ZHOU Kai, LI Daojing, CUI Anjing, HAN Dong, TIAN He, YU Haifeng,DU Jianbo, LIU Lei, ZHU Yu, and ZHANG Running

1.National Key Laboratory of Microwave Imaging Technology, Aerospace Information Research Institute of Chinese Academy of Sciences, Beijing 100190, China; 2.School of Electronic, Electrical and Communication Engineening, University of Chinese Academy of Sciences, Beijing 100049, China; 3.Science and Technology on Electromagnetic Scattering Laboratory, Beijing Institute of Environmental Features, Beijing 100854, China; 4.General Design Department,China Academy of Space Technology, Beijing 100094, China

Abstract: The spaceborne synthetic aperture radar (SAR)sparse flight 3-D imaging technology through multiple observations of the cross-track direction is designed to form the crosstrack equivalent aperture, and achieve the third dimensionality recognition.In this paper, combined with the actual triple star orbits, a sparse flight spaceborne SAR 3-D imaging method based on the sparse spectrum of interferometry and the principal component analysis (PCA) is presented.Firstly, interferometric processing is utilized to reach an effective sparse representation of radar images in the frequency domain.Secondly, as a method with simple principle and fast calculation, the PCA is introduced to extract the main features of the image spectrum according to its principal characteristics.Finally, the 3-D image can be obtained by inverse transformation of the reconstructed spectrum by the PCA.The simulation results of 4.84 km equivalent crosstrack aperture and corresponding 1.78 m cross-track resolution verify the effective suppression of this method on high-frequency sidelobe noise introduced by sparse flight with a sparsity of 49% and random noise introduced by the receiver.Meanwhile, due to the influence of orbit distribution of the actual triple star orbits, the simulation results of the sparse flight with the 7-bit Barker code orbits are given as a comparison and reference to illuminate the significance of orbit distribution for this reconstruction results.This method has prospects for sparse flight 3-D imaging in high latitude areas for its short revisit period.

Keywords: principal component analysis (PCA), spaceborne synthetic aperture radar (SAR), sparse flight, sparse spectrum by interferometry, 3-D imaging.

1.Introduction

The spaceborne synthetic aperture radar (SAR) [1] forms a large equivalent aperture in the cross-track direction and obtains a 3-D image of the observed scenario [2,3] by multiple flights, which has important application value in the fields of terrain mapping and battlefield reconnaissance [4].Formation satellite can obtain multiple baselines in the cross-track direction through only one flight, which can effectively shorten the satellite revisit time, greatly increase the number of the flights and expend the synthetic aperture length of the cross-track direction.Therefore, with the development of small formation satellite [5−7], sparse flight will be the main way to realize 3-D imaging in the future.

At present, Tomographic SAR (Tomo-SAR) [8,9] imaging technology utilizes compressive sensing (CS)[10−12] in sparse signal processing theory to estimate the position and scattering intensity from the sparse observation data of the same scenario.This kind of method can solve the problem of signal reconstruction under sparse sampling of the cross-track direction, but in order to avoid the coupling of the echo signal between the crosstrack direction and the range direction, the length of the aperture in the cross-track direction is limited by the optimal vertical baseline length [13], which is usually less than 1 km and makes the corresponding resolution lower,which ultimately affects the quality of the 3-D image.For example, the equivalent cross-track aperture is nearly 270 m in Terra-SAR-X space-borne data imaging experiment and the corresponding 3 dB resolution is 33 m [14−16].

