SHI Hongyin,XIA Saixue,and TIAN Ye

School of Information Science and Engineering,Yanshan University,Qinhuangdao 066004,China

1.Introduction

In the military,inverse synthetic aperture radar(ISAR)imaging technology has been the focus of the study at home and abroad.It can be all-weather,all-day and longrange imaging and has a very high information acquisition ability[1–5].For synthetic aperture radar(SAR)system[6–8],the target is stationary and the radar platform is moving.The motion parameters of the radar system are basically known.However,the target is moving and the radar platform is stationary for ISAR system.The traditional ISAR algorithm[9]is suitable for slow moving targets imaging,but it will not be suitable for the target with complex motion,such as high-speed motion,maneuver and rotation,which needs to carry on the motion compensation to the complex movement.The target of complex motion should be motion compensated.In[10],the motion compensation technique for ISAR imaging based on the parameter estimation has been studied.Some existing motion compensation techniques have some drawbacks,such as the cross-correlation method[11,12]and the minimum entropy method[13–15].The cross-correlation method has a small scope of measuring velocity and a low accuracy,and the computational cost of the minimum entropy method is large.

Recently,phase retrieval theory[16–20]has been widely used to rebuild the targets in many areas.Phase retrieval is an imaging method from the magnitude of its Fourier transform which is popularly used in various areas such as optical imaging,crystallography,and astronomy.The principle of phase retrieval is to recover the original signal only from the amplitude measurement of the signal in a certain transformation domain.Phase retrieval algorithms can be divided into two categories:alternative projection phase retrieval algorithm and phase retrieval algorithm based on sparse representation.The hybrid input output (HIO) [21] algorithm and oversampling smoothness(OSS)algorithm[22]are classic alternative projection algorithms.The OSS phase retrieval algorithm iterates back and forth between the image and Fourier domains.For the phase retrieval algorithm of sparse signals,Schniter[23]proposed a compression phase retrieval algorithm that uses a small amount of data to reconstruct the target. Some other methods have been proposed in[24 – 27].Schulz’s image recovery from autocorrelations[28],Fienup’s reconstruction from the modulus of the transform[29]and Phillips’maximum likelihood estimator based on Gaussian statistics[30]are three important algorithms in phase retrieval.From the perspective of the space geometry model and the imaging mechanism of ISAR,ISAR echoes can be consi-dered as Fourier transform of space targets.Thus the ISAR imaging of maneuvering targets can be taken as a broadly self-focusing problem.

If phase information is ignored or phase information is considered to be destroyed,the ISAR imaging and autofocusing problem can be equivalent to phase retrieval.In the optical imaging problem,there is no target phase information in the optical imaging and the detection equipment can only measure the amplitude or intensity information of its Fourier transform.While for the ISAR system,the target motion leads to phase errors,which can be still exploited.However,the original OSS algorithm can obtain good imaging results under the condition of Poisson noise and will not achieve better retrieval performance in the case of random noise[22].

Based on the above analysis,combined with the phase retrieval principle and part of the priori information of the classical imaging algorithm,this paper presents an ISAR autofocusing imaging method based on the improved OSS algorithm.The algorithm can realize high resolution imaging of high speed maneuvering targets.The effectiveness of the proposed algorithm is proved by simulation results.

The remainder of this paper is organized as follows:In Section 2,a brief description of Fourier phase retrieval theory is shown.The reconstruction algorithm is provided in detail in Section 3.Simulation analysis is presented in Section 4 and conclusions are given in Section 5.

2.Fourier phase retrieval theory

Suppose that one-dimensional discrete real field distribution function of an object is x∈CN,which can be directly extended to a higher dimension.The one-dimensional discrete Fourier transform(FT)of x is expressed as

where M is the M-point discrete FT of the function,and M＞N.The formula can be rewritten as

The phase retrieval algorithm only uses the known FT information|X(k)|to recover the Fourier phase information φ(k),and then restore the distribution function?x through the inverse Fourier transform(IFT).

An effective iterative phase retrieval method named OSS algorithm based on alternate projections is used in this paper.The OSS phase retrieval algorithm iterates back and forth between the image and Fourier domains.Fig.1 shows the schematic of the algorithm in[22].

Fig.1 Block scheme of the OSS algorithm

(i)xi(n)is the input signal with the random phase.The calculation result xi+1(n)of step(v)will replace it in the future iterations.

(ii)Obtain a Fourier pattern Xi(K)after applying the Fourier transform to the input data.

