T d φ R due to the factthat k ρ T and k ρ R vary slowly along the direction of k x T and k x R , respectively, for a limited angle extent subtended by thecircular-arc array. Then, representing the spatial frequencies inthe cylindrical coordinates, (18) is rewritten as s ( k T , k R ,θ T , θ R , k z T , k z R ) = k ρ T k ρ R · (21) (cid:90) (cid:90) G ( k ρ T , k ρ R , φ T , φ R , k z ) cos φ T cos φ R · e j k ρT R cos( θ T − φ T ) e j k ρR R cos( θ R − φ R ) d φ T d φ R . Clearly, the integrals over φ T and φ R can be represented by thefollowing convolutions with respect to θ T and θ R , respectively. s ( k T , k R , θ T , θ R , k z T , k z R ) (22) = k ρ T k ρ R G ( k ρ T , k ρ R , θ T , θ R , k z ) cos θ T cos θ R (cid:126) T e j k ρT R cos θ T (cid:126) R e j k ρR R cos θ R , D Array aperture 𝑅 Δ𝜃 𝑅 𝑅 O 𝑥𝑦 Rx 𝑅 𝜃 𝜃 Rx Fig. 5. Illustration of the sampling criterion of circular-arc MIMO array. where (cid:126) T and (cid:126) R denote convolutions in the θ T and θ R domains, respectively.Note from (22) the scattering coefficients of target inthe spatial frequency domain can be obtained throughdeconvolutions of the two exponential functions. We take theFourier transforms of both sides of (22) with respect to θ T and θ R , respectively. With the help of the Fourier property ofconvolution, we can write s ( k T , k R , ξ T , ξ R , k z T , k z R ) = k ρ T k ρ R (cid:101) G ( k ρ T , k ρ R , ξ T , ξ R , k z ) · (23) H (1) ξ T ( k ρ T R ) e j πξ T / H (1) ξ R ( k ρ R R ) e j πξ R / , where ξ T/R denotes the Fourier domain for θ T/R , (cid:101) G ( · · · ) represents the Fourier transform of G ( · · · ) cos θ T cos θ R withrespect to θ T and θ R , and H (1) ξ T/R is the Hankel function ofthe first kind, ξ T/R order [23]. H (1) ξ T/R ( k ρ T/R R ) = F θ T/R [ e j k ρT/R R cos θ T/R ] e − j πξ T/R / . (24)Thus, dividing both sides of (23) by the two Hankelfunctions and exponentials, and performing the inverse Fouirertransforms with respect to ξ T and ξ R , we obtain G ( k ρ T , k ρ R , θ T , θ R , k z ) = 1 k ρ T k ρ R cos θ T cos θ R · (25) F − ξ T/R (cid:34) s ( k T , k R , ξ T , ξ R , k z T , k z R ) e − j πξ T / e − j πξ R / H (1) ξ T ( k ρ T R ) H (1) ξ R ( k ρ R R ) (cid:35) . Finally, interpolations based on (20a) and (20b) anddimension reduction [7] are implemented to change G ( k ρ T , k ρ R , θ T , θ R , k z ) into G ( k x , k y , k z ) , over whichthe 3-D inverse FFT is performed to obtain g ( x, y, z ) . B. Sampling criteria
We assume that the transmit array is undersampled, and thereceive array is fully sampled. First, the requirement for theinter-element spacing of the receive array is considered. Toavoid aliasing, the phase difference between two neighboringreceive antennas has to be less than π rad [7]. Consideringthe illustration in Fig. 5, we have k ( R − R ) ≤ π , where R = (cid:113) R + D − R D cos θ ≈ R − D cos θ , R = Fig. 6. 1-D images along the horizontal direction by using circular-arc MIMOarrays with N t transmit antennas and 31 receive antennas, and by using amonostatic array with 61 transmit-receive pairs. (cid:113) R + D − R D cos θ ≈ R − D cos θ , and θ = θ +∆ θ R . Thus, k ( R − R ) ≈ k D sin θ sin ∆ θ R ≤ k D sin ∆ θ R ≈ k D ∆ θ R ≤ π, due to cos ∆ θ R ≈ and sin ∆ θ R ≈ ∆ θ R with a small inter-element spacing ∆ θ R of the receive array. Then, we obtain ∆ θ R ≤ λ min D , (26)where λ min denotes the minimum wavelength of the workingEM waves, and D is the maximum dimension along x direction.For the undersampled transmit array, there is no limit to theinter-element spacing as long as two antennas at least are putat the both ends of the receive array to ensure a comparableresolution with a monostatic array. Fig. 6 shows a comparisonof 1-D imaging results obtained using a monostatic array with61 transmit and 61 receive antennas as a benchmark, andusing circular MIMO arrays with N t transmit antennas and31 receive antennas. Note that the resolutions of MIMO arraysare slightly worse than that of the monostatic array due to theconvolution in (22), which is equivalent to adding a weightingfunction (shaped like a triangle) to the spatial frequency data.On the other hand, the weighting function with respect to N t = 2 is shaped somewhat like a ‘U’-shape, which results ina slightly narrower main lobe but higher sidelobes than thoseof the other cases (such as N t = 3 or ). However, on thewhole, the results in Fig. 6 have comparable resolutions.As for the requirement for the mechanical scanning stepalong the vertical direction, according to e − j k z z (cid:48) in (17), thescanning step should satisfy k z max ∆ z (cid:48) ≤ π to avoid aliasingeffects. Based on (13), we have k z max = 2 k max sin Θ z , then, ∆ z (cid:48) ≤ λ min Θ z , (27)where Θ z denotes the minimum one between the antennabeamwidth and the angle subtended by the scanning lengthfrom the center of imaging scene. C. Resolutions
First, consider the cross-range resolution along the directionof the circular-arc array. Note that the cross-range resolution
TABLE IIS
IMULATION P ARAMETERS
Parameters ValuesRadius of the circular array ( R ) is determined by the extent of the corresponding spatialfrequency, which leads to, δx = πk x max , (28)based on the exponential function exp( − j k x x ) , where k x max denotes the maximum of k x .According to (19a), and the array configuration in Fig.1, we note that k x max = k x T max + k x R max with k x T max = k x R max ≈ k c sin(Θ h / with the help of (16a), (16b), and(20a), where k c denotes the center wavenumber of the workingwaves, and Θ h represents the angle subtended by the arrayaperture, assuming that the beamwidth of each antenna canfully illuminate the target in the horizontal direction. Thus, δx = π k c sin Θ h = λ c Θ h . (29)This is the same with that of the monostatic imaging scenario,which has been verified by the results in Fig. 6.Similarly, the resolution along the z direction is given by, δz = πk z max = λ c Θ z , (30)where Θ z denotes the minimal one between the anglesubtended by the scanning length and the beamwidth of theantenna element.Finally, the down-range resolution can be expressed as δy = c B , where B denotes the bandwidth of the working EM waves.III. R ESULTS
In this section, we show more results of the proposedmethod in comparison with BP. The parameters for simulationsare presented in Table II. (a) (b)(c) (d)(e) (f)Fig. 8. 2-D images with respect to the three coordinate planes by the proposedalgorithm: (a), (c), (e); and by BP: (b), (d), and (f).
First, we provide simulations of point targets to comparefocusing property with the BP algorithm. The 3-D imagingresults are shown in Fig. 7. To view the details, the 2-D imageswith respect to the three coordinate planes are demonstratedin Fig. 8. Clearly, the results of the proposed algorithm havesimilar performance with those of BP. Fig. 9 shows the 1-D images where the main lobes and sidelobes of the pointtargets can be clearly demonstrated. Note that the resolutionsalong the horizontal cross-range and down-range directions inFigs. 9(c) and 9(d), respectively, of the proposed algorithmare slightly worse than those of BP, probably due to theinformation loss caused by the transformation from the polarto Cartesian grids.Finally, we provide the results by using FEKO - acomputational electromagnetics software [25], to simulatethe scattered EM waves. The reconstructed images by theproposed algorithm and BP are shown in Figs. 10(a) and10(b), respectively. Note that the focusing performance of theproposed algorithm is very close to that of BP, indicatingfurther the effectiveness of the approach.IV. C
ONCLUSIONS
The paper presented a near-field 3-D MMW imagingscheme based on a circular-arc MIMO array associated withmechanical scanning along its perpendicular direction. The (a)(b)(c)Fig. 9. 1-D images corresponding to the (a) height, (b) horizontal, and (c)range dimensions, respectively.(a) (b)Fig. 10. Imaging results by (a) the proposed algorithm, and (b) BP. transmit and receive antennas are uniformly distributed overan arc of a circle. The circular-arc MIMO array can providemore even illuminations of the imaging area than the linear orplanar MIMO arrays, which may offer better observation ofthe human body and any concealed items. Further, the imagingalgorithm was presented based on the spatial frequency domainprocessing. Numerical experiments demonstrated the efficacyof the proposed imaging technique. R