In this paper, combined with the actual triple star orbits (ATSO), the spotlight sparse flight 3-D imaging problem of spaceborne SAR is studied.Firstly, the ωKalgorithm is utilized to complete the large range migration correction in the cross-track direction and the 3-D imaging result of a 4.84 km equivalent cross-track aperture under sparse sampling is realized.Secondly, referring to the method of literature [17] to [21], interferometry is exploited to obtain a sparse frequency spectrum.In addition, principal component analysis (PCA) [22,23]is utilized to extract the main features in the image spectrum, and the 3-D image reconstruction result of the observation scenario can be obtained by inverse transformation of the reconstructed spectrum.Finally, we employ scenario simulation to verify the effective suppression of this method on high-frequency sidelobe noise introduced by sparse flight with a sparsity of 49% and random noise introduced by the receiver.Compared with the spaceborne SAR sparse flight 3-D imaging method based on back projection (BP) and CS [24], the method presented in this paper has low computational complexity as well as short running time.Furthermore, the simulation results of sparse flight with 7-bit Barker code [25,26] orbits(7BCO) are given as a reference and comparison for simulation results of sparse flight with ATSO to demonstrate t he significance of orbits distribution to imaging quality.

2.3-D imaging geometry

The 3-D side look imaging geometry under sparse flight of spaceborne SAR is depicted in Fig.1.In this geometry,the Cartesian coordinatex-y-zwith the center of the target scenario as origin is established, where thexaxis,yaxis, andzaxis denote the ground range direction, azimuth direction, and elevation direction of the 3-D scenario space, respectively.The satellite moves in a uniform motion and in a straight line along the azimuth direction with the speedV.In order to enlarge the synthetic aperture in the cross-track direction, the satellite has multiple flights in the ground range direction.

Fig.1 3-D imaging geometry under sparse flight of spaceborne SAR

Considering the existence of the incident angle, we set a 3-D imaging coordinate systems-y-rto form a side look imaging model in the ground range direction, where theraxis represents the radar line of sight direction (range direction), and thesaxis represents the vertical line of sight direction (cross-track direction), which is perpendicular to the azimuth-range plane, and theyaxis is consistent withyaxis in thex-y-zcoordinate system.θ shows the incident angle of the radar,Hshows the height of the satellite platform, andR0shows the closest distance between the radar and the centre of the scenario.The spaceborne SAR forms a synthetic aperture in the cross-track direction through the projection from the sparse flights in the ground range direction.

According to the length of the cross-track aperture formed by the sparse flights, we can describe the 3dB theoretical resolution of cross-track direction ρsand elevation direction ρhby

where λ denotes the radar wavelength,Lsdenotes the cross-track equivalent synthetic aperture length, anddmindenotes the minimum sampling interval of the sparse flights in the cross-track direction.According to the minimum baseline length of sparse flight, we can describe the maximum unambiguous rangesmaxin the cross-track direction,hmaxin the elevation direction, andxmaxin the ground range direction by

Assume that the number of full flights in the crosstrack direction isM, the number of sparse flights isN,and the sparse rate (without flights) can be defined as

Usually, the target has a small distribution range in the elevation direction, and the variation in the elevation of the ground object in the same azimuth-range direction cell is small.Therefore, it is possible that it can be sparsely sampled and imaged at a sample interval which is larger than the antenna size (such as the level of 100 m)in the cross-track direction.Consequently, when the distance resolution ρris high, the requirement for the unambiguous range in the elevation direction can be reduced.

While realizing the 3-D imaging by using the ωKalgorithm, the large sampling interval and imaging in the cross-track direction can be explained by the theory of spotlight imaging for small scenarios.In this situation, it is necessary to introduce de-chirp and up-sampling processing in the cross-track direction to remove obscurity introduced by large sampling interval.In microwave imaging system, whiledminindicates the sampling interval,the size of the scenario in the ground range direction should be smaller thanxmax.Assume that the antenna size in the cross-track direction isD, which is much smaller thandmin.As a result, the beam coverage in the ground range directionSxis defined as.When the beam coverage corresponding to the antenna size in the ground range direction and the distribution range of the targets in the ground are wide, the de-chirp and upsampling results in the cross-track direction of 3-D imaging target signals may be influenced by them.In order to avoid this influence, the range direction selection method could be introduced because of the high resolution in the range direction of spaceborne SAR.