(iii)Retain the phase information of Xi(K),but replace the magnitude of Xi(K)with the known Fourier magnitude|Y(K)|.Then form a new complex-valued function(K),where|Y(K)|represents the magnitude of the ISAR echo signal.

where γ represents a finite support and β is a parameter between 0.5 and 1.

(v)Calculate the next iteration image xi+1(n).

The smoothing filter W(K)is only applied to the density outside the support.By changing parameter∂,the width of the Gaussian filter can be tuned to reduce the influence of high-frequency information outside the support,while the density inside the support is not disturbed.

3.ISAR imaging theory based on the improved OSS

3.1 Maneuvering targets echo signal analysis

Analyze the geometric model of the maneuvering target[11]in Fig.2.The point scatterer P(xn,yn)on the target has translational motion and rotational motion.The middle of the target is selected as the phase center and is assumed to be the origin.

Fig.2 ISAR geometry for a maneuvering target

Making R(t)into their Taylor series,they can be represented to yield

where,vtand atare the target’s translational velocity and the acceleration respectively.The initial range of the target is Roand the uniform angular velocity is ω.For simplifying this phase,(6)can be shown as

Assume that the initial angle of the target with respect to the radar line of sight axis is 0.The rotation angle θ(t):

When the target is far away from the radar,the distance from point P to the radar can be approximated as

where rPis the imaging plane rectangular coordinates of point P.The distance between the radar and the target is greater than the size of the target geometry.r(t)can be approximated as

ωt is small in a relatively short period of time,so r(t)can be written as follows:

Therefore,the backscattered echoes from all the scatterers can be theoretically approximated as

Here k=2πf/c.c is the electromagnetic wave speed and Anis the scattering intensity.

|Es(k,t)|is the maneuvering target’s magnitude.

From(13),the ISAR echo module is not affected by the radial component without noise and using the phase retrieval algorithm can realize ISAR high resolution imaging.

From(14),the ISAR echo module changes a lot under the condition of random noise.Thus it is not suitable for directly applying the phase retrieval algorithm under this condition.For this case,this paper elaborates a new algorithm in the following section.This algorithm can achieve high resolution imaging and can improve the accuracy of the phase recovery algorithm.

3.2 Model

where u is the frequency(fast time)coordinate and v is the azimuth(slow time)coordinate.This is appropriate for rigid-body translational motion in the plane of apparent rotation.The model above is appropriate only for spatially invariant phase errors,as would be caused by translational motion of the target.Other motions,such as rotational acceleration,may cause spatially variant errors,such as formatting errors.Spatially variant errors require a more sophisticated algorithm for correction,and they are beyond the scope of what we are considering here.

Remove noise before applying the phase retrieval algorithm,and then return to(15).

3.3 ISAR autofocus imaging method based on phase retrieval

According to phase retrieval theory,when the raw data magnitude of the moving target is approximate to the stationary one in the ISAR system,the phase retrieval algorithm can realize an autofocus imaging.In this paper,an algorithm is proposed in the case of ISAR imaging with noise.The initial phase is random in the OSS phase retrieval algorithm.In order to improve the accuracy of ISAR imaging,using the imaging phase of the traditional algorithm replaces the random phase.Select the support domain according to the size of the blurred target in this article.And it also needs to capture input information according to the size of the fuzzy image,which will be used in the OSS phase retrieval method.After a certain number of iterations,the image is focused.The block scheme of the proposed method can be seen in Fig.3,where FFT and IFFT represent fast Fourier transform and inverse fast Fourier transform respectively.

Fig.3 Block scheme of the proposed method

3.4 Support domains

Special attention should be paid to the definition of the support constraint as well as possible.Support constraints should be as tight(small)as possible,while still including all the main lobe returns from the target.If the support constraint is large,the algorithm tends to stagnate without correcting the phase error.If the support constraint is too small,the algorithm will inadvertently attempt to truncate the real part of the image,and it may perform poorly.In ISAR imaging,blurred imaging can be obtained by range Doppler(RD).The size of the support domain is estimated from the RD imaging results.

4.Simulation analysis

To evaluate the performanceof the proposed approach,we conductseveral experiments.A hypotheticalairplanecomposed of point scatterers and the amplitude of the raw data are shown in Fig.4.Table 1 shows the radar parameters that are used in the simulation.

Fig.4 A hypothetical airplane and the amplitude of the raw data

Table 1 Radar parameters for step frequency continuous wave illumination

The translational velocity and acceleration of the moving target do not affect the amplitude of the raw data,which is consistent with the above theoretical deduction.The echo amplitude values in Figs.5–8 are the same as in Fig.4(b).

