sar imaging via modern 2-d spectral estimation …...ieee transactions on image processing, vol. 7,...

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 7, NO. 5, MAY 1998 729

SAR Imaging via Modern 2-DSpectral Estimation Methods

Stuart R. DeGraaf,Member, IEEE

Abstract—This paper discusses the use of modern two-dimen-sional (2-D) spectral estimation algorithms for synthetic aper-ture radar (SAR) imaging. The motivation for applying powerspectrum estimation methods to SAR imaging is to improve reso-lution, remove sidelobe artifacts, and reduce speckle compared towhat is possible with conventional Fourier transform SAR imag-ing techniques. This paper makes two principal contributionsto the field of adaptive SAR imaging. First, it is a comprehen-sive comparison of 2-D spectral estimation methods for SARimaging. It provides a synopsis of the algorithms available,discusses their relative merits for SAR imaging, and illustratestheir performance on simulated and collected SAR imagery.Some of the algorithms presented or their derivations are new,as are some of the insights into or analyses of the algorithms.Second, this work develops multichannel variants of four relatedalgorithms, minimum variance method (MVM), reduced-rankMVM (RRMVM), adaptive sidelobe reduction (ASR) and spacevariant apodization (SVA) to estimate both reflectivity intensityand interferometric height from polarimetric displaced-apertureinterferometric data. All of these interferometric variants arenew. In the interferometric context, adaptive spectral estimationcan improve the height estimates through a combination of adap-tive nulling and averaging. Examples illustrate that MVM, ASR,and SVA offer significant advantages over Fourier methods forestimating both scattering intensity and interferometric height,and allow empirical comparison of the accuracies of Fourier,MVM, ASR, and SVA interferometric height estimates.

Index Terms—Adaptive imaging, superresolution, syntheticaperture radar, 2-D spectral estimation.

I. INTRODUCTION

SYNTHETIC aperture radar (SAR) imaging can be viewedas a parameter estimation problem in which one seeks

to estimate the scene reflectivity intensity versus slant-planelocation, i.e., an intensity image. Interferometric SAR systemsalso seek to estimate scattering height out of the slant-plane,which is proportional to the phase difference, pixel by pixel,between registered images formed from a pair of coherent,vertically displaced measurement apertures. Here we discussthe limitations of conventional Fourier methods for estimatingintensity and interferometric height images, and the rationalefor employing alternative two-dimensional (2-D) spectral esti-mation methods. Section II provides a synopsis of 2-D spectral

Manuscript received July 6, 1995; revised January 30, 1997. This workwas supported by the Advanced Research Projects Agency (DOD), AdvancedSystems Technology Office, under ARPA Order A284, issued by U.S. ArmyMissile Command under Contract DAAH01-93-C-R178. The associate editorcoordinating the review of this manuscript and approving it for publicationwas Dr. John W. Adams.

The author is with the Northrop Grumman Corporation, Electronic Sensors& Systems Division, Baltimore, MD 21203 USA (e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(98)03081-4.

estimation algorithms, discusses their relative merits for SARimaging, and compares scalar imagery produced by thesealgorithms for both synthetic point scattering data and datacollected of two commercial ships near Toledo, Ohio. Thissection also presents a theoretical model for the impact ofadaptive sidelobe reduction (ASR) filter order and constrainton target-to-clutter ratio (TCR) that sheds light on strate-gies for selecting these parameters. Section III generalizesfour algorithms—minimum variance method (MVM), reducedrank MVM (RRMVM), ASR, and space variant apodization(SVA)—for application to interferometric (and polarimetric)data, and uses data collected of the area around the Universityof Michigan football stadium to illustrate that MVM, ASRand SVA (but not RRMVM) offer significant advantages overFourier methods for interferometric SAR imaging. This sectionalso provides an empirical comparison of the accuracies ofFourier, MVM, ASR, and SVA interferometric height esti-mates, and discusses their impact on absolute and relativecontrast. Section IV summarizes and draws conclusions. AllSAR data shown were collected by the Wright Laboratory-ERIM Data Collection System (DCS) SAR, which is installedon a CV-580 aircraft, and operates at a variety of frequencybands, resolutions, and polarizations, in spotlight and stripmapmodes. The DCS supports research in multichannel SAR,where the channels can represent multiple frequencies, mul-tiple polarizations, and/or multiple interferometric apertures.Finally, the Appendix contains a table that summarizes thenumerous acronyms used throughout the text.

Fourier SAR imaging exploits the Fourier transform pairrelationship between signal history measurements (polar-to-rectangular [1] or range migration [2], [3] formatted) andscene reflectivity. Reference [3] provides a comprehensivediscussion of SAR imaging methods. Fourier imaging exhibitsseveral drawbacks for imaging interferometric intensity andphase/height. First, as the collection apertures are of finite sizein -space (wavenumber space), the spatial resolution affordedby Fourier imaging is inherently limited. Typically, Taylor orKaiser–Bessel weightings are employed to control impulseresponse (IPR) peak sidelobe and integrated sidelobe level.The artifacts (poor resolution and/or sidelobe artifacts) inducedin Fourier SAR imagery by a fixed system IPR are oftenundesirable. Second, finite resolution leads to the classicalcoherent imaging speckle phenomenon, which is caused byscintillation of independent unresolved scattering elements.Complex circular white Gaussian noise, with varianceis a common signal history domain model for the scatteringfrom a patch of homogeneous clutter. The corresponding

1057–7149/98$10.00 1998 IEEE

730 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 7, NO. 5, MAY 1998

unweighted (and not zero-padded) Fourier transform imageis also complex circular white Gaussian noise. The magnitudeof this Fourier image is Rayleigh distributed, with mean andstandard deviation that are both proportional tothe intensityis exponentially distributed. The classic description of SARimage speckle as “multiplicative noise” stems from the factthat the Fourier transform is not an appropriate estimatorfor On the other hand, a power spectral density (PSD)estimator is appropriate for estimating PSD estimatorsare also appropriate for estimating the scattering intensityof deterministic scatterers. Third, thermal noise, horizontallyunresolved scatterers, and IPR integrated sidelobe level allcontribute to interferometric phase noise which degrades theaccuracy of interferometric height estimates. Generally, heightaccuracy improves with increasing signal-to-noise ratio (SNR).In this case, we are referring to thermal noise that is inde-pendent in the two interferometric apertures; clutter, whichis also a noise process, is very strongly correlated betweenthe apertures. Often, some form of smoothing or filtering isnecessary to reduce the thermally induced phase/height noiseassociated with clutter and target scattering.

Modern spectral estimation techniques offer attractive alter-natives to Fourier SAR imaging. These nonlinear techniquesoffer the promise of improved resolution and contrast, andreduced speckle. Improvements in resolution and reductions insidelobe artifacts arise through adaptive interference nulling,linear predictive modeling, signal-clutter subspace decompo-sition, or parametric sinusoidal signal history (point scatterer)modeling. Speckle reduction arises through the signal historydomain averaging implicit in PSD image estimation. Contrastimprovement arises through signal-clutter subspace decom-position or algorithm singularities. Similarly, interferometricvariants of some of these algorithms offer the promise of re-duced phase/height noise and improved phase/height accuracythrough enhanced resolution and averaging.

II. 2-D SPECTRAL ESTIMATION ALGORITHMS

In this section, we summarize the available 2-D spectralestimation algorithms, review the theory behind them, anddiscuss their relative advantages for SAR imaging. In addition,we illustrate their performance on both simulated and collectedSAR data.

A. Synopsis of Algorithms

Table I summarizes the rationale and formulation of avariety of 2-D spectral estimation algorithms. The table groupsthe algorithms on the basis of their rationale. Many of thesealgorithms are discussed in [4]–[7], and in a vast spectral es-timation and array processing literature, for example [8]–[32].However, RRMVM [44], ASR [45], [46], and SVA [48], [49]are new, as is much of the literature describing the applicationof spectral estimation algorithms to radar cross-section (RCS)analysis and imaging [33]–[49]. Moving from the top of thetable toward the bottom, the degree to which the algorithmsexploit a point scattering (sinusoidal signal history) modelincreases. As the algorithms exploit this model to a higherdegree, their resolution and accuracy improves, provided the

model is valid. However, the high-performance algorithms areless robust when the model is compromised. While FourierSAR imagery is often characterized by “prominent points,”the sinusoidal signal history model can be compromised inthe cross-range dimension by a variety of common physicalphenomena: glints, sliding speculars, creeping waves, reso-nances and motion-induced phase errors. Similarly, frequency-dependent scattering amplitude, characteristic of certain typesof scattering, can compromise the sinusoidal model in therange dimension for systems that exploit a large fractionalbandwidth.

Some of the mathematical notation in Table I is fundamen-tal, and is defined here. The rest is defined in the subsectionsthat describe the individual algorithms. The elements of thesignal history data vector are the rectangularly formatted2-D radar signal history samples. (We assume that the resam-pling involved in polar format or migration processing hasalready occurred, and do not address it here.) One can eitherraster-scan the Cartesian signal history samples into the signalhistory vector, or choose another convenient lexicographicordering. The elements of vector are the samples ofthe complex-valued 2-D unit sinusoid that corresponds tothe scattering from a point target at locationthe elementsconstitute the exponential coefficients of a 2-D Fourier trans-form tuned to spatial location If the 2-D signal historydata is raster-scanned into the vector then canbe described as a Kronecker product of one-dimensional (1-D) Fourier transform vectors. Obviously, the lexicographicordering of the elements of and must be consistent.Signal history correlation matrix represents an estimateof the expected matrix The following paragraphsprovide an overview of the algorithms to establish theirconceptual relationships and discuss issues that are commonto the algorithms.