3.Sparse flight sampling mode

Taking the actual satellite flight orbit design with a height of 500 km [24] as an example, part of orbits data in this literature are selected and formatted to ATSO, and the minimum interval among them is 110 m when they are set as transmission-self and receive-self mode.The number of satellites full flights tracks is 45 and the interval between two adjacent tracks is 110 m.Fig.2(a) shows the sparse flight of the ATSO in the cross-track direction,where the horizontal axis denotes the spatial position number of sparse flights in the cross-track direction and the vertical axis denotes the normalization time in the azimuth direction.The blue and white bars represent the effective and empty flights, respectively.And the number of sparse flight tracks is 23, and corresponding sparsity rate is 49% according to (3).Fig.2(b) delineates the normalized autocorrelation function (ACF) of this sparse flight sequence.It can be seen that this sequence has limited randomicity, its main-lobe energy is dispersive with a few cuspidal sidelobes.

Fig.2 Orbit distribution of ATSO and its ACF

As a comparison, Fig.3 depicts the orbit distribution and the ACF of sparse flight based on the 7-bit Barker code ([1110010]).Similarly, we set 45 tracks as full flights, and every six tracks are regarded as 1 bit in the 7-bit Barker code and the last three tracks are set as empty.There are 24 sparse flight tracks in all with a sparsity of 47%.It can be seen from Fig.3(b) that the main lobe energy of the ACF is concentrated and the sidelobes is low, which illuminates the randomicity of the 7BCO is better than ATSO.

Fig.3 Orbit distribution of 7BCO and its ACF

4.3-D imaging processing based on frequency domain PCA

4.1 Direct imaging

This paper uses the 3-D ωKalgorithm [27] as the imaging method (hereinafter called direct imaging, DI).Under the condition that the orbit distribution interval in the cross-track direction is relatively large, the de-chirp and up-sampling operation in the cross-track direction of echo data is necessary.Based on the sparse flight of ATSO, if the sparsely spatial sampled echo signal is directly imaged, part of the signal energy may leak into side lobes and is distributed in the entire frequency band, so the sparsely sampled DI results need to be further processed.

4.2 Sparse spectrum of interferometry and PCA processing

Although CS is a kind of sparse sampling imaging method, beside the disadvantages mentioned in the introduction, it also has high computational complexity as well as long running time [18].On the contrary, PCA technology is relatively simple in principle and implementation.Moreover, [13] has made it clear that PCA processing and CS processing have similar performance on reconstruction under sparse sampling.The PCA can simplify as well as find out the representative elements and structures of the original data through reducing its dimension to remove noise and redundancy.Therefore, the PCA can be introduced into the SAR image reconstruction process in sparse sampling situation.

Firstly, we assume that the 3-D DI result α in the spatial domain can be described as

whereArepresents the amplitude matrix of the image α and φrrepresents the initial phase matrix, and φdrepresents the phase matrix caused by the distance between the radar and the target.Due to the existence of φr, the frequency spectrum of the target scenario usually has a large bandwidth [28], which indicates that when sparsely sampled, the broad-band energy will be mixed with sidelobe noise [18], and the desired signal cannot be extracted completely and exclusively by the PCA.Fortunately,the initial phase φrcan be removed approximately by constructing a reference image αrefby the DI results of partial sparse flight, which can be described as

whereA′,, andrepresent the similar meanings in(4) of the reference image, respectively.Because of the approximate view angle, φrandare approximately equal, so after interferometry, the initial phase is strikingly eliminated

Hence, by interferometry, the sparse representation with concentrative energy in the frequency domain is achieved [18].The frequency spectrum can be divided into two parts: one is caused by side lobes and receiver noise, with small amplitudes and randomly distributed in the whole frequency band; the other is in the main lobe determined by the target scenario, with large amplitudes and concentrated in the low frequency band.We take the later as the main feature.

Secondly, we implement the PCA processing in frequency domain to extract principal components[29].Suppose thatB∈CNa×Nr×Ncis the 3-D frequency spectrum of image αinafter 3-D Fourier transformation, whereNa,Nr,Ncrepresent the frequency points in azimuth,range and cross-track directions, andBnr∈CNc×Nais an azimuth and cross-track 2-D frequency spectrum ofnrpoints in the range direction.Consider the covariance matrix ofBnrbetween different frequencies in the crosstrack direction isCnr∈CNc×Nc, then the eigen-decomposition can be described as

wherePis the eigenvector matrix ofCnrand λ1≥λ2≥···≥λnc≥···≥λNcare eigenvalues ofCnr.

