ISAR imaging based on improved phase retrieval algorithm

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

Obtain the Fourier pattern(K)by applying the FT toW(K)is a normalized Gaussian function in Fourier domain,defined as

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.

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.

12）农业部令第52号《高致病性动物病原微生物实验室生物安全管理审批办法》（2005-05-20施行）。

(iv)ApplyIFTto(K),and obtainanewimage(n).Revise (n)based on HIO equation,and generate the new

聚丙烯腈纤维在高边坡预应力锚索孔道压浆中的应用…………………………………………… 蔡宣炎，张杜娟（11-55）

自然资源系统入围山东省庆祝改革开放40周年感动山东人物（群体）的是：淄博市原山林场党委书记孙建博，莱西市国土资源局党组原书记、原莱西县委组织部副部长周明金，省地质科学研究院科技创新团队等3个人物（群体）。

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).

这里只介绍下位机软件的设计，主要包括：串口初始化、SHT11初始化、温湿度原始值测量、温湿度以及露点计算和数据传送。程序流程如图5所示。

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

经济时代背景下，旅游企业的人力资源管理方式要进行创新，使人才能够发挥自身作用。人力资源管理切忌墨守成规，束缚人才的个性。旅游企业中每一名员工都应有自身独有的特点，这样才能形成多元化的企业形式。只有为人才提供更多的发展空间，才能够使人力资源管理得到创新，使旅游行业得到更好的发展。

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

3.ISAR imaging theory based on the improved OSS

3.1 Maneuvering targets echo signal analysis

Assume that(u,v)is the ideal signal history and the signal history(phase history)experiences phase errors φe(u,v).The aberrated signal history can be expressed as

在建立高职校企混合所有制二级产业学院的过程中，通过建立市场化运行机制，以市场为导向，紧跟国家产业政策和市场供需变化。通过市场来配置二级产业学院的资源，在满足教育教学的前提下，通过市场向社会要效益，减少不必要的课程设置和师资浪费等教学成本过高问题。[4]

Fig.2 ISAR geometry for a maneuvering target

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

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

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

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.

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.

“你讲的也许没错，可是这对我来说却是一个错误。你知道当年和我一起值班，看见盗贼逃跑，后来被开除的那个年轻人在哪里，他现在过得怎样了吗？”

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

以唐山市某大型商业住宅的供配电设计为样本，该住宅小区总建筑面积约为8×104 m2，包括1栋商业楼、2栋10层商住楼，6栋24层高层住宅楼。设计范围从10 kV配电线路开始，从高压开闭所引来2条主线，其中1条运行，1条备用；低压侧引至小区各单元计量表，为一户一表[2]。

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

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.

各级地方政府亲自去海外向离散人群筹款的行为在北伊洛戈省非常普遍，海外离散人群通常是地方政府筹款时首先想到的对象，因为他们给人以普遍富裕的刻板印象。在当地，普通人的生活依然艰苦，农业机械化程度较低，大量农民依然在进行手工劳作，遇到灾害的抵抗能力也很低。政府和机构雇员虽然和农民相比能够做到“旱涝保收”，但工资较低，让不少人发出“这里生活太艰苦”的感慨。海外工作的工资大约是国内工资的3倍以上。去海外谋生，一般都能感到短时间内生活水准的明显提高。最直观的感受是美元、英镑来到这里直接乘上好几倍，立刻变为有钱人。

在Amos 24.0中输出标准化路径系数图,结果显示,免费开放感知对公园满意度、地方依赖以及地方认同有正向影响,公园满意度对地方依赖和地方认同有正向影响,但是“免费开放→地方认同”和“满意度→地方认同”的路径系数太小,地方依赖对地方认同有正向影响且路径系数最大,与预测测量模型大致相同．在研究后对模型进行修改,将“满意度→地方认同”以及“免费开放→地方认同”这两条路径删除,得到现模型(图3)以及变量相互之间的关系(表3),并进行拟合度检验,所有指标都达到标准水平(表4)．

