RESEARCH Open Access Compressive SAR raw data with

Wang et al. EURASIP Journal on Wireless Communications and Networking 2012, 2012:258http://jis.eurasipjournals.com/content/2012/1/258

RESEARCH Open Access

Compressive SAR raw data with principalcomponent analysisWei Wang, Baoju Zhang* and Jiasong Mu

Abstract

The theory of synthetic aperture radar (SAR) signal model is briefly introduced which is constructed with a series ofecho signals in range direction. The procedure of principal component analysis (PCA) is presented which is used astransformation basis to sparsify the signals. The joint compressive sensing (CS) and PCA algorithm is devised torealize SAR raw data compressive measurement. A SAR raw data for a point target is simulated and used to verifythe performance of the joint CS and PCA algorithm. The numerical experimental results demonstrate that the PCAmethod has good sparse performance and the joint CS and PCA algorithm is possible to online compressivelymeasure the SAR raw data.

Keywords: Synthetic aperture radar, Compressive sensing, Principal component analysis, Sparsity

IntroductionSynthetic aperture radar (SAR) is active and coherentmicrowave radar, which produces high spatial resolutionimages from a moving platform—an airplane or a satel-lite. SAR systems take advantage of the long-rangepropagation characteristics of radar signals and the com-plex information processing capability of modern digitalelectronics to provide high-resolution imagery. SARtechnology has provided terrain structural informationto geologists for mineral exploration, oil spill boundarieson water to environmentalists, sea state and ice hazardmaps to navigators, and reconnaissance and targeting in-formation to military operations [1]. And there are manyother applications or potential applications.A SAR receiver consists of a high-rate A/D converter

followed by pulse compression. The resolution of SARhas been limited by the sampling rate of the fastest avail-able A/D. And the performance of SAR system is furtherdegraded by the trade-off that exists between the rateof sampling and the number of quantization level ofan A/D. As the development of SAR imaging technol-ogy, the dimension of SAR imagery will become largerand larger and the resolution will become higher andhigher [2]. The traditional SAR data acquisition followsthe Nyquist sampling theorem and it increases the scale

* Correspondence: [email protected] of Physical and Electronic Information, Tianjin Normal University,Tianjin, China

© 2012 Wang et al.; licensee Springer. This is aAttribution License (http://creativecommons.orin any medium, provided the original work is p

of SAR system for obtaining more and more data. Thussome problem such as data processing and transmissionin real time would occur. Hence, how to compress theimagery data effectively during the acquiring processand reconstruct them accurately is a hot spot [3].A new theory of compressive sensing (also known

under the terminology of compressed sensing, compres-sive sampling, CS) in signal processing provides a funda-mentally new approach to data acquisition whichoverwhelm the common Nyquist sampling theory.Consider a signal that is sparse in some basis (oftenusing a wavelet-based transform coding scheme), thebasic idea of CS is projecting the high-dimensional signalonto a measurement matrix, which is incoherent withthe sparsifying basis, resulting to a low-dimensionalsensed sequence. Then with a relatively small number ofappropriately designed projection measurements, theunderlying signal may be recovered exactly [4,5]. In con-trast to the common framework of first collecting asmuch data as possible and then discarding the redundantdata by digital compression techniques, CS seeks tominimize the collection of redundant data in the acquisi-tion step. Because of the special advantages of CS, manyreferences about SAR imagery raw data compressing andreconstruction based on CS have been researched.Pruente [6] introduced CS to the field of ground

moving target indication using multichannel stripmapSAR-data. And the results showed that the approach

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was stable regarding noise and missing data. Althoughthe statistics of SAR images imply that there was nobasis or dictionary where the data could be assumedsparse, Rilling et al. [7] investigated two simple techni-ques, post processing and hybrid compressed sensing,to enhance the quality of the reconstructed SAR imagein the non-compressible areas which could not be welldescribed by a sparse approximation. Baraniuk [2] usedCS ideas for SAR imaging and proposed two potentialimprovements: (i) eliminated the matched filter (MF) inthe radar receiver and (ii) reduced the required sam-pling rate of the receiver A/D converter. And the authoralso indicated that there were a number of challengersto be overcome before an actual CS-based radar systemwould become a reality. Jiao et al. [8] proposed a newpursuit algorithm called Bayesian evolutionary pursuitalgorithm for the CS reconstruction problem of SARimages with weak sparsity. To reduce the amount ofstored SAR data, Sujit [1] proposed a method based onCS theory. According to the three steps in CS theory, in[3], DWT was utilized to make SAR imagery sparse andthe random Gauss matrix after approximate QR decom-position was employed to complete the low-dimensionmeasurement for sparse results based on the originalSAR imagery divided into several sub-imageries. Fur-thermore, a modified OMP algorithm was proposed toreconstruct SAR imagery. In [9], the echo model ofSAR imagery with multiple ships was converted into aCS model with orthogonal basis and used the CS-basedmethod to achieve higher azimuth resolution. In [10], asparse reconstruction method for SAR imaging basedon CS theory was presented and compared with thetraditional MF-based RD imaging method, the proposedmethod significantly suppressed the sidelobe and greatlyimproved the imaging performance of SAR when thetarget space was sparse.There are three main procedures should be executed