The fast Fourier transform (FFT) and ASR algorithm bothproduce coherent (complex-valued) imagery. These coherentimages represent the outputs of banks of 2-D narrowbandfilters, where each filter output is tuned to a given spatiallocation. The images are a convolution between the scene re-flectivity and IPR. The FFT filters are fixed, and the FFT IPR isspace invariant. The ASR filters are adaptive, and the ASR IPRis space variant. SVA is a special case of ASR that employs asingle degree of freedom, together with a constraint motivatedby the oscillatory nature of the sidelobes of a sinc IPR.

The periodogram, MVM, and RRMVM techniques all pro-duce power spectral density (positive semidefinite, real-valued)imagery. These PSD images represent the average, or expectedvalue, of the output energies of banks of 2-D narrowbandfilters, where each filter output is tuned to a given spatiallocation. The periodogram filters are fixed, while the MVMand RRMVM filters are adaptive. In each case, a correlationmatrix whose entries are estimates of the correlationsbetween signal history domain data samples, must be estimatedfrom the signal history data. MVM requires a full-rank,nonsingular correlation matrix estimate, which implies a largeamount of averaging, while RRMVM and the periodogramcan accommodate a reduced-rank, singular correlation matrixbased on a small amount of averaging.

DEGRAAF: SAR IMAGING 731

TABLE ISYNOPSIS OF 2-D SPECTRAL ESTIMATION ALGORITHM RATIONALE AND FORMULATION

RRMVM, ASR, and SVA share the spirit of MVM in thatthey seek to maximize adaptively the signal-to-interferenceratio (SIR). However, they are “singular” methods in that theyoptimize SIR on the basis of low-rank (unit-rank for ASRand SVA) signal history correlation matrices. In sequence,RRMVM, ASR, and SVA utilize progressively fewer adaptivedegrees of freedom, and computational complexity decreasesdramatically. Stankwitz [49] has developed a nonlinear band-width extrapolation technique based on the premise that SVApreserves sinc IPR mainlobes, but eliminates their sidelobes.Super SVA combines inverse filtering with this nonlinearbehavior of SVA to obtain bandwidth extrapolation.

Eigenvector (EV) and multiple signal classification (MU-SIC) methods are signal-clutter subspace decomposition vari-ants of MVM that cause the image peaks corresponding tohigh target-to-clutter ratio (TCR) point scatterers to becomevery sharp and bright. MUSIC explicitly whitens, or equalizes,the clutter eigenvalues, while EV does not.

Autoregressive linear prediction (ARLP) methods predictsignal history samples as linear combinations of the neigh-boring signal history samples, and select the predictor filtercoefficients to minimize average prediction error. Based onthe assumption that the prediction error signal is an innova-tions process, i.e., white noise, the PSD estimate equals theminimized prediction error energy divided by the magnitudesquared of the transfer function. ARLP imagery benefits fromreciprocal root mean square (RRMS) averaging across multipleprediction elements to reduce spurious peaks and improvecontrast. RRMS averaging of the ARLP spectra across all pre-diction elements yields one of Pisarenko’s spectral estimates.

The Tufts–Kumaresan ARLP (TKARLP) method is basedon a signal-clutter subspace decomposition in the frameworkof ARLP. In this case, the clutter contribution to the correlationmatrix is omitted to boost the apparent TCR. Further, theTKARLP prediction filter is chosen on the basis of the pseudo-inverse of the signal portion of a singular correlation matrix,which allows larger subaperture sizes to be used, therebyimproving resolution. TKARLP imagery also benefits fromRRMS averaging across multiple prediction elements.

In those instances where a point scattering model is validin both range and cross-range dimensions, the parametricmaximum likelihood (PML) [18], [19], [22], [26] methodprovides extremely accurate estimates of both the locationof scattering points and their complex scattering amplitudes.This method is not suitable for general SAR imaging, butmay be valuable for specialized SAR analysis tasks, includingautomatic target recognition (ATR).

Four methods can be employed to estimate the correlationmatrix from the signal history data [4], [5]: the covari-ance method (subaperture averaging), the modified covari-ance method (forward-backward subaperture averaging), thebiased or unbiased correlation method (block-Toeplitz en-forcement), and, for oversampled data, decimation averaging.Block–Toeplitz enforcement can be combined with the co-variance or modified covariance methods. Unfortunately, thecorrelation method yields poor results for SAR imaging (andmany other spectral estimation applications), which negatesits computational appeal. We employ the modified covariancemethod of averaging in conjunction with all 2-D spectralestimation algorithms that utilize a correlation matrix.


Fig. 1 illustrates that the unidirectional (forward) correlationmatrix estimate is an average of

outer-products which represent all possible overlappingsample subapertures within the full sample

aperture

(1)

Recall that the individual subaperture vectors are formed byraster-scanning the sample 2-D subapertures into

element vectors. For forward-backward averaging, thecorrelation matrix estimate is an average of outer-products,

(2)

where resembles an identity matrix, but the diagonal isoriented from NE to SW rather than NW to SE;represents the reversal (rotation by 180of subapertureForward-backward averaging exploits the fact that a 2-Dsinusoid evolves in one spatial direction in the same manner asthe conjugate sinusoid evolves in the opposite spatial direction,while conjugating and reversing the clutter/noise contributioneffectively yields an independent realization, thereby doublingthe amount of averaging that occurs.

Most SAR clutter exhibits texture over large spatial scales.Nevertheless, treating returns from locally homogeneous clut-ter (such as unresolved blades of grass or tree canopy) aswhite noise enables us to invoke the well-known theoryof Fourier SNR processing gain to understand SAR imagecontrast, as well as the impact of subaperture averaging onSAR image contrast. SAR imaging engineers generally defineTCR in the image domain as the ratio of a target peakintensity to the surrounding clutter variance, which can beviewed as a measure of contrast. On the other hand, spectralestimation researchers typically define TCR (or SNR) in thesignal history domain, where it represents the ratio of sinusoid(point scatterer target) squared amplitude to total clutter (noise)variance. The 2-D sample FFT normally usedto form a SAR image adds the point scattering sinusoidphasors coherently, but adds the independent clutter samplesincoherently. Thus, the image domain TCR is timesthe signal history domain TCR. The improvement factor of

the area of the aperture, is usually known as theSNR processing gain. Here, for obvious reasons, we adoptthe terminology TCR gain. Based on asymptotic theory, it iswell known that the TCR gains of MVM and ARLP equalthe subaperture area and that the resolutioncapability of these algorithms improves with increasing TCRand subaperture size. Thus, we have two compelling reasonsto employ large subapertures in obtaining our correlationmatrix estimate. At the same time, we need to performenough averaging to obtain nonsingular and statistically robustestimates. The resolution and TCR gain of EV and MUSICincreases when the accuracy of the correlation matrix estimate

Fig. 1. Correlation matrix estimate is average of subaperture outer products.All possible subapertures are extracted from the full aperture.

improves [15]; for this reason, forward-backward averaginghas a particularly positive impact on EV and MUSIC imagecontrast (much more so than with MVM).

In the context of 2-D SAR data, we have found em-pirically that subaperture sizes of 40–50% (i.e.,

work best for MVM, Pisarenko,EV, and MUSIC, while 50–60% subapertures work best forTKARLP. In conjunction with forward-backward averaging,40% subapertures provide a factor of 4.5 times as manysubapertures in the correlation average as the dimension ofthe correlation matrix; with 50% and 60% subapertures, thefactors are 2 and 0.889, respectively. Using 40–50% forward-backward subapertures with ARLP leads to poor contrastdue to a large number of spurious peaks; however, RRMSaveraging over prediction element, a la Pisarenko’s method,reduces the occurrence of these spurious peaks dramatically.

In practice, all of the methods that require evaluation,inversion, or eigendecomposition of a full-rank signal his-tory correlation matrix are computationally intensive, withcomplexity on the order of for typical amountsof averaging, Tufts–Kumaresan ARLP shares this order ofcomputational complexity. To apply these algorithms to typicalSAR scenes, it is necessary to employ a decimation andmosaicing strategy. As shown in Fig. 2, the signal history isdecimated down by factors and in range and cross-range dimensions. Decimation is performed so as to obtainthe downsampled signal history measurements that correspondto a series of small overlapping subimages or regions ofinterest (ROI) within the entire SAR scene. Neglecting thesmall overlap, the decimation factors correspond to the numberof image chips in range and cross-range. The computationalcomplexity for each subimage drops by a factor ofIf the subimages are computed serially, then the overallimprovement factor drops to Of course, the individ-ual chips can be computed in parallel. Overlapped polyphasebandpass filterbanks provide better mosaicing performancethan the Fourier approach illustrated in Fig. 2, and allow useof smaller chip sizes.

B. Mathematical Algorithm Formulations

Following is a brief mathematical review of each of theadaptive SAR imaging methods. The derivation presented herefor ASR is new in that it solves the joint I/Q optimization prob-lem exactly, rather than approximately as in [45]. In addition,


Fig. 2. Decimation/mosaicing strategy employed to evaluate large adaptive images. Downsampled signal histories corresponding to each image chip areimaged adaptively and mosaiced into a composite image. In practice, the chips overlap, so that edge effects can be trimmed. Each chip signal historyyields a correlation matrix by averaging subapertures, as in Fig. 1.

the new derivation generalizes ASR and SVA for multichannelapplications. This paper does not discuss the separate I/Qchannel implementations of ASR and SVA, since they areneither well suited for interferometric imaging nor optimumfor multichannel imaging. While RRMVM was developed si-multaneously and in parallel by the author and Benitz [44], themathematical details of the algorithm have not been publishedbefore. Our presentation of ARLP and TKARLP methodsemphasizes the twist of varying the prediction element, andleads to a previously unreported connection between ARLPand one of Pisarenko’s methods. Varying the ARLP predictionelement has been touched upon in [7], [13], [27]. Also, theauthor is unaware of any previous work that describes varyingprediction element in conjunction with TKARLP. The remain-ing algorithms and their brief derivations and algorithms arenot new, but are included because they provide the foundationfor multichannel interferometric imaging (MVM) or are usefulfor completeness, logical flow, and consistency of presentation(FFT, periodogram, EV, and MUSIC, and PML).