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

当前，智慧商圈的建设不是只停留在提供商圈电子地图和WIFI的基础设施上，而是要实现集智慧商务、智慧生活、智慧管理和智慧服务于一体的转型升级。目前的商圈大多都已提供WIFI等公众服务，但与“智慧”的定义仍有差距。为使商圈“智慧”化，需将商圈智慧系统建设与商圈周边的智慧园区建设和智慧交通建设等相结合,形成合力，适应并利用当前的环境，推进数据管理和共享。

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

Parameter Symbol Value Target’s initial position in range/km R0 16 Target’s rotational velocity/(rad/s) w 0.02 Starting frequency/GHz f0 9 Frequency bandwidth/MHz B 125 Pulse repetition frequency/KHz PRF 35 Number of pulses Npulse 128 Number of bursts Mburst 128

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).

当前，我国正处于发展的关键时期，一些西方国家不断加大对我国实施“西化”“分化”的力度，不断策划针对中国的渗透、颠覆和破坏活动。同时，国内社会矛盾进入多发期和凸显期，内部和外部安全风险联动态势增强，我国化解安全风险难度加大。新时代，统一战线在维护国家安全中要立足自身特点，选择符合自身优势的着力方向。

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 1 Imaging 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 2 Imaging 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.

先天性肥厚性幽门狭窄是幽门环肌肥厚增生、幽门腔狭窄导致不全梗阻，是新生儿期常见疾病。依据地理、时令和种族，有不同的发病率。欧美国家较高，约为2.5‰～8.8‰，亚洲地区相对较低，我国发病率为3‰，以男性居多，男女之比为4:1～5:1，有些地区甚至高达9:1。多见于第一胎，占总病例数的40%～60%。本病病因目前不清楚，有几种假说：肌层先天性发育异常、神经发育异常、遗传因素、内分泌因素、环境因素、肌肉酶、血型等与发病有一定关系。

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 3 Comparison 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

Algorithm Iteration Success rate Classical OSS algorithm 200 500 0.11 0.12 The proposed algorithm 200 500 11

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.

References

[1]HU J,ZHANG J,ZHAI Q,et al.ISAR imaging using a new stepped-frequency signal format.IEEE Trans.on Geoscience&Remote Sensing,2014,52(7):4291–4305.

[2]MARTORELLA M.Analysis of the robustness of bistatic inverse synthetic aperture radar in the presence of phase synchronization errors.IEEE Trans.on Aerospace and Electronic Systems,2011,47(4):2673–2689.

[3]LV X,XING M,WAN C,et al.ISAR imaging of maneuvering targets based on the range centroid Doppler technique.IEEE Trans.on Image Processing,2010,19(1):141–153.

[4]ZHOUX,WEIG,WUS,etal.Three-dimensional ISARimaging method for high-speed targets in short-range using impulse radar based on SIMO array.Sensors,2016,16(3):364.

[5]ZHENGJ,LIU H,LIAO G,et al.ISAR imaging of targets with complex motions based on a noise-resistant parameter estimation algorithm without nonuniform axis.IEEE Sensors Journal,2016,16(8):2509–2518.

[6]IVANOV A Y,KUCHEIKO A A.Distribution of oil spills in inland seas based on SAR image analysis:a comparison between the Black Sea and the Caspian Sea.International Journal of Remote Sensing,2016,37(9):1–14.

[7]HERRMANN F,BAGHDADI N,BLASCHEK M,et al.Simulation of future groundwater recharge using a climate model ensemble and SAR-image based soil parameter distributions—a case study in an intensively-used mediterranean catchment.Science of the Total Environment,2016,543(Pt B):889–905.

[8]ZHAO Z,JIAO L,ZHAO J,et al.Discriminant deep belief network for high-resolution SAR image classification.Pattern Recognition,2017,61:686–701.