in CS theory which is the sparse expression of originalsignal, low-dimensional sampling, and precisely recon-struction of original signals. According to the above re-search achievements in SAR raw data compressingbased on CS theory, we can see that the SAR imagerydata usually have poor sparsity feature and looking forsuitable sparse transformation basis for SAR images isextremely significant. The two-dimensional SAR imageformed is interpreted in the directions of range and azi-muth. The range is the direction of signal propagationand the azimuth is the direction parallel to the flightpath. In this article, the SAR signals are thought as com-posed with distributed signals of the range samples andprincipal component analysis (PCA) can be used to findtransformations that sparsify the range samples. The re-mainder of the article is organized as follows. In the fol-lowing section, the SAR raw data model is analyzed. In

Section “Joint PCA and CS for SAR raw data compres-sion”, we summarized a mathematic description of CSand PCA theory and the sparse transform procedure forSAR raw data was proposed. Simulation results arepresented in Section “Simulation results and analysis”.Finally, the article is ended with some conclusion.

SAR signal model and imaging theorySAR is a high-resolution imaging radar technology. Forthe conventional standard SAR imaging, the high-resolution of range direction is obtained by the pulsecompression of chirp signal and the high resolution ofazimuth direction is received through the synthetic aper-ture formed by platform movement [11].

SAR signal modelSuppose SAR platform works in stripmap mode. TheSAR platform flights along a track parallel to azimuthdirection at altitude H with a constant velocity. Assum-ing the radar transmits a linear frequency modulated(LFM) pulse train as

f tð Þ ¼Xþ1

n¼�1rect t � nTrð Þ exp

� j2π f0 t � nTrð Þ½ � þ 12α t � nTrð Þ2

� � ð1Þ

where α is the LFM, f0 denotes carrier frequency, Tr ispulse repetition time and pulse width is τ, and rec(t)denotes the unit rectangular function.So, the SAR echo signal at time t can be expressed as

s tð Þ ¼ ∬Dσ x

0; r

0� �

Wα x� x0

� �Wr r � r

0� �

�f t � 2r x0; r

0� �c

!dx

0dr

0

ð2Þ

where D denotes the radar beam irradiation area, σ(x, r)represents the backscattering coefficients of target point(x,r), Wα :ð Þ, Wr :ð Þ are antenna gains of azimuth directionand range direction, respectively, and r(x, r) states thedistance between radar platform with target point (x, r).The actual received echo signals are still pulse train

and they have two kinds of time concept. The first oneis fast-time which is included in the complex amplitudeof the echo signal and reflects the echo signal variedtrend. The fast-time signal is the range dimension signalactually and determines the radar range resolution char-acteristics. The other one is slow-time which is includedin signal detention and reflects position changing ofradar platform. The slow-time signal is also called azi-muth dimension signal and determines the radar azi-muth resolution characteristics.


SAR imagingAssuming Nr and Na denote sample size of range andazimuth dimension signal, respectively, then the matrixof SAR echo signals can be written as

S ¼ ½s 1; 1ð Þ; s 1; 2ð Þ; . . . ; s 1;Nrð Þ; s 2; 1ð Þ; s 2; 2ð Þ; . . . ;�s 2;Nrð Þ; . . . ; s Na; 1ð Þ; s Na; 2ð Þ; . . . ; s Na;Nrð Þ�

ð3Þ

where s i; 1ð Þ; s i; 2ð Þ; . . . ; s i;Nrð Þ denotes the echo signalat time i which are sampled following Shannon’s cele-brated theorem. SAR imaging is to extract the two-dimensional distribution of target scattering coefficientσ(x, r) from echo signal train S. The most universal meth-ods for SAR imaging are based on MF theory to deal withrang and azimuth dimension signal separately, such asRang-Doppler algorithm, Chirp-Scaling algorithm, Spec-tral Analysis algorithm, etc. [10]. However, there are manyobvious defects in these traditional SAR imaging meth-ods. For instance, according to the Nyquist sampling the-ory, a large number of echo samplers must be gathered inorder to raise SAR range resolution which will lead to anexcessive burden of SAR acquisition and storage system.Therefore, how to realized high-resolution SAR fromlow-dimension samplers is extremely significant. Theemergence of CS theory makes it possible.