1) Fourier Transform: Fourier transform image formationevaluates a linear combination of the signal history samplesof the form is a real-valued diagonal matrixwhose entries correspond to a separable or nonseparableweighting function used to control the IPR mainlobe/sidelobetradeoff. Unweighted FFT image formation corresponds toevaluating a bank of matched filter outputs, each filter beingmatched to a particular spatial location. In the simple case of asingle sinusoid in white Gaussian noise or clutter, this matchedfilter maximizes SIR. However, in more complicated scenarios,energy from scatterer B can leak through the sidelobes (or evenmainlobe) of the IPR tuned to scatterer A, thereby corruptingthe estimate of energy scattered from A.

2) Periodogram: The periodogram recognizes the stochas-tic nature of the clutter data and seeks to estimate the average

power output of the Fourier transform, i.e., the power spectraldensity One can viewthis averaging as the logical extension of “independent lookaveraging,” which is often performed in SAR to reduce imagespeckle. Of course, subaperture averaging entails a loss ofresolution because the subapertures are smaller than the fullaperture. For this reason, the periodogram is of little practicalinterest.

3) Minimum Variance Method:To maximize expected SIRat each spatial location, Capon’s minimum variance method[7] evaluates a different linear combination of the signalhistory samples, of the form where the space-variantweighting vector is complex-valued. Both the amplitudeand phase of the components of can differ from thoseof the weighted Fourier transform vector Thus MVMdoes more than merely change the real-valued weights in aFourier transform. To insure that the sinusoidal signal returnedfrom a point scatterer at location is passed with unit gain,MVM imposes the constraint Since theoutput consists of desired and undesired energy, and sincethe signal passes with unit gain, MVM maximizes SIR byselecting to minimize the expected output energy,

(3)

Solving the constrained optimization via the method of La-grange multipliers yields

(4)

and corresponding optimized output energy, or spectral esti-mate

(5)


Evaluating and inverting the correlation matrix dominatesthe computational complexity of MVM; subsequent evaluationof the quadratic form in (5) involves computing a pair of 2-DFFT’s.

4) Reduced Rank Minimum Variance Method:Thereduced-rank minimum variance method [44] also providesa power spectral density estimate that maximizes theexpected SIR. While MVM does this on the basis of anonsingular, invertible correlation matrix, RRMVM seeks todo this on the basis of a singular, noninvertible correlationmatrix. RRMVM circumvents the difficulty of a singularcorrelation matrix by invoking an additional constraint,namely, The impact of theadded constraint is to add a scaled identity matrix to thesingular correlation matrix. Thus the optimum RRMVMweight vector is

(6)

where the Lagrange multiplier satisfies

(7)

Substituting expression (6) into definition (3) yields theRRMVM PSD image

(8)

In the case of full-rank MVM, represents thereciprocal of TCR gain against white clutter. Thus, one caninterpret the factor in the new constraint as a factor by whichRRMVM can degrade white TCR gain in its efforts to optimizeSIR. In conjunction with singular correlation matrices, theeffect of the new constraint is to prevent the output energyfrom always being zero in spite of the original unit-gainsignal constraint. Use of allows the algorithm nodata adaptation; in this case and the weight vectoris colinear with In conjunction with signal historyaveraging, causes RRMVM to reduce to the unweightedperiodogram. In the case of a unit-rank correlation matrix (asingle full aperture, no averaging), choosing causesRRMVM to further degenerate to the magnitude-squared ofthe unweighted Fourier transform. Use of allows dataadaptation, i.e., weighting vectors that are not colinear with

As is clear from comparing (8) and (5), when averagingis sufficient to insure a full-rank correlation matrix, relaxingthe new constraint, i.e., using and causesRRMVM to degenerate to MVM.

We define the data subaperture matrixto have columnscorresponding to the small number of forward-backward datasubapertures, such that It is computationallyadvantageous to evaluate the right singular vectors andsingular values of or equivalently, theeigendecomposition of which is dimension-ally smaller than and where Exploiting the ma-trix inverse lemma

we obtain

which, when substituted into (7) and (8) forms the basis of thefollowing efficient computational procedure.

1) Evaluate “kernel” correlation matrix2) Evaluate eigendecomposition3) Evaluate eigen images, i.e., Fourier transforms of data

mapped onto eigenvectors,4) Evaluate Lagrange multiplier by solving

5) Evaluate output

When a small amount of averaging is performed, as isthe intent of the algorithm, evaluating the eigen images,step (3), dominates both the computational complexity andstorage requirements of RRMVM. The burden of evaluatingand storing all of the eigen images makes RRMVM unattrac-tive when a significant amount of averaging is performed.However, our experience suggests that, in very low-rankscenarios, in which either a handful of forward-backwardnearly full-aperture subapertures or a handful of forward-backward nonoverlapping subapertures are used, the dominanteffect of RRMVM is to threshold the corresponding un-weighted (sinc IPR) Fourier image. In scenes where thescattering intensity spans a large dynamic range, selectinga constraint that is large enough to eliminate the sidelobeartifacts of prominent scatterers causes weaker scatterers tobe eliminated. In conjunction with limited averaging, wehave observed significant resolution enhancement by RRMVMonly in simple, simulated scenarios. However, in cases wherethe dynamic range is limited, RRMVM produces cleanerlooking intensity imagery than ASR. We have been unableto establish a criterion for selecting reasonable choices ofthe constraint coefficient as a function of the amount ofaveraging (i.e., correlation matrix rank) that insures consistentRRMVM image characteristics from one SAR scene to thenext.

Benitz’s high definition imaging (HDI) method [52] intro-duces a clever subspace constraint into RRMVM that preventsthe weight vector from becoming orthogonal to the subaperturedata. HDI constrains the degree to which the weight vectordeviates from the projected Fourier vector, rather than thenorm of the weight vector. Together with use of 80% sub-apertures (more averaging), these modifications eliminate theweak signal suppression problem that plagues RRMVM.


5) Adaptive Sidelobe Reduction:The adaptive sidelobe re-duction algorithm, extends RRMVM to its logical conclusion,namely maximizing the SIR on the basis of a single, full-aperture realization, or unit-rank correlation matrix. In theprocess, ASR overcomes the weak signal suppression (WSS)and computational complexity drawbacks of RRMVM byrestricting the number of adaptive degrees of freedom. Todo this, ASR imposes a structure on the functional form ofthe weighting vector. For purposes of clarity, we present theASR algorithm in a 1-D context, and subsequently discussits application to 2-D data. For a single full-aperture we canexpand the complex-valued (i.e., omit the magnitude squared)starting point of MVM and RRMVM, namely (3), as

(9)

where we have explicitly broken the complex weighting co-efficients into an amplitude term and an exponential Fourierphase factor. Here, we require to be real-valued;thus, unlike MVM and RRMVM, ASR does not perturb theFourier transform phase factors. The form of this equationresembles that of a weighted inverse discrete Fourier transform(interpolated by a factor except that the weightingcoefficients can depend on the spatial tuning locationTorestrict the number of adaptive degrees of freedom further,ASR requires that the weighting coefficients be of the form

(10)

where the ASR order is small This weightedsum-of- -cosines form parallels that of a Taylor weightingfunction of order except that we allow the coefficients tovary with output sample The rationale for this choice isthat, for integer interpolation (zero-pad) factors, substitutingexpression (10) into (9) yields

(11)

where is the unweighted (sinc IPR) inverse Fouriertransform image. Equation (11) says that the ASR imagecan be evaluated by applying a space-variant, symmetric,noncausal finite impulse response (FIR) filter to the sinc IPRFourier image. One can either wrap the lags that go off oneedge of the scene around to the other, or treat the missingsamples as zero; we let them wrap. The unit-gain signalconstraint, is satisfied automatically, regardlessof the filter coefficients This is clear from the fact thatthe sinc IPR centered at interpolated sampleexhibits its zerosat samples Thus, we maximize the single-realizationSIR by selecting the FIR coefficients to minimize

As with RRMVM, in spite of the unit-gain signal constraint,an additional constraint must be invoked to insure a nonzeroASR image. We define the vector of ASR filter coefficientsas and impose a constrainton the -norm of the filter vector, This

constraint limits cancellation of the desired spatial frequencysinusoid that can arise from modulating and amplifying the si-nusoids from neighboring spatial frequencies via the weightedcosine form of the weighting function.

To formulate the solution for the ASR filter coefficients, it isconvenient to treat the complex-valued sinc IPR Fourier image

and the ASR output image as real-valued, two-channel (I and Q) vector images, and respectively.We also define a real-valued lag matrixwhere and are -element vectors of the I andQ channel lags, respectively, combined symmetrically aroundoutput sample Using this notation, we express (11) as

(12)

and suppress the understood dependence onEquation (12)also expresses the operation of a single ASR filter on-channel SAR data (such as interferometric, whereif we increase the dimensions ofand and the number ofcolumns in the lag matrix, to

We select the ASR filter vector to minimize the outputenergy (or average multichannel energy) subject to

The solution is

(13)

where the Lagrange multiplier satisfies the constraint

(14)

In the underdetermined case, where it is compu-tationally advantageous to evaluate the right singular vectorsand values of and exploit where

together with the matrix inverse lemma to obtain

and thus

(15)

(16)

(17)

Thus, the following procedure implements the ASR algorithmefficiently for the underdetermined case. Note that the outputis zero whenever the constraint is inactive.