Fig.5 Imaging results with the target simulation parameters vt=5 m/s,at=0.04 m/s2,ω=0.02 rad/s

Fig.6 Imaging results with the target simulation parameters vt=50 m/s,at=0.04 m/s2,ω=0.02 rad/s

Fig.7 Imaging results with the target simulation parameters vt=5 m/s,at=0.6 m/s2,ω=0.02 rad/s

Fig.8 Imaging results with the target simulation parameters vt=35 m/s,at=2 m/s2,ω=0.02 rad/s

Experiment 1Imaging results with different target parameters

As can be seen from Fig.5,the RD algorithm is only applicable to the approximately stationary target imaging.It needs the motion compensation algorithm for the moving target imaging.When the radial velocity is big and the radial acceleration is small,using the cross-correlation method could be able to improve imaging effect from Fig.6.When the radial velocity is small and the radial acceleration is slightly larger,it can get ISAR defocused imaging by using the cross-correlation method from Fig.7.When the radial velocity and acceleration are larger,the compensation estimation of the cross-correlation method is greatly reduced and ISAR imaging is blurred from Fig.8(b).Compared with the cross-correlation method,the minimum entropy method can obtain better imaging effect,but the proposed method in this article outperforms the minimum entropy method in Figs.5–8.The minimum entropy method must be set as an appropriate search range and step length,otherwise it is not possible to image well,or even if it can be a good imaging,it will take more time to image.Through the simulation results and analysis,the proposed method can realize high resolution imaging of high-speed maneuvering targets.

Experiment 2Imaging results with different signalnoise ratios

The amplitude of the raw data changes under the condition of random noise is shown in Fig.9.The amplitude of the raw data becomes poor with the decrease of the signal-to-noise ratio.It needs to remove the effect of random noise before applying the phase retrieval algorithm.Through the comparison of the above figure,the amplitude of the raw data obtained by this proposed algorithm is closer to the amplitude value without noise.The effectiveness of this proposed method is illustrated by imaging results(vt=35 m/s,at=0.06 m/s2,ω=0.02 rad/s)in Figs.10–12.

Fig.9 Amplitude of the raw data under different signal-noise ratios(vt=5 m/s,at=0.06 m/s2,ω=0.02 rad/s)

Figs.10–12 are the simulated ISAR imaging reconstructed by three algorithms.The conventional imaging algorithm cannot realize ISAR autofocus imaging when the speed of the maneuvering target is fast.From Fig.10(a),Fig.11(a)and Fig.12(a),random noise aggravates the blurred imaging.The OSS algorithm cannot achieve high resolution imaging under the condition of random noise in Fig.10(b),Fig.11(b)and Fig.12(b).However,the proposed algorithm can obtain ISAR autofocus imaging of the high speed moving target under the condition of random noise in Fig.10(c),Fig.11(c)and Fig.12(c).When the signal-noise ratio is small,it can preferably reflect the validity of the proposed algorithm in Fig.12.

Fig.10 Imaging under 1.32 dB signal-noise ratio

Fig.11 Imaging under 0.60 dB signal-noise ratio

Fig.12 Imaging under–0.59 dB signal-noise ratio

Experiment 3Comparison of the success rate of the algorithms

To illustrate the feasibility of the proposed method,the improved OSS algorithm is compared with the traditional OSS algorithm.The experimental results are shown in Table 2,which are received from 100 independent experiments.

Table 2 The success rate with different iterations

The standard for the success of the target reconstruction is that all scattering points in the model can be clearly presented.

The experimental results show that the success rate of the proposed algorithm based on prior phase information is much higher than the OSS algorithm(for ISAR imaging problems).

5.Conclusions

In this paper,a Fourier phase retrieval method for maneuvering target’s refocusing in ISAR has been proposed.Combined with the classical OSS phase retrieval algorithm and the prior phase information that ISAR imaging system can provide,it realizes high resolution reconstruction of high speed maneuvering targets and the success rate is 100%.Even in the presence of random noise,the above conclusion still holds.The method can effectively eliminate defocusing caused by the radial motion.At the premise of high accuracy,this method can realize the high resolution imaging of high speed maneuvering targets under the condition of random noise for ISAR imaging.The results of the study provide a new way of thinking for the ISAR imaging problem.The next work is to verify the phase retrieval algorithm for ISAR imaging in practice.

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