Joint PCA and CS for SAR raw data compressionIn this section, we first review the basic theory of CSand PCA and we subsequently illustrate the sparsetransformation procedure for SAR raw data.

Basic theory of CSRecently, CS has received considerable attention. Thetheory of CS states that a signal may be captured with asmall set of non-adaptive, linear measurements as longas the signal is sparse in some basis (such as DCT, wave-let) [12,13]. We can write in matrix notation:

y ¼ Φx ¼ ΦΨθ ¼ Θθ ð4Þwhere Φ is an M × N measurement matrix with M<N, y2 RM is the vector of samples observed. θ is the trans-form coefficient vector of signal x under orthonormalbasis ψ, i.e., θ ¼ ΨTx.Known the observed vector y and the measurement

matrix Φ, sparse transformation basis ψ, the recovery ofthe unknown signal x could be achieved by searching forthe l0 -sparsest representation that agrees with the mea-surements [14]:

θ_ ¼ argminkθk0

Subject to

y ¼ ΦΨθ ð5Þwhere the l0 pseudo-norm k � k0 corresponds to thenumber of non-zero elements. As it is well known,

this is a daunting NP-complete combinatorialoptimization problem which cannot be solved directlyin practice. Instead of solving the l0 -minimizationproblem, non-adaptive CS theory seeks to solve the‘closest possible’ tractable minimization problem, i.e.,the l1-minimization:

θ_ ¼ argminkθk1

Subject to

y ¼ ΦΨθ ð6ÞThis modification leads to a much simpler convex

problem, but it involves expensive computations whenapplied to large-scale signals. Therefore, a second ap-proach using iterative greedy methods has been pro-posed, such as matching pursuit (MP), orthogonalmatching pursuit (OMP), StOMP, Subspace Pursuit (SP),and CoSaMP.

PCAFrom the recovery algorithm of CS, it can be shown ob-viously that the appropriate sparse transformation basisψ can improve recovery precision and reduce the com-putations. In this article, the PCA is the technique pro-viding the transformation matrix ψ.PCA is a mathematical procedure that uses an orthog-

onal transform to convert a set of observations of pos-sibly correlated variables into a set of values of linearlyuncorrelated variables called principal components. Thenumber of principal components is less than or equal tothe number of original variables.Assuming xi 2 RN is the samples at discrete time

i=1,2,. . .,K with a fixed sampling rate. The measure-ments mean vector and covariance matrix of xi can bedefined as

�x ¼ 1K

XKi¼1

xi

C ¼ 1K

XKi¼1

xi � �xð Þ xi � �xð ÞT ð7Þ

Considered the above equations, the orthonormalmatrix P can be constructed whose columns are the uni-tary eigenvectors of covariance matrix C, placed accord-ing to the decreasing order of the corresponding eigenvalues. Then we can define

zi ¼ PT xi � �xð Þ ð8Þwhere PT is the inverse matrix of P. Due to the construc-tion of P, we have the entries of zi are ordered as follows:zi 1ð Þ≥zi 2ð Þ≥⋯≥zi Nð Þ If the instances x1; x2; . . . ; xK arecorrelated, there exists an M<N such that j >M we have


zi jð Þ is negligible. In other words, the N-dimensionalvector zi can be considered as an M sparse vector andthe value of M depends on the level of correlationamong the gathered samples x1; x2; . . . ; xK [15,16].

Joint CS and PCA for compressive SAR imagingGiven the SAR imaging theory and Equation (3), we canrewrite the echo signals as si 2 CNr ; i ¼ 1; 2; ::::;Na , si ¼s i; 1ð Þ; s i; 2ð Þ; . . . ; s i;Nrð Þ½ � . Due to the echo signals re-flect the backscatter coefficients of target, the instancess1; s2; . . . ; sNa are temporally correlated, that is, the PCAalgorithm can be used to sparsify the echo signal. Con-sidered that the echo signals are complex, we can writethe compressive measurement of SAR echo signalthrough CS and PCA as follows.

yrei ¼ Φ Re sið Þ � Re �sð Þ½ � ¼ ΦPreθrei ð9Þ

yimi ¼ Φ Im sið Þ � Im �sð Þ½ � ¼ ΦPimθimi ð10Þ

The real and imaginary part of echo signal is calcu-lated, respectively. Where �s denotes the mean vector ofSi, y

re, yim denote the low-dimensional measurements,Pre, Pim represent the orthonormal transformation basis

constructed based on Equation (7), θrei and θimi are thesparse coefficients.As above, we can obtain the recovery sparse vector

θire and θiim through acquired low-dimensional mea-surements and recovery algorithms such as OMP. Thenthe estimations of si can be written as follows.

si ¼ Re sið Þ þ jIm sið Þ¼ Re �sð Þ þ Preθ

re

i

h iþ j Im �sð Þ þ Pimθ

im

i

h ið11Þ

In short, we can achieve high-resolution SAR fromlow-dimension samplers based on joint CS and PCAschemes. Furthermore, it is noted that real SAR echosignals are in general non-stationary. This means thatthe statistics that we need to use in the PCA must beacquired at runtime and it makes that an online estima-tion of these parameters possible.