1) Evaluate SVD (right singular vectors only) ofnote

2) Project data onto right singular vectors,3) Evaluate Lagrange multiplier by solving

4) Evaluate output, ifotherwise.


Evaluating the SVD in step (1) dominates the computationalcomplexity of the underdetermined ASR algorithm.

In the overdetermined case, where it is computa-tionally advantageous to evaluate the eigendecomposition of

to obtainand thus

(18)

(19)

(20)

Thus, the following procedure implements the ASR algorithmefficiently for the overdetermined case.

1) Evaluate eigendecomposition of2) Project lags onto eigenvectors,3) Evaluate4) Evaluate Lagrange multiplier by solving

5) Evaluate output,

Projecting the lags onto the eigenvectors, step (2), dominatesthe computational complexity of the overdetermined ASRalgorithm.

To apply the ASR algorithm to 2-D SAR data, we have twooptions. First, we can implement distinct 1-D ASR filters to therows, then columns, or columns, then rows of an image, andselect the ordering that achieves the minimum output energy.This approach yields a suboptimum (but very good) separable2-D ASR filter. Finding the optimum separable ASR filterinvolves solving a set of nonlinear equations, and is difficult.The second option is to implement a nonseparable 2-D ASRfilter. To preserve the computational simplicity and spirit ofthe 1-D ASR algorithm, we employ 2-D weighting functionsof the form

(21)

In this manner, we obtain a 2-D FIR filter implementation.However, since the weights are nonseparable functions of the2-D lags, we do not encounter nonlinear equations in solvingfor the filter coefficients. In fact, by defining -element vectors in the obvious way, the formalism of our 1-Dsolutions can be used directly to obtain a 2-D solution. Fora given order the separable ASR filter offersdegrees of freedom in each direction, while the nonseparableASR filter offers a total of degrees offreedom.

A simple argument, based on a single isolated scalar pointscatterer, establishes the minimum constraint value that willinsure that input sinc IPR sidelobes are eliminated in the ASRoutput image. The argument is based on the fact that the distanton-axis sidelobes, rather than near-in or off-axis sidelobes,are the most difficult to eliminate. For both separable andnonseparable ASR implementations, it can be shown thatemploying a filter constraint of (or larger) willeliminate sidelobes.

Similarly, we have gained insight into the impact ofunder-determinedASR implementation (separable versus nonsepara-ble), order and constraint by developing approximate formulaefor the image domain TCR by assuming scalar data of a singlepoint scatterer in additive white Gaussian clutter. Omitting thedetails, the excess TCR gain afforded by underdeterminedASR (in excess of the usual Fourier TCR gain) is determinedby the product of the number of ASR degrees of freedomand the squared constraint, i.e., as well as by the inputimage domain TCR, as in (22), shown at the bottom of thepage, where As a function ofincreasing input TCR, the excess gain function saturates at avalue dictated purely by the product i.e., (23), shownat the bottom of the page.

For a nonseparable ASR filter the excess TCR gain is,directly

(24)

On the other hand, for a separable ASR filter, we get a cascadeof excess gain from running in the row and column directions

(25)

(22)

(23)


Fig. 3. Comparison of theoretically predicted and observed TCR gain provided by the ASR algorithm as a function of input TCR for nonseparable andseparable filters using constraints satisfyingE = Mc

2= 1; 2:

TABLE IISTRATEGIES TO SELECT ASR CONSTRAINT VERSUSORDER TO EITHER

ELIMINATE SIDELOBES OR ACHIEVE SPECIFIED PEAK TCR GAIN

Equation (23) indicates that underdetermined nonseparable andseparable ASR filters for which the product isa constant, offer the same maximum level of excess TCRgain; however, since the separable ASR filter cascades thesegains, it attains high levels of excess gain at lower input TCRthan the nonseparable ASR filter. Fig. 3 illustrates theoreticallypredicted and observed excess TCR gain versus input TCRfor nonseparable and separable filters using andWhile the theoretical predictions are higher than we observe inpractice, the predictions accurately reflect the general behaviorof excess TCR gain with respect to order, constraint and inputTCR. Thus our approximate formulae afford useful insightinto the impact of ASR implementation, order and constrainton image domain TCR.

Table II summarizes strategies for selecting ASR constraintto either eliminate sidelobes or fix the maximum excessTCR gain. The separable sidelobe and excess TCR gainstrategies are consistent, in that they share the same power-of-order dependence. In contrast, the nonseparable sidelobe andexcess TCR gain strategies are inconsistent. If one implementsthe sidelobe rejection strategy with both nonseparable andseparable filters, the nonseparable filter will suppress clutterto a greater degree.

6) Space Variant Apodization:The SVA algorithm is a spe-cial case of ASR that exhibits minimal impact on clutter.Its dominant impact is to remove the sinc sidelobe artifactspresent in unweighted Fourier imagery. SVA exploits a sepa-rable ASR filter of order one (hence overdetermined even fora single channel) together with a filter coefficient positivityconstraint. This constraint reflects the oscillatory nature of asinc IPR. When the pixel being operated upon is a sidelobe of

sinc IPR, the neighboring lags are of opposite sign, and addingthem to (rather than subtracting them from) the input reducesthe output energy. An additional ramification of this constraintis that it prevents SVA from sharpening the interpolated sincmainlobe, since in this case the neighboring lags share thesign of the mainlobe, and a negative coefficient is necessaryto reduce the output energy. Since the SVA algorithm usesonly a single filter coefficient, it is more direct to employcomplex notation rather than the separate I/Q channel notationemployed in deriving the higher-order ASR algorithm. Notethat Stankwitz [48] also describes separate I/Q and uncoupled2-D versions of SVA.

While the SVA algorithm typically utilizes a constraintvalue of here we provide the explicit formulationof SVA for arbitrary positive As with the ASR notation,we suppress the understood dependence on spatial locationMultichannel SVA evaluates where each elementof lag vector represents a single complex-valued lag, andminimizes the output energy with respect to subject tothe dual constraints and Using the method ofLagrange multipliers, the solution is obtained via the followingprocedure.

1) Evaluate unconstrained solution

2) Evaluate output:if , then use andelse, if use

and

else, use and

The computational burden of the SVA algorithm is trivial,being dominated by evaluating the dot products andin step 1.

7) Super Space Variant Apodization:Super SVA [49] is anonlinear bandwidth extrapolation method that is based on thepremise that SVA preserves sinc IPR mainlobes, but eliminatestheir sidelobes. The method obtains bandwidth extrapolation


Fig. 4. Illustration of super SVA bandwidth extrapolation byp2 for a single point scattering target.

based on this nonlinear operation, combined with inversefiltering.

Fig. 4 shows a block diagram of the basic super SVAalgorithm, and illustrates how it extrapolates the signal historyof a single point scatterer by a factor of As shown,the nonlinear SVA operation increases the original bandwidthof the point scatterer, but introduces a magnitude taper thatincludes nulls. This taper corresponds to the transform of asinc mainlobe, and can be computed easily and analyticallyfor any integer input interpolation factor. The magnitude tapercan be equalized over an aperture somewhat larger than afactor of (the position of the first null depends on the inputinterpolation factor) in each direction without encountering thesingularity. In practice, however, it is convenient to extrapolateby a factor of and repeat the operation twice to obtainextrapolation.

Various refinements can be incorporated into the superSVA algorithm. One can replace the center portion of theextrapolated data with the measured data and perform nonex-trapolative iterations to smooth the junction between themeasured and extrapolated signal history. One can employseparate I/Q, rather than joint I/Q SVA filters, or 2-D uncou-pled SVA filters. Finally, on a pixel-by-pixel basis, one canmerge the basic SVA or super SVA images on the basis ofselecting the algorithm that yields a smaller magnitude result.All of these nuances can qualitatively improve the nature ofthe super SVA result, particularly in clutter areas. We utilizedreplacement and nonextrapolative smoothing in our subsequentexamples.

However, one cannot accurately characterize an SVA imageof distributed or very closely spaced scatterers as the con-volution of impulses with a sinc mainlobe. In the author’sexperience, super SVA is most effective for isolated pointtargets where the convolutional model is accurate. Underthese conditions, the algorithm affords image sharpening andTCR gain consistent with the extrapolated aperture size. Thealgorithm can also resolve points that are not resolved in the

initial sinc IPR Fourier image. The resolution capability ofsuper SVA has yet to be determined or analyzed theoretically.

8) Autoregressive Linear Prediction:The autoregressivelinear prediction method [7] predicts a particular sampleinthe signal history subaperture as a linear combination of theremaining samples, i.e.,

Note that the prediction filter in question is a true 2-D filter,rather than a succession of 1-D predictions in the row andcolumn directions. Let then the

prediction error can be written aswhere and is theerror prediction filter. The errorprediction coefficients are chosen to minimize the averageprediction error energy over all subapertures within the fullsignal history aperture

(26)

subject to the normalization constraintwhere is a vector of zeros, with a single unit-valued entrycorresponding to the predicted element. In this context, the useof forward-backward averaging corresponds to minimizing theaverage of forward and backward prediction errors while usingthe spatially reversed and conjugated forward error predictionfilter for backward error prediction. Solving the constrainedoptimization via the method of Lagrange multipliers yields

(27)

and a minimum prediction error energy of

One can invert the prediction process, and view the predic-tion error as noise driving an all-pole (autoregressive) filterwhose output is the data sample being predicted. Based on


the assumption that the prediction error signal is an innova-tions process, i.e., white noise, the PSD estimate equals theminimized prediction error energy divided by the magnitudesquared of the transfer function. However, it is known thatthe PSD should be chosen as the square root of this quantityto obtain correct scaling [7], [13]. Thus, the ARLP spectralestimate is

- (28)

Evaluating and inverting the correlation matrix (or at leastsolving the equations dominates the computationalcomplexity of ARLP.