[9]FAN L,SHI S,LIU Y,et al.A novel range-instantaneous-Doppler ISAR imaging algorithm for maneuvering targets via adaptive Doppler spectrum extraction.Progress in Electromagnetics Research C,2015,56:109–118.

[10]WANG Y,ABDELKADER A C,ZHAO B,et al.ISAR imaging of maneuvering targets based on the modified discrete polynomial-phase transform.Sensors,2015,15(9):22401–22418.

[11]OZDEMIR C.Inverse synthetic aperture radar imaging with MATLAB algorithms.Wiley,2012,60(10):5191–5200.

[12]CHEN C C,ANDREWS H C.Target-motion-induced radar imaging.IEEE Trans.on Aerospace and Electronic Systems,1980,AES-16(1):2–14.

[13]ZHANG S,LIU Y,LI X.Fast entropy minimization based autofocusing technique for ISAR imaging.IEEE Trans.on Signal Processing,2015,63(13):3425–3434.

[14]XI L,LIU G,NI J.Autofocusing of ISAR images based on entropy minimization.IEEE Trans.on Aerospace and Electronic Systems,1999,35(4):1240–1252.

[15]SHIN S Y,MYUNG N H.The application of motion compensation of ISAR image for a moving target in radar target recognition.Microwave&Optical Technology Letters,2008,50(6):1673–1678.

[16]FOGEL F,WALDSPURGER I,D’ASPREMONT A.Phase retrieval for imaging problems.Mathematical Programming Computation,2016,8(3):1–25.

[17]CAHILL J,CASAZZA P,DAUBECHIES I.Phase retrieval in in fi nite-dimensional Hilbert spaces.Transactions of the American Mathematical Society,2016,arXiv:1601.06411v2.

[18]QIU T,BABU P,PALOMAR D P.Prime:phase retrieval via majorization-minimization.IEEE Trans.on Signal Processing,2016,64(19):5174–5186.

[19]CHEN P,FANNJIANG A,LIU G R.Phase retrieval with one or two diffraction patterns by alternating projections with the null initialization.Physics,2017,16(2):1–40.

[20]CAND`ES E J,LI X,SOLTANOLKOTABI M.Phase retrieval from coded diffraction patterns.Applied&Computational Harmonic Analysis,2015,39(2):277–299.

[21]TAKAJO H,TAKAHASHI T,SHIZUMA R T.Further study on the convergence property of the hybrid input-output algorithm used for phase retrieval.Journal of the Optical Society of America A,1999,15(11):2163–2168.

[22]RODRIGUEZ J A,XU R,CHEN C C,et al.Oversampling smoothness:an effective algorithm for phase retrieval of noisy diffraction intensities.Journal of Applied Crystallography,2013,46(2):312–318.

[23]SCHNITER P,RANGAN S.Compressive phase retrieval via generalized approximate message passing.IEEE Trans.on Signal Processing,2015,63(4):1043–1055.

[24]SHECHTMAN Y,ELDAR Y C,COHEN O,et al.Phase retrieval with application to optical imaging:a contemporary overview.IEEE Signal Processing Magazine,2014,32(3):87–109.

[25]FANNJIANG A.Absolute uniqueness of phase retrieval with random illumination.Inverse Problems,2012,28(7):75008–75027.

[26]SHENOY B A,MULLETI S,SEELAMANTULA C S.Exact phase retrieval in principal shift-invariant spaces.IEEE Trans.on Signal Processing,2016,64(2):406–416.

[27]NETRAPALLI P,JAIN P,SANGHAVI S.Phase retrieval using alternating minimization.IEEE Trans.on Signal Processing,2015,63(18):4814–4826.

[28]SCHULZ T J,SNYDER D L.Image recovery from autocorrelations.Optical Society of America,1992,9(8):1265–1272.

[29]FIENUP J R.Reconstruction of an object from the modulus of its Fourier transform.Optics Letters,1978,3(1):27–29.

[30]PHILLIPS JD,CAIN SC.Image synthesis from aseries of coherent frames of pupil intensity.Proc of SPIE,2007:671207.