Computational imaging with low-order OAM beams at microwave frequencies

With the distinguished wavefront characteristics of vortex electromagnetic wave carrying orbital angular momentum (OAM), the OAM beams have been exploited for radar imaging in recent years. In this paper, the computational imaging model is built using OAM wave, which enables the target reconstruction with limited measurements. The measurement matrix is designed, and the target reconstruction method is proposed in the Cartesian coordinate. Simulation results indicate that the proposed computational imaging approach is robust against noise influence. Furthermore, the outdoor experiments are carried out, for the first time, to validate the super-resolution imaging ability of this novel technique. Experimental results show good agreement with theoretical analyses. This work can advance the development of OAM-based sensing technology.

Scientific RepoRtS | (2020) 10:11641 | https://doi.org/10.1038/s41598-020-68586-y www.nature.com/scientificreports/ where u(t − dT r ) denotes the rectangular pulse function, and T r is the pulse repetition interval. f 0 is the signal frequency of the first pulse, and f is the step frequency. φ n indicates the azimuthal position of the nth antenna, D is the number of pulses, and N is the number of antennas. l denotes the OAM index. Without loss of generality, the target is assumed to be located at the position P(x, y, z) , and the echo s r (t, l) can be given by where τ = 2 (x − a cos φ n ) 2 + (y − a sin φ n ) 2 /c is the time delay, and φ n = 2π(n − 1)/N , n = 1, 2, . . . , N . a is the radius of the UCA, and σ stands for the target scattering coefficient.
When multiplying Eq. (2) by the reference term exp(−i2πf 0 t) , it leads to To reconstruct the target, the computational imaging equation is written as where S r is the echo vector, S is the measurement matrix, and σ is the unknown scattering coefficient vector. n indicates the unknown noise vector. Based on Eqs. (3) and (4), the computational imaging equation can be rewritten as follows where the target area is discretized into Q grids, and the target scatters are assumed to be located at the center of each cell. P is the sampling points of the echo in the time domain, and L is the total number of OAM modes used in the imaging. σ i , i = 1, 2, . . . , Q denotes the scattering coefficient of the ith grid. Though the OAM eigenvalues are orthogonal to each other, the rows of the measurement matrix are coherent since the step frequency signal is used. Thus, Eq. (5) is usually an ill-posed inverse problem, i.e., the number of parameters to be solved is more than the measured information. The stable solutions cannot be obtained by the conventional least-square method 20 .
Generally, the matched filtering approach can solve the unstable phenomenon caused by the ill-posed problem of the imaging equation. And the estimation σ for the target scattering coefficients can be expressed as . . . The method in Eq. (6) is robust against the noise, however, the main lobes of the reconstructed profiles usually widen and the side lobes always exist. For radar imaging applications, the sparse characteristics of the imaging scene are usually assumed to be the prior information, which can be exploited to reduce the solution instability of the imaging equation and improve the target reconstruction performance. This can be presented as an optimization process where γ is a scale weighting coefficient.    The comparison of the target reconstructed results using plane wave and OAM beams is shown in Fig. 2. Results indicate that two targets in the same range cannot be correctly reconstructed in the cross-range dimension when the plane wave is transmitted. In contrast, the helical wavefront of OAM beams can enhance the imaging resolution. Furthermore, two targets are placed closer to each other in the same cross range, and the reconstructed results with different OAM modes are shown in Fig. 3. It is clear from Fig. 3 that more OAM modes are expected to be used to achieve higher resolution.
To analyze the noise influence on the imaging performance, the mean square error (MSE) 21 as a function of the signal-to-noise ratio (SNR) is shown in Fig. 4. In the simulation, the imaging scene is the same as that in Fig. 3, and 100 Monte trials are performed. Results demonstrate that better reconstruction performance can be achieved as the SNR increases. experimental results. In this paper, the outdoor experiments are performed, shown in Fig. 5. The array radar placed on a platform is exploited to generate OAM beams. The corner reflectors are placed in the scene for imaging. The main system parameters are listed in Table 1. Considering the outdoor imaging scenario, the noise always exists due to the realistic atmosphere, which will have negative effects on the target reconstruction. To reduce the noise influence, the matched filtering is performed on the echo vector S r and the measurement matrix S , and then Eq. (4) leads to where A indicates the matched filtering vector. Since the noise is incoherent, only the signal will be enhanced. www.nature.com/scientificreports/ The distance between radar and the scene center is set as 79 m, firstly. Two corner reflectors are placed in the same range, and the distance is 2 m in the cross-range domain. The SBL and OMP methods are used to reconstruct the targets, shown in Fig. 6. Compared with the result using plane waves in Fig. 6b, higher resolution can be achieved by transmitting OAM beams, shown in Fig. 6c,d. It can be seen from Fig. 6 that the spatial cross-range resolution for OAM-based imaging is about ρ a = 2 m , whereas the resolution for conventional imaging is decided by the aperture ρ a = R · /(2a) , i.e., 78 · /(2a) ≈ 3 m , where R denotes the target's position in the range domain, is the signal wavelength corresponding to the center frequency ( f c = 9.5 GHz ) of the transmitted signal. Thus, the target cannot be reconstructed using plane waves, shown in Fig. 6b. Further, due to the distinguished positions and the manufacture error of the two corner reflectors, the backward scattering coefficients are usually different, and thus the color scales in Fig. 6c,d are different. Furthermore, the distance between the radar and the scene center is set as 433 m. In Fig. 7, the distances between two corner reflectors are 2 m and 4 m in the range domain and cross-range domain, respectively. In Fig. 8, two corner reflectors are placed in the same range, and the distance is 3 m in the cross-range domain. Results in Figs. 7 and 8 validate the effectiveness of the proposed imaging method, which shows good agreement with the simulation results in Fig. 2. It can be seen from Fig. 8 that the spatial resolution in the cross-range dimension is ρ a = 3 m , whereas the spatial resolution for plane wave is ρ a = R · /(2a) = 431 · /(2a) ≈ 17 m . Thus, higher cross-range resolution can be achieved by the proposed imaging method, in this paper.