A more common, but equivalent formulation of the ARLPalgorithm [4], [5] involves solving the normal equations

where the reduced correlation matrix is obtained fromthe full correlation matrix by omitting prediction elementrow and column and is obtained by extracting column

from the full correlation matrix, omitting the th row.Similarly, the prediction filter is missing the th unit entryelement of the error prediction filter One reason for thepopularity of this less conceptually direct formulation is that, inthe special case of Toeplitz correlation estimates (correlationmethod) and a causal prediction filter, the normal equationsbecome the Yule–Walker equations, which make the use ofefficient numerical methods, such as Durbin’s recursion, moredirect. In addition, the normal equation formulation is usefulfor describing the Tufts–Kumaresan ARLP algorithm.

9) ARLP Spectral Averaging and Pisarenko’s Method:Thesignal history sample being predicted need not have anyparticular spatial relationship to the samples being used topredict it. In other words, the filter need not be causal,semicausal, etc. ARLP imagery based on any one choice ofprediction element may exhibit spurious spiky behavior andelliptical, rather than circular, contours. The first- and second-quadrant averaging proposed in [27] reduces (but does noteliminate) the spurious spikiness, and makes the countoursmore circular:

-

In this case, prediction elements and are chosen suchthat they lie in orthogonal quadrants with respect to the centerof the subaperture. Even with this averaging, the choice oforthogonal prediction elements remains arbitrary. A logicalgeneralization of the orthogonal-quadrant RRMS averagingconcept is to RRMS average ARLP spectra/images across all

possible prediction elements, i.e.,

-

where indicates the diagonal portion of sois a diagonal matrix of prediction errors.

If one makes the assumption that the individual ARLP filtersyield the same prediction error energy, i.e., thenthe RRMS ARLP image reduces to one of a family of estimatessuggested by Pisarenko [10], as follows:

- (29)

In practice, the difference between RRMS ARLP imagery andPisarenko imagery is negligible, validating the assumption.

10) Signal-Clutter Subspace Decomposition:Here we con-sider the idealized form and properties of the signal historycorrelation matrix [14], [15], [20] based on an assumption ofa superposition of sinusoids (point scatterers) embedded inadditive white Gaussian clutter

The ideal (infinite averaging) correlation matrix takes the formof an identity matrix scaled by the clutter variance, combinedwith a quadratic term that combines outer-products ofconstituent sinusoidal signal vectors weighted by a so-calledcoherence matrix

The diagonal terms of the coherence matrixare the sinusoidenergies. By definition, the correlation matrix is positive defi-nite and Hermitian; consequently, its eigenvectors form an or-thonormal basis for the -dimensional space spanned byits columns. Further, the correlation matrix can be expressedas a sum of outer-products of the eigenvectors weighted bytheir associated eigenvalues. Theeigenvectors associatedwith the largest eigenvalues span the subspace defined by thesinusoid vectors, i.e., the columns of the signal matrixTheremaining eigenvectors are orthogonal to the signalvectors, and display eigenvalues equal to the clutter variance.The signal subspace eigenvalues are larger than the cluttervariance, but their values are complicated functions of thepoint scattering energies and differential locations.


The ideal correlation matrix inverse can be expressed similarly,but using the reciprocal eigenvalues.

However, establishing the order i.e., the number of pointscatterers, is not as simple as determining the number oflarge eigenvalues that deviate from a constant clutter floor,for several reasons. First, the correlation matrix estimateis imperfect, and statistical perturbations cause the cluttereigenvalues to spread about the clutter energy. Second, acollection of equal energy point scatterers typically producesa range of signal eigenvalues, that can vary from near theclutter energy to well above it; unequal scattering amplitudesexacerbate this spread. Third, realistic SAR clutter is not white.

There are two simple methods for establishing the modelorder. First, one can assume/decree the value ofa priori,and assume that the largest eigenvalues correspond to the pointscatterers. Second, one can assume that the ensemble of pointscatterers contributes a fixed fractionof the total signal his-

tory energy, and select the order so that

where we have assumed that the eigenvalues aresorted from smallest to largest, i.e.,

More complicated information-theoretic criteria for choosing order also exist. We defer furtherdiscussion of order selection until we show examples withactual SAR data.

11) Eigenvector and Multiple Signal Classification:TheEV [15] and MUSIC [14] methods are variants of MVMthat seek to drive its denominator toward zero when theanalysis sinusoid vector aligns with one of the truesinusoid signal vectors, thereby giving rise to sharp imagepeaks. The height of these peaks is a measure of “pointiness”rather than of scattering intensity. Since the correlation matrixeigenvectors that span the clutter subspace are orthogonal tothe signal vectors, EV and MUSIC truncate the eigendecom-position of the correlation matrix inverse to include only thoseeigenvector outer products that lie in the clutter subspace.While EV employs the reciprocals of the measured cluttereigenvalues, MUSIC exploits the white clutter assumptionfurther, by replacing the measured clutter eigenvalues with aconstant (we use the arithmetic mean of the measured cluttereigenvalues):

(30)

(31)

Evaluating and performing the eigendecomposition of thecorrelation matrix dominates the computational complexityof EV and MUSIC; subsequent evaluation of the quadraticform in (30) or (31) involves computing a pair of 2-D FFT’s.

MUSIC is not generally suitable for SAR imaging becausewhitening the clutter eigenvalues destroys the spatial inho-mogeneities associated with terrain clutter or other diffusescattering in SAR imagery.

12) Tufts–Kumaresan ARLP: The Tufts-Kumaresanmethod [16] is a variation of ARLP that seeks to force theprediction filter to lie in the signal subspace, or equivalently,the error prediction filter to lie in the clutter subspace.TKARLP differs from ARLP in two respects. First, it allowsthe use of larger subapertures, such that the correlationmatrix becomes singular, by employing the Moore–Penrosepseudoinverse in calculating the prediction filter. Use of largersubapertures improves resolution. Second, the correlationmatrix is truncated to omit the clutter contribution. Whilethis truncation improves the apparent TCR, and improves theaccuracy of point scattering peak locations in the imagery, itdoes not improve the image domain TCR. Thus, the TKARLPprediction filter is the minimum-norm solution to

which is

(32)

where the eigendecomposition of the reduced correlation ma-trix is One obtains the corresponding errorprediction filter by inserting a one in the th entry of theprediction filter and obtains the TKARLP image

- (33)

Evaluating and performing the eigendecomposition of thecorrelation matrix or the right singular vectorsin the SVD of the corresponding reduced forward-backwardsubaperture data matrix dominates thecomputational complexity of TKARLP.

As with ARLP, we find that TKARLP imagery benefitsfrom RRMS averaging over multiple prediction elements, forexample

-

and

- (34)

For TKARLP we have been unable to derive a nonbrute forcemethod of RRMS averaging across all prediction elements.


Fig. 5. Comparison of 2-D spectral estimation techniques for imaging synthetic point scatterers. Image-domain TCR (with sinc IPR) is 33 dB.

13) Parametric Maximum Likelihood:The PML algorithmfits a complex-valued superposition of sinusoids to the signalhistory data, and selects the amplitude, phase, and location ofthe point scatterers to minimize residual energy. For sinusoidsin white Gaussian clutter, the PML algorithm achieves theCramer–Rao lower bound on estimation accuracy [19], [22],[26]. The signal model is

while the estimated signal model is

where

The residual and its energy are


Given a set of point scattering location estimates, we minimizethe residual energy with respect to the complex amplitudes viaclosed-form least-squares regression

which yields the residual that now must be minimized withrespect to the spatial scattering locations

We employ quasi-Newton methods to solve this nonlinearoptimization problem [18] in conjunction with interactivegraphics that lets us monitor and guide the convergenceof the algorithm from a set of selectable initial locationestimates. In some cases, a variant of the CLEAN algorithm[53] used in radio astronomy can provide adequate initiallocation estimates.

C. Simulated Point-Scattering Results

Preliminary assessment of the benefits and limitations of2-D spectral estimation algorithms for SAR imaging can bemade using simulated point scatterers. These simple exam-ples illustrate the characteristic properties of these imagingmethods.

Fig. 5 compares imagery of a collection of 36 equal-amplitude, randomly phased simulated point scatterers,configured to spell in additive white Gaussian clutter;the Fourier image domain TCR is 33 dB (the product of the6 dB signal history domain TCR and the 27 dB compressiongain afforded by the 24 24 signal history sample size).This is a higher TCR than is typical of many SAR imagingscenarios. The SVA image essentially resembles sinc mainlobecontributions (from an unweighted Fourier image) of the pointscatterers, and is sharper than the baseline Taylor weightedFourier imagery. The nonseparable underdetermined ASR andRRMVM images are slightly sharper than the SVA image, andhave enhanced the TCR due to their singular nature. Thesuper SVA extrapolated signal history has been imaged bothby Taylor weighted FFT and SVA. Both super SVA imagesshow improved TCR compared with the nominal Fourier orSVA images because their increased apertures afford 6 dBgreater TCR gain. Similarly, the super SVA images do abetter job of resolving the point scatterers than the Fourier,RRMVM, ASR, or SVA images. Use of 50% subapertureaveraging in our MVM, EV, MUSIC, ARLP, and Pisarenkoexamples (60% for TKARLP) penalizes these algorithmswith roughly 6 dB loss of TCR gain compared to the fullaperture Fourier transform. Nevertheless, the MVM imageis much sharper than the Fourier image, and the specklevariability of the clutter background is reduced. EV andMUSIC improve upon the resolution of MVM and boostthe apparent TCR, more than compensating for the 6 dBaveraging loss. While MUSIC yields a benign, flat background,EV clutter variations track those afforded by MVM. TheARLP image exhibits numerous spurious peaks in the clutterbackground (despite first- and second-quadrant averaging).