Then we can chose the firstnc′big eigenvalues according to cumulative contribution of interest, and get the new frequency spectrumafter feature extraction by the firstnc′eigenvectorsPnc′.

By repeating the above operation at all range frequencies, the new 3-D spectrumB′ without sidelobe noise and receiver noise can be obtained.

Finally, the 3-D spatial domain image can be obtained by the 3-D inverse Fourier transformation (PCA after interferometry, hereinafter called IPCA).Fig.4 shows the flowchart of IPCA 3-D imaging processing under sparse sampling.Fig.4 (a) is the DI flowchart, and Fig.4 (b) depicts the whole processing flow in this paper.

Fig.4 Flowchart of IPCA 3-D imaging processing under sparse sampling

Additionally, it should be noted that the echo of the construction for reference image should be selected as continuous flights as possible to ensure that the reference image is not affected by sidelobe noise.If the selected flights contain many empty flights, it will introduce sidelobe noise into the main image, the corresponding reference image and the main image will be less correlated,and the interferometry effect will not be obvious, making the spectrum reconstructed by IPCA still contain much aliasing, which reduces the reconstruction quality of the 3-D image [24].

5.Simulation analysis

The simulation is carried out in the coordinate systems-yr, and the equivalent imaging model is spotlight downward looking.The observation scenario is set in coordinate systemx-y-z, and its size is 50 m × 50 m × 50 m.The scenario model contains a circular cone (20 m in radius and 50 m in height), as shown in Fig.5 (the system parameters in Table 1).In order to increase the complexity of target scenario, we add a straight liner target (50 m long,in red color) along thexdirection under the cone.

Fig.5 Observation scenario in x-y-z coordinate system

Table 1 3-D imaging simulation parameters of spaceborne sparse flight in side-look model

5.1 Sparse flights imaging results of 7BCO

Given the parameters and observed scenario above, Fig.6 illustrates the simulation results of DI via full flight(without sparse samling), DI and IPCA via sampling mode of 7BCO (depicted in Fig.3(a)).The 3-D imaging results (the second row in Fig.6) are depicted in thes-y-rcoordinate system, which can be transformed to thex-y-zcoordinate system by coordinate transformation.And the cross-track and along-track 2-D slices (the first row in Fig.6) are cut from 3-D imaging result corresponding to the same mid-position of cone in the range direction.

Fig.6 (a) indicates that the DI results via full flight are close to the true value of the target scenario, which can be used as a comparison for image reconstruction quality evaluation with sparse sampling.Through sampling mode of 7BCO, Fig.6 (b) can also be seen as acceptable results without too many high sidelobes, and the details as well as continuity of the observed scenario are also well preserved in the image.At this time, the IPCA method is utilized to reconstruct the results (Fig.6 (c)), where the DI result of the flights from the 1st to the 18th in Fig.3(a)is used as reference image for interferometry.It can be seen that scenario details are more abundant and the continuity is better compared with Fig.6(b).Meanwhile, the energy of the target contour is increased, which means t he sidelobe noise is effectively suppressed.

Fig.6 Different imaging results via full flight / sparse flight with sampling mode of 7BCO