EFERENCES[1] A. Moreira, P. Prats-Iraola, M. Younis, G. Krieger, and etc., “A tutorialon synthetic aperture radar,”
IEEE Geosci. Remote Sens. Mag. , vol. 1,pp. 6–43, March 2013.[2] S. Di Meo, P. F. Esp´ın-L´opez, A. Martellosio, and etc., “On thefeasibility of breast cancer imaging systems at millimeter-wavesfrequencies,”
IEEE Trans. Microw. Theory Tech. , vol. 65, pp. 1795–1806, May 2017.[3] M. Amin,
Through-the-Wall Radar Imaging . Taylor & Francis, 2010.[4] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Near field imagingat microwave and millimeter wave frequencies,” in , pp. 1693–1696, June 2007.[5] D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensionalmillimeter-wave imaging for concealed weapon detection,”
IEEE Trans.Microw. Theory Tech. , vol. 49, pp. 1581–1592, Sep. 2001.[6] S. S. Ahmed, A. Schiessl, F. Gumbmann, M. Tiebout, S. Methfessel,and L. Schmidt, “Advanced microwave imaging,”
IEEE Microw. Mag. ,vol. 13, no. 6, pp. 26–43, 2012.[7] X. Zhuge and A. G. Yarovoy, “Three-dimensional near-field MIMOarray imaging using range migration techniques,”
IEEE Trans. ImageProcess. , vol. 21, pp. 3026–3033, June 2012.[8] Y. ´Alvarez, Y. Rodriguez-Vaqueiro, B. Gonzalez-Valdes, and etc.,“Fourier-based imaging for subsampled multistatic arrays,”
IEEE Trans.Antennas Propag. , vol. 64, no. 6, pp. 2557–2562, 2016.[9] G. Gennarelli and F. Soldovieri, “Multipath ghosts in radar imaging:Physical insight and mitigation strategies,”
IEEE J. Sel. Topics Appl.Earth Observations Remote Sens. , vol. 8, no. 3, pp. 1078–1086, 2015.[10] Y. ´Alvarez, Y. Rodriguez-Vaqueiro, B. Gonzalez-Valdes, and etc.,“Fourier-based imaging for multistatic radar systems,”
IEEE Trans.Microw. Theory Tech. , vol. 62, no. 8, pp. 1798–1810, 2014.[11] X. Zhuge and A. G. Yarovoy, “A sparse aperture MIMO-SAR-basedUWB imaging system for concealed weapon detection,”
IEEE Trans.Geosci. Remote Sens. , vol. 49, pp. 509–518, Jan 2011.[12] F. Gumbmann and L. Schmidt, “Millimeter-wave imaging withoptimized sparse periodic array for short-range applications,”
IEEETrans. Geosci. Remote Sens. , vol. 49, pp. 3629–3638, Oct 2011.[13] J. Gao, Y. Qin, B. Deng, and etc., “Novel efficient 3D short-rangeimaging algorithms for a scanning 1D-MIMO array,”
IEEE Trans. ImageProcess. , vol. 27, pp. 3631–3643, July 2018.[14] J. Gao, B. Deng, Y. Qin, and etc., “An efficient algorithm for MIMOcylindrical millimeter-wave holographic 3-D imaging,”
IEEE Trans.Microw. Theory Tech. , vol. 66, pp. 5065–5074, Nov 2018.[15] H. Gao, C. Li, S. Wu, and etc., “Study of the extended phase shiftmigration for three-dimensional MIMO-SAR imaging in terahertz band,”
IEEE Access , vol. 8, pp. 24773–24783, 2020.[16] S. S. Ahmed, A. Schiessl, and L. Schmidt, “A novel fully electronicactive real-time imager based on a planar multistatic sparse array,”
IEEETrans. Microw. Theory Tech. , vol. 59, pp. 3567–3576, Dec 2011.[17] X. Zhuge and A. G. Yarovoy, “Study on two-dimensional sparseMIMO UWB arrays for high resolution near-field imaging,”
IEEE Trans.Antennas Propag. , vol. 60, pp. 4173–4182, Sep. 2012.[18] K. Tan, S. Wu, Y. Wang, S. Ye, and etc., “A novel two-dimensionalsparse MIMO array topology for UWB short-range imaging,”
IEEEAntennas Wireless Propag. Lett. , vol. 15, pp. 702–705, 2016.[19] K. Tan, S. Wu, Y. Wang, S. Ye, J. Chen, X. Liu, G. Fang, and S. Yan,“On sparse MIMO planar array topology optimization for UWB near-field high-resolution imaging,”
IEEE Trans. Antennas Propag. , vol. 65,no. 2, pp. 989–994, 2017.[20] J. Wang, P. Aubry, and A. Yarovoy, “3-D short-range imaging withirregular MIMO arrays using NUFFT-based range migration algorithm,”
IEEE Trans. Geosci. Remote Sens. , vol. 58, no. 7, pp. 4730–4742, 2020.[21] S. Wu, H. Wang, C. Li, and etc., “A modified omega-k algorithm fornear-field single-frequency MIMO-arc-array-based azimuth imaging,”
IEEE Trans. Antennas Propag. , pp. 1–1, 2021.[22] Z. D. Qin, J. Ylitalo, and J. Oksman, “Circular-array ultrasoundholography imaging using the linear-array approach,”
IEEE Trans.Ultrason. Ferroelectr. Freq. Control , vol. 36, no. 5, pp. 485–493, 1989.[23] M. Soumekh, “Reconnaissance with slant plane circular SAR imaging,”
IEEE Trans. Image Process. , vol. 5, no. 8, pp. 1252–1265, 1996.[24] I. Cumming and F. Wong,