The Pisarenko image exhibits fewer spurious clutter peaksand more accurate target peaks than the ARLP image. ThePisarenko image is slightly sharper than the MVM image, butits clutter variability is higher; Pisarenko clutter variationstrack those afforded by MVM. TKARLP provides excellentlocalization of the point scatterers, but loses roughly 20 dBof TCR and introduces a herringbone texture into the clutter.RRMS averaging TKARLP imagery across all predictionelements, improves contrast, eliminates the herringbone cluttertexture, and reduces the bias of the peak locations in thebutdegrades point resolution slightly. PML provides extremelyaccurate estimates of point scattering amplitude, phase, andlocation, as is evident by comparing them against the truepoint locations and amplitudes. All of these methods, withthe possible exception of ARLP, provide a truer picture ofthe underlying point scatterers than does the baseline Taylorweighted Fourier image.

Fig. 6 compares the same types of imagery of the samesimulated point scatterers when the Fourier image domainTCR is 13 dB, which is a more typical TCR than the 33 dBof Fig. 5. As is well known, the resolution of the MVM, EV,MUSIC, ARLP, TKARLP and Pisarenko methods is coarserat lower TCR than at high TCR. In contrast, the resolution ofRRMVM, ASR, SVA, and super SVA is largely independentof TCR. With the TCR already low, the compression lossessuffered by MVM, ARLP, and TKARLP are problematic.The EV and MUSIC algorithms offer better contrast, but arebeginning to place their 36 peaks in erroneous locations. ASR,RRMVM, and super SVA provide better definition of thepoints in the and M than EV or MUSIC.

Both ASR and RRMVM tend to suppress clutter scattering,thereby relatively enhancing stronger scattering. In contrast,EV enhances strong point scatterers, while leaving clutterscattering relatively unaffected. While RRMVM, ASR, SVA,and super SVA do not offer the resolution enhancement thatEV can at high TCR, our experience is that they are morerobust than EV at low TCR. ASR enhances locally prominentscatterers over a spatial extent dictated by the order of the ASRfilter, and super SVA enhances locally prominent scatterersby increasing the aperture and compression gain. In contrast,RRMVM and EV enhance only the strongest scatterers in thescene.

The TKARLP and PML algorithms provide the best local-ization of the point scatterers. Indeed, much literature existsto illustrate that they approach and/or attain the Cramer–Raobound on estimation accuracy for sinusoids in white clutter.However, the electromagnetic notion of a point scatterer is,arguably, more of a mathematical and conceptual constructthan a physically meaningful entity. The usual purpose androle of SAR imaging is far more complicated than the simpleproblem of estimating optimally the location of and ampli-tude of point scatterers. While neither TKARLP nor PML iswell suited nor intended for imaging realistically complicatedscenes, these algorithms may be valuable for specialized SARanalysis tasks, such as target classification in ATR.

Table III summarizes the dominant order of computationalcomplexity (for single channel data), TCR gain, and ad-vantages or disadvantages of the 2-D spectral estimation


Fig. 6. Comparison of 2-D spectral estimation techniques for imaging synthetic point scatterers. Image-domain TCR (with sinc IPR) is 13 dB.

algorithms for SAR imaging. and are the sizes ofthe SAR signal history in the range and cross-range direc-tions, while and are the sizes of the signal historysubapertures used to compute the correlation matrix. Forperspective, is often on the order of one million.and are small amounts of averaging in the range andazimuth directions, satisfying

and Recall that the computationalburden of the order complexity algorithms can bereduced by employing a decimation and mosaicing strategy.

D. Collected Scalar SAR Results

Naturally, a more complete assessment of the benefits andlimitations of 2-D spectral estimation algorithms for SAR


TABLE IIISYNOPSIS OF2-D SPECTRAL ESTIMATION ALGORITHM COMPLEXITY, TCR PROCESSINGGAIN, ADVANTAGES, AND DISADVANTAGES FOR SAR IMAGING

imaging necessitates the use of collected data. Here, we utilizeKu-band data collected by the WL-ERIM DCS radar of twocommercial ships docked near Toledo, OH, together with somecalibration trihedrals. The same signal history, which affords auniformly weighted Fourier image resolution of one meter, wasused to produce all images shown. The rectangularly formattedsignal history employed is 400 400 samples. We computedTaylor weighted and unweighted Fourier images using a zero-pad factor of four. We exploited signal history decimationand image mosaicing to compute MVM, ARLP, Pisarenko,and EV images based on 40% subaperture forward-backwardaveraging. Each decimated signal history aperture was 2525; the subaperture size was 10 10. The overall imageswere computed as mosaics of overlapping 100100 samplesubimages. The final image sizes for all methods is 16001600 samples, of which 1000 1200 samples are shown.

Fig. 7 illustrates the baseline Taylor weighted 35 dB,order 5) Fourier image on a relative 60 dB gray scale.Sidelobes are visible in the water from the trihedral near thewater’s edge. Also note the water tower in the center-topportion of the image; the tower lays over toward the radar,which is looking from the right. All subsequent images shouldbe compared against this baseline.

Fig. 8 illustrates the MVM image on a relative 45 dBscale. The reduced dynamic range (compared to the 60 dBscale used for the Fourier image) was chosen to preserve theapparent contrast ratio. The reduced TCR gain afforded bythe 40% subapertures accounts for 8 dB of the difference;we hypothesize that motion compensation errors, i.e., along-track phase errors, together with MVM’s increased sensitivity

to such errors, accounts for the remaining 7 dB loss incontrast. Aside from contrast, the differences between MVMand Fourier imagery are startling. First, MVM improves thesharpness of the trihedrals and resolution of detail on theships, yet displays less IPR scintillation (breakup) along thecontinuous bulkheads and gunwales of the ships. Second, thereare no sidelobe artifacts in the MVM image. Third, MVMreduces clutter speckle. The MVM image has a more “optical”quality than the Fourier image.

Fig. 9 illustrates the RRMVM image, evaluated using aforward-backward subaperture size of 398 398 samplesout of an aperture of 400 400 samples and a constraintof 1.0002, on a relative 70 dB scale. This example highlightsa problem that can arise with the algorithm when a large dy-namic range of scattering amplitudes exists within the scene. Inthis case, RRMVM eliminates much of the detail of the ships’structure between the bow and stern, yet at the same time,fails to eliminate the near-in sidelobes of the bright trihedralscatterers. To first order, the effect of RRMVM is to thresholdthe uniformly-weighted Fourier image. RRMVM exhibits toomany adaptive degrees of freedom for the small amount ofaveraging and highly singular correlation matrix employed.The result is global weak signal suppression. While thissuppression can be reduced by using smaller subapertures, i.e.,more averaging, RRMVM rapidly loses its computational ad-vantage when a significant amount of averaging is performed.

Fig. 10 illustrates the underdetermined nonseparable (order2, constraint .5, eight degrees of freedom) ASR image ona relative 70 dB gray scale. Sidelobes are no longer visiblein the water from the trihedral near the water’s edge despite


Fig. 7. Baseline Taylor weighted(�35 dB n = 5) Fourier imagery.

Fig. 8. 40% subaperture MVM imagery.


Fig. 9. 398/400 subaperture RRMVM(c = 1:002) imagery.

the 10 dB increase in dynamic range of the ASR display.The trihedrals and prominent scatterers on the ships are moresharply defined. In addition, the contrast of the trihedrals withrespect to the surrounding clutter is improved by roughly 8dB, although the variance of the clutter speckle (on a dBscale) is increased. While ASR eliminates sidelobes withoutsuppressing weak scatterers globally, it does suppress weakscatterers locally, on the scale of the FIR filter size. Cluttersuppression, local weak signal suppression, and increasedclutter speckle are related phenomena caused by the completelack of averaging in the spectral estimate, despite the smallnumber of adaptive degrees of freedom.

Fig. 11 illustrates the SVA image on a relative 60 dBgray scale. Sidelobes are no longer visible in the water fromthe trihedral near the water’s edge, yet the trihedrals andprominent scatterers on the ships are more sharply defined. Forcomparison, Fig. 12 illustrates the unweighted Fourier imageon a relative 60 dB gray scale. The dominant impact of SVAis to eliminate the sidelobe artifacts while leaving the sinc IPRmainlobe and clutter largely intact.

Fig. 13 illustrates the ARLP image on a relative 35 dB scale.The image shown is the RRMS average of imagery basedon first- and second-quadrant predictors, as described in thetext. The ARLP image is qualitatively very poor. Comparedto the MVM image, the ARLP image offers less contrast anddisplays spurious diagonal texture (both caused by ubiquitousspurious peaks), and exhibits sidelobelike artifacts that extendthroughout the scene.