5.2 Sparse flight imaging results of ATSO

As for actual satellite orbit, Fig.7 illustrates the simulation results of DI via full flight, DI and IPCA via sampling mode of ATSO (depicted in Fig.2 (a)).It can be seen that the DI results via sampling mode of ATSO(Fig.7(b)) are heavily aliased in the cross-track direction,and the scattering intensity of the target is drowned by the high sidelobe noise.Then, we use the IPCA method to reconstruct the results (Fig.7(c)), where the DI result of the flights from the 23rd to the 27th in Fig.2(a) is used as reference image for interferometry.Compared with Fig.7(a) and Fig.7(c), although the IPCA results have not reached the effect of DI results via full flight, the energy of the target contour is higher than DI results via sampling mode of ATSO.It means that it is easier to distinguish the target from the noise background, which verifies the effectiveness of the IPCA method on suppressing high-frequency sidelobe noise introduced by sparse flight.However, because of the influence of orbits distribution of the ATSO, the low-frequency sidelobe noise of the image is relatively large.Therefore, these sidelobe noise will be included in the main component and utilized to reconstruct the image when using the IPCA method, so the suppression effect on low-frequency sidelobe noise is not obvious.

Fig.7 Different imaging results via full flight / sparse flight with sampling mode of ATSO

In addition, compared with Fig.6, the IPCA method based on sampling mode of the 7BCO can obtain better reconstruction results because of its better orbit distribution depicted in Fig.3, but this kind of orbit is difficult to be used for spaceborne SAR in practice for its dense flights.In the future, in order to obtain better 3-D imaging results of spaceborne SAR sparse flight through the IPCA method, it is significant to optimize the orbit design to find a sparse sampling sequence with better orbit distribution and practical application.

5.3 Sparse flight imaging results with receiver noise of ATSO

To further illuminate the suppression effect of the IPCA method on noise introduced by receiver, Fig.8 illuminates the simulation results of DI via full flight, DI and IPCA via sampling mode of ATSO with artificially added random noise, which makes the SNR of raw data of receiver −36 dB, and the corresponding SNR of 2-D image of azimuth and slant range direction is higher than 15 dB, and 3-D image SNR is higher than 25 dB.Meanwhile, Fig.9 illustrates the simulation results before and after PCA processing of the cross-track and azimuth 2-D spectrum corresponding to the same range frequency.As a result, whether viewed from the spatial image or spectrum before and after IPCA, it can be seen that this method can effectively suppress the random noise introduced by the receiver and high-frequency sidelobe noise introduced by sparse flight, and reconstruct the spatial image or spectrum with the main energy only.

Fig.8 Different imaging results via full flight / sparse flight with sampling mode of ATSO with noise in echo

Fig.9 Cross-track and along-track 2-D spectrum before and after PCA processing with noise in echo

5.4 Image quality assessment

This paper utilizes root mean square error (RMSE) [30]to quantitatively evaluate the reconstruction results.RMSE is usually used to measure the deviation between the observed value and the true value, which is generally considered as

whereNdenotes the total number of SAR image resolution units,a˜ denotes the elevation position of the full flight imaging result, andadenotes the elevation position of the reconstruction result.Taking the full flight imaging result in Fig.6(d) and Fig.8(d) as the true value,it can be seen that the RMSE (shown in Table 2) of the IPCA imaging results are all better than DI imaging results regardless of the sparse flight mode or noise situation.Additionally, because of the better orbit distribution of 7BCO, the RMSE of its DI results and IPCA reconstruction results are both smaller than ATSO.This result verifies the effectiveness not only of the IPCA method on reconstruction results under sparse sampling,but also of good orbit distribution of sparse flight orbit on reconstruction results.

Table 2 Error analysis of 3-D imaging results under different imaging methods

6.Conclusions

For spaceborne SAR sparse flight 3-D imaging, the method based on interferometry and frequency domain PCA is presented in this paper to ameliorate imaging quality,with other situation results put into comparison.The simulation results of 4.84 km equivalent cross-track aperture and corresponding 1.78 m cross-track resolution show that the combination of sparse flight with ATSO and the image reconstruction based on interferometry and PCA can attain better image quality than the DI results under 49% sparse sampling by effective suppression on high-frequency sidelobe noise introduced by sparse flight and random noise introduced by the receiver.Due to the influence of orbit distribution of the ATSO, this paper also gives the simulation results of the sparse flight with the 7BCO, which indicates the significance of the optimized design of orbit distribution for spaceborne SAR sparse flight 3-D imaging results based on IPCA.