Discussion
To conclude, the computational imaging method based on low-order OAM beams has been developed. The imaging model in the Cartesian coordinate was built firstly, and the SBL and OMP approaches were introduced to reconstruct the targets. Outdoor experiments of the OAM-based imaging were carried out, for the first time, to validate the effectiveness of the proposed method. Results indicated that the OAM beam in single mode can be used to reconstruct the target, which can achieve higher resolution than the plane wave. Furthermore, the proposed matched filtering method can enhance the target reconstruction performance for outdoor imaging scene.

Methods
Two different signal processing approaches are employed to reconstruct the target. When the prior information of the samples and the sparse parameters are set, the Bayesian interference can be used to realize the sparse reconstruction and parameter learning, i.e., the sparse Bayesian learning (SBL) 21,22 . Though the columns of the measurement matrix are coherent to each other, the SBL method can still maintain good performance.
Let h(S r ; σ ) be the likelihood function. For an arbitrary probability density function g(z) , it leads to where where the E-step and M-step are in the following (9) ln h(S r ; σ ) = g(z) ln h(S r ; σ )dz = F(g, σ ) + KL(g||h) (10) F(g, σ ) = g(z)ln h(S r , z; σ ) g(z) dz www.nature.com/scientificreports/ Furthermore, the orthogonal matching pursuit (OMP) 23 is also applied to reconstruct the target. Let {x 1 , . . . , x PL } be a sequence of measurement vectors in R Q . The main procedure can be given by where i denotes the iteration number, and S i is the ith row of the measurement matrix S . r i = S r − a i is the residual, and a i = S i x i is the approximation of the echo vector S r . The nonzero elements of the estimation σ is recovered from x i .