Fig. 14 illustrates the Pisarenko image on a relative 35 dBscale. Compared to the MVM image (Fig. 8), Pisarenko loses

roughly 10 dB of compression gain or contrast. We suspect thisoccurs because Pisarenko is even more sensitive than MVMto along-track phase errors. Otherwise, there appears to belittle difference between the Pisarenko and MVM imagery.However, by averaging over all possible ARLP predictors, thePisarenko image greatly reduces spurious diagonal texture andimproves image contrast compared to that afforded by a singlepair of orthogonal prediction elements.

Fig. 15 illustrates the EV image on a relative 55 dB scale.We employed a fixed order of ten point targets for each mosaicchip. The EV image is very similar to the MVM image, but EVgains roughly 10 dB of contrast by enhancing the prominentpoint scatterers; several scatterers on the ships are moresharply defined by EV. The EV image also enhances randompoints in mosaic chips comprised of homogeneous clutter;in such cases, there are no dominant signal eigenvalues, yetcertain eigenvalues are decreed to be signal, and EV enhancesthe associated points. Use of energy-based order selectioncriteria results in poor quality imagery where the boundariesbetween mosaic chips become clearly (and distractingly) visi-ble, particularly between benign homogeneous chips and chipscontaining a few dominant scatterers. Both clutter level andtextural mismatches occur with energy-based order selectioncriteria. SAR target detection, target recognition, and sceneanalysis problems usually involve spatially distributed objects.Order selection criteria that are based solely on discontinuitiesin or thresholding of the eigenvalue spectrum fail to satisfyour desire to enhance scatterers on the basis of their localprominence or relative spatial position. Further, order selectionis complicated by the arbitrary manner in which an object or


Fig. 10. Nonseparable ASR (order= 2; c = 0:5) imagery.

Fig. 11. SVA imagery.


Fig. 12. Unweighted (sinc IPR) Fourier imagery.

Fig. 13. 40% subaperture ARLP (first- and second-quadrant average) imagery.


Fig. 14. 40% subaperture Pisarenko imagery.

terrain can span multiple mosaic chips. More sophisticated or-der selection criteria are necessary to fully realize the potentialof EV for SAR imaging. The logical intersection between EVorder selection and SAR target detection is very intriguing.

Fig. 16 illustrates the SVA imagery of the super SVAextrapolated signal history on a relative 66 dB gray scale.Compared with the SVA imagery, the super SVA imageryappears somewhat sharper, the clutter has a more fine-grainedspeckle, and the extrapolated aperture provides 6 dB greaterTCR gain. Sidelobes of the trihedrals and other prominentpoint scatterers are heavily attenuated, although low-levelresidual sidelobes of the strongest scatterers are visible.

Figs. 17 and 18 illustrate heavily interpolated slices throughone of the trihedrals on the causeway, and facilitate compar-ison of the sharpness, but not strictly the resolution, affordedby each algorithm. Fig. 17 demonstrates that SVA effectivelyyields a sinc IPR mainlobe while eliminating the sinc side-lobes. ASR realizes a slight sharpening of the sinc mainlobe,together with a reduction in the background clutter. Both SVAand ASR offer lower sidelobes and sharper mainlobe than thebaseline Taylor weighted imagery. Super SVA sharpens themainlobe by roughly a factor of two. Fig. 18 demonstratesthat MVM, EV, ARLP, and Pisarenko all produce 3 dB peakwidths that are roughly five times sharper than the sinc IPRmainlobe. The relatively poor contrast ratios, compared toMVM, exhibited in the ARLP and Pisarenko slices reflect thegreater sensitivity of these algorithms to measurement phaseerrors.

Finally, Fig. 19 illustrates the application of PML to thestern of the upper ship. The image on the right displays the

point scattering estimates (dots) superimposed on the Taylorweighted residual image. In this case 100 points were fit tothe image. Clearly, one might choose to fit more points to theship structure. One could conceive of an application where theprecise point scattering estimates afforded by PML would beuseful, but the estimates do not, by themselves, constitute auseful image.

III. I NTERFEROMETRIC AND POLARIMETRIC

MVM, RRMVM, ASR, AND SVA

We developed multichannel interferometric versions ofMVM, RRMVM, ASR, and SVA based on our belief thatthese methods could improve upon the accuracy of Fourierheight estimates, through a combination of statistical averagingand/or interference nulling and resolution enhancement. ARLPand the signal-noise subspace decomposition methods offerno plausible interferometric generalizations. The followingsubsections generalize the four algorithms and illustrateresults.

A. Algorithm Generalizations

Interferometric SAR systems exploit vertically displacedapertures and to collect registered, phase coherent sig-nal histories, and Conventional interferograms arethe product of the Fourier image from aperture timesthe conjugate of the Fourier image from aperture i.e.,

The magnitude of the interferogramcorresponds to scattering intensity, while the phase is pro-portional to scattering height out of the slant-plane. Fig. 20


Fig. 15. 40% subaperture EV (fixed order 10 per mosaic chip) imagery.

Fig. 16. Super SVA�2 extrapolated SVA imagery.


Fig. 17. Interpolated slices through trihedral on causeway, “easy” methods.

Fig. 18. Interpolated slices through trihedral on causeway, “hard” methods.

illustrates the geometric and conceptual basis of interferomet-ric SAR imaging.

Both target and clutter returns are virtually perfectly corre-lated across the two interferometric apertures. Thermal noise,which is independent for the two apertures, together withunresolved scatterers and IPR sidelobe leakage, contributesto interferometric phase noise, thereby degrading the accuracyof interferometric height estimates. Generally, height accuracyimproves with increasing SNR or clutter-to-noise ratio (CNR).Often, interferograms must be lowpass filtered (spatially aver-aged) to reduce the phase/height noise associated with clutterand target scattering. Filtering the interferogram, rather thanthe phase, weights strong scatterers with less phase noisemore heavily than the weak scatterers with more phase noise.Unfortunately, spatial averaging of interferograms degradesresolution.

A fully polarimetric SAR system collects the full 2 2scattering matrix. Without loss of generality, we assume alinearly polarized basis, i.e., horizontal and vertical receiveand transmit polarizations, providing two co-polarized receive-transmit pairs HH and VV, and two cross-polarized receive-transmit pairs HV and VH. To apply MVM and RRMVMto polarimetric data, we assume that we seek a polarimetricspanlike intensity that represents the energy received by a re-ceive antenna whose polarization is aligned with the scattered

field, averaged over all possible transmit polarizations. Thepolarimetric span weights the energy on the four polarizationchannels equally. Thus, we apply the same processing toall polarimetric channels, and maximize average SIR acrossthe four polarization channels. Similarly, to apply MVM andRRMVM to interferometric data, we assume balanced andindependent thermal noise levels for the two interferometricchannels and To preserve interferometric phase, we applythe same processing to channelsand and maximize SIRon the basis of the average of the two interferometric channels.In a fully polarimetric and interferometric system, there are atotal of eight data channels. An average correlation matrix

(35)

across the interferometric and polarimetric channels arisesnaturally as we seek to minimize average output energy,or maximize average SIR. Thus, from (4) and (6), the in-terferometric and polarimetric MVM and RRMVM filters,respectively, are

(36)


Fig. 19. PML estimates and Taylor weighted residual for stern of top ship; 100 points.

Fig. 20. Geometric and conceptual foundation of interferometric SAR imaging.

and

(37)

where the Lagrange multiplier solves

(38)

In both cases, we average the output interferogram acrossall available forward-backward subapertures, as well as allpolarization channels to insure target visibility, i.e.,

where the polarimetrically averaged interferometric correlationmatrix is

and superscript denotes a polarimetric interferogramimage. Thus, the interferometric MVM and RRMVM images,respectively, are

(39)

and

(40)


Recall that RRMVM exhibits global weak signal suppres-sion in scalar SAR data because of excess degrees of freedom.In an interferometric context, we find that these excess degreesof freedom destroy the interferometric height information, aswell. Consequently, we have not found the RRMVM algorithmto be useful in the context of interferometric SAR. This isalso why we show no subsequent interferometric results forRRMVM.

To apply ASR to interferometric data, we again assumebalanced and independent thermal noise levels for the twointerferometric channels and and seek a spanlike output.Consequentlywe apply the same ASR filter to the polarimetricand interferometric channels. In this case, the multichannelASR algorithm outlined in the previous section employs 16channels (two I and Q channels for each polarization andinterferometric channel), and selects the optimum filter basedon all 16 channels simultaneously. The ASR polarimetricinterferogram is simply

(41)

As described in Section II-B6, multichannel SVA is a specialcase of ASR in which a separable order one filter is employedtogether with a requirement that the single filter coefficient bepositive-valued.

B. DCS IFSAR Examples

Here we compare Fourier, MVM, ASR, and SVA polari-metric interferograms using X-band data collected of the areaaround the University of Michigan football stadium by theERIM-WL DCS IFSAR. The same signal histories, whichafford a baseline (Taylor weighted) Fourier image resolution ofone meter, were used as input to each of the adaptive imagingalgorithms. The rectangularly formatted interferometric signalhistories employed were 600600 samples. We computed theFourier interferogram using Taylor weighting dB peaksidelobe, order 5) and a zero-pad factor of two. We computedthe underdetermined nonseparable ASR interferogram using

(24 filter taps for 16 I/Q channels) and a constraintfrom eight sinc IPR Fourier images, each interpolated

by a factor of two. We exploited signal history decimationand image mosaicing to compute an MVM interferogrambased on 40% subaperture forward-backward averaging. Eachdecimated signal history aperture was 5050; the subaperturesize was 20 20. The overall MVM interferogram wascomputed as a mosaic of overlapping 100 100 samplesubimages. The final interferogram image size for all threemethods is 1200 1200 samples, of which 850 1050samples are shown.