Simulation results and analysisTo illustrate the effectiveness of the proposed joint CSand PCA algorithm, we test its performances on simu-lated SAR imaging of point target. In the experiments,we chose random Gaussian matrix for measurementmatrix Φ and OMP for recovery algorithm. Due to therandomness of measurement matrix Φ, all the experi-ment results are the mean value of ten times computa-tions. All the computations are run on a 2.60-GHz InterCore i5 laptop with 4-GB RAM.

In this section, we simulate the SAR echo signals froma point target with carrier frequency f0 ¼ 1 GHz , pulserepetition time Tr ¼ 0:02 s , and pulse width τ ¼ 5 μs .In Figure 1, we show the real part of SAR raw data withNr=Na= 512, SAR image after range compression andSAR image after range and azimuth compression.To test the feasibility of the joint CS and PCA scheme,

we test the performance of the algorithm based on theabove real part of SAR raw data. In Figure 2, it is showedthat the trend of reconstruction error versus the num-ber of CS observations. The reconstruction error is

defined asE¼k Re Sð Þ�Re Sð Þð Þk2

kRe Sð Þk2 , where S is the original

matrix of SAR echo signals according to Equation (3)and Ŝ is the reconstructed matrix of SAR echo signalson the basis of Equations (3) and (11). The number ofCS observations is the amount of random measuringwith CS measurement matrix Φ for per echo signal si.From Figure 2, it can be seen clearly that we can gethigh compression effect with CS and PCA and the PCAalgorithms have good performance to sparsify the SARraw data.As is clear from the PCA procedure, the statistics

(sample mean and covariance matrix) must be gener-ated at runtime. Hence, in order to implement jointCS and PCA for real signals, we should implementour algorithm at two phases: training phase and subse-quent monitoring phase. In training phase, we collecta series of signals to compute samples mean andorthonormal transformation matrix. Then, in the mon-itoring phase, it is compressive measurement of subse-quent received signals using the statistics computed inthe previous phase [15]. To test the effective of the al-gorithm of the two phases, we simulate the trend ofreconstruction error E versus the number of trainingsignals. In Figure 3, with the constant number of CSobservation, the simulation results show that the ten-dency of reconstruction error is decreasing with theincreasing amount of training signals. And we can alsosee that the fluctuation is smaller when a properamount of training signals is used to compute the sta-tistics. From the experimental results, we can stressthat the joint CS and PCA algorithm can readily beapplied to online compressively measuring SAR rawdata.

ConclusionIn this section, we analyze the structure of SAR imagingdata which can be processed through range directionand azimuth direction, separately. And the PCA methodis used to sparsify the range direction raw data of SARand the data are compressed by CS theory. From thesimulation results, it demonstrates that the PCA method

Figure 1 SAR data of point target. (a) Real part of SAR raw data. (b) Compressive SAR images.


10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4

The number of CS Observation

Rec

onst

ruct

ion

Err

or

Figure 2 Performance of joint CS and PCA for SAR raw data.


has good sparse performance for SAR raw data whichare temporal correlated in range dimension and the jointCS and PCA algorithm is possible to online compressivemeasure the SAR raw data.However, there are some challenges to be overcome

before a high efficient joint CS and PCA algorithm

100 150 200 250 3000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

The column number

reco

nstr

ucti

on e

rror

Figure 3 The outline performance of joint CS and PCA algorithm.

for real-time SAR image compression become a real-ity. First, how to decide an appropriate amount ofecho signals to determine the statistics should beanalyzed in order to lessen computation burden. Sec-ond, for a large number of SAR echo signals, weshould decide which part of echo signals is temporal

350 400 450 500 550 for compute statistics


correlated, and it is the premise to use joint CS andPCA algorithm.

Competing interestsThe authors declare that they have no competing interests.

AcknowledgmentsThis research was supported by the High Education Science & TechnologyFoundation Planning Project of Tianjin (20100716), the Tianjin YoungerNatural Science Foundation (12JCQNJC00400), and the Tianjin NaturalScience Foundation (10JCYBJC00400).

Received: 18 July 2012 Accepted: 24 July 2012Published: 16 August 2012

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doi:10.1186/1687-1499-2012-258Cite this article as: Wang et al.: Compressive SAR raw data withprincipal component analysis. EURASIP Journal on Wireless Communicationsand Networking 2012 2012:258.

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RESEARCH Open Access Compressive SAR raw data with