Figs. 21–24 illustrate the intensity and phase/height of theFourier, MVM, SVA, and ASR interferograms, respectively.The relative dB scales for interferogram intensities have beenchosen to preserve apparent contrast across the algorithms.

Compared against Fourier intensity contrast, MVM losesroughly 10 dB of contrast due to the loss of TCR gain affordedby the 40% subapertures, SVA contrast is comparable, whileASR improves contrast by roughly 10 dB, approximatelyconsistent with theoretical predictions. Qualitatively, theMVM intensity imagery looks more “optical” than theFourier, displaying both less speckle and sharper resolutionof prominent structures. MVM intensity of linear structures isnot broken up by IPR scintillation, while the Fourier intensityis. The SVA intensity imagery is somewhat sharper than theFourier imagery, and comparably speckled. The ASR intensityimagery is more speckled and sharper than either the Fourieror SVA imagery. To reduce the phase noise of the Fourier,ASR and SVA estimates, we applied a 33 ideal averagingfilter to these interferograms prior to displaying their phases(but not intensities). Without such smoothing, phase noiseobscures the ASR, and to a lesser extent the SVA and Fourier,height signatures. In contrast, we did not average the MVMinterferogram to display its phase; all averaging associatedwith MVM phase estimate occurs in the signal history domain.The principal difference between the Fourier and MVM phaseimagery is that the Fourier imagery is much noisier, despitefiltering. In addition, the MVM phase signature displays muchsharper definition of the structural detail around the perimetersof the stadium and Crisler arena than the Fourier interferogram.The SVA phase imagery offers slightly sharper detail than theFourier phase imagery, as well as slightly less noise. TheASR phase imagery has a dramatically different characterfrom that of the Fourier or MVM phase imagery. While theASR height is noisier than the Fourier height, the spatialstructures are much better defined in the ASR height image.This is largely due to the fact that the ASR interferogramphase is randomized in the relatively low return areas thatsurround locally prominent scatterers.

Typically, the quality of SAR intensity imagery is judgedon the basis of metrics like image contrast, IPR peak sidelobelevel, and IPR integrated sidelobe level, or other criteria estab-lished by SAR image analysts. Estimation theoretic concepts,such as bias and variance, are less commonly discussed in aSAR context. Here, we attempt to establish crudely the relativeperformance of the imaging algorithms in such a context, byexamining various statistics of the log (dB) intensity overhomogeneous areas; homogeneous SAR clutter intensity is fre-quently modeled as log-normal. Table IV compares the mean,standard deviation, coefficient of skewness, and coefficient ofkurtosis of five 50 50 sample regions in the Fourier, MVM,SVA, and ASR interferogram dB intensity. The coefficientof skewness [50] is proportional to the third central momentof a histogram or density, and is a normalized measure ofits symmetry about its mean; a Gaussian or other symmetricdensity exhibits zero skewness. The coefficient of kurtosis [50]is proportional to the fourth central moment of a histogram ordensity about its mean, and is a normalized measure of itscompactness; the coefficient for a Gaussian density is three,while that of a more heavily or lightly tailed density is higheror lower, respectively. The first four regions are spatiallyhomogeneous clutter areas: tree crown, football field, grass,and parking lot; the fifth region is inhomogeneous, containing


Fig. 21. Baseline Taylor weighted(�35 dB n = 5) Fourier interferogram.


Fig. 22. 40% subaperture MVM interferogram.


Fig. 23. SVA interferogram.


Fig. 24. Nonseparable ASR (order= 4; c = 0:25) interferogram.


TABLE IVSTATISTICS OF POLARIMETRIC INTERFEROMETRICdB INTENSITIES OVER

HOMOGENEOUSCLUTTER AREAS AND BUILDING STRUCTURE

the building whose ridgelines are clearly visible at the upperright end of the stadium.

Across the clutter regions, the MVM dB intensities arebetter-fit by log-normal distributions than are the Fourier dBintensities. Further, the MVM clutter distributions are tighterthan the corresponding Fourier distributions, yet equally wellseparated. The Fourier mean dB intensities are roughly 6.6dB higher than the corresponding mean MVM dB intensities,suggesting a normalization bias. At the same time, the MVMstandard deviations are roughly 2.2 dB less than the corre-sponding Fourier standard deviations, reflecting a reductionin speckle across the clutter regions. The MVM dB intensitydistributions are less skewed than the Fourier distributions, andalso exhibit kurtoses that are closer to three. The MVM dBintensity distribution is also tighter and more Gaussian thanthe Fourier over the building.

Across the clutter regions, the SVA and Fourier dB intensitystatistics are very similar, with the SVA distributions slightlybroader than the Fourier distributions. The Fourier mean dBintensities are roughly 3.3 dB higher than the mean SVA dBintensities, again suggesting a normalization bias. The standarddeviations of the SVA intensities are roughly 0.4 dB higherthan the Fourier standard deviations. The skewnesses andkurtoses of the SVA dB intensities are generally somewhatmore consistent with Gaussian distributions than the Fourierstatistics.

Since we employed an underdetermined ASR filter, therewas a risk that some of the ASR interferogram intensitieswould be zero, causing the statistics of the dB intensities tobe undefined. While this did not occur, the underdeterminedASR filter does impact the dB intensity distribution in twoundesirable ways. First, the ASR mean intensities are less thanthe Fourier mean intensities by roughly 9–15 dB, where thedifference is larger for strong return areas than for weak returnareas; these differences are not the result of a normalizationbias. Second, the ASR standard deviations are much larger

TABLE VABSOLUTE AND NORMALIZED (CFAR) MEASURES OFCONTRAST

BETWEEN BUILDING AND SPECIFIED CLUTTER TERRAIN

than the Fourier standard deviations, reflecting the fact thatunderdetermined ASR exacerbates speckle. Together, thesefactors may diminish the differentiability of natural and man-made objects in the ASR scene. For example, the Fouriermean intensity difference between building and field is 2.1 dB,the ASR difference is 1.2 dB, a loss of 3.3 dB in absolutecontrast. On the other hand, if we evaluate the mean valuesof the interferogram intensity, rather than of the dB intensity,and compare the ratio of building mean intensity to field meanintensity, then ASR provides a contrast improvement of 3.0dB over Fourier; this is the sense in which our TCR analysissuggests that ASR improves contrast. Underdetermined ASRdecreases absolute dB contrast because it surrounds prominentpoints with low return areas, which are accentuated by the lognonlinearity.

Table V compares two measures of contrast between thebuilding and homogeneous clutter patches. The first metric,the difference between mean building and clutter values, is ameasure of absolute contrast. The second metric, the averagevalue of the constant false alarm rate (CFAR) detection statistic[51] between the building and the clutter patch, normalizesthe absolute contrast by the clutter standard deviation. In bothcases, the mean and standard deviations are of dB intensities.Larger values indicate that the building is more detectablein the context of a log-normal CFAR detector, while lowervalues indicate it is less detectable. SVA improves absolutecontrast (with respect to Fourier) in all cases. However, whenthe Fourier image exhibits strong positive absolute contrast,SVA decreases the CFAR statistic slightly. In complementaryfashion, MVM decreases the absolute contrast (with respectto Fourier) in all cases. However, when the Fourier imageexhibits strong positive absolute contrast, MVM increases theCFAR statistic dramatically. Underdetermined ASR has thedubious distinction of decreasing both absolute and CFARcontrast in all cases.

To compare the interferometric height accuracy of themethods, we evaluated the mean and standard deviation ofthe polarimetric interferogram phase over four flat regions (30


TABLE VISTATISTICS OF POLARIMETRIC INTERFEROMETRIC PHASE OVER HOMOGENEOUS LEVEL AREAS

30 pixels) within the stadium scene: a roof, the footballfield, two other grassy regions, a low-return parking lot (whichwe expect to be noisy). Table VI compares these statistics forFourier, MVM, SVA, and ASR imagery, with and withoutinterferogram averaging by an ideal 33 smoothing kernel.

With the exception of the low-return parking lot, there isgenerally good agreement (less than 1.4difference) betweenthe unfiltered and filtered Fourier, SVA and MVM meanphases. Also, filtering has relatively little impact on theFourier, SVA, and MVM mean phases. While filtering reducesthe standard deviation of the Fourier phases by a factor of1.7, it reduces the standard deviations of the MVM phases byonly a factor of 1.1; filtering reduces the standard deviationof the SVA phases by a large factor of 2.8. The unfilteredFourier phase standard deviations are roughly 3.0 times aslarge as those of the unfiltered MVM phase standard devia-tions. Thefiltered Fourier phase standard deviations remainroughly 1.9 times as large as those of theunfiltered MVMphase standard deviations. Although the unfiltered SVA phasestandard deviations are significantly larger than the unfilteredFourier standard deviations, the filtered SVA phase standarddeviations are smaller than the filtered Fourier standard devi-ations. The Fourier, SVA, and MVM interferogram phases areequally accurate, yet the MVM phases are both less noisy andresolve finer structural detail; if spatial averaging is performed,the SVA phases are also less noisy and sharper than thecorresponding Fourier phases.

In contrast, the unfiltered ASR mean phases can be as muchas 6.0 different from the unfiltered Fourier mean phases, anddisplay huge standard deviations. However, filtering the ASRinterferogram improves its phase enormously, reducing itsphase standard deviations by roughly a factor of 3.6 and caus-ing its mean phase to be less than 3.5from the filtered Fouriermean phase. The reason that filtering the interferograms hasa relatively greater impact on underdetermined ASR phasethan Fourier or MVM phase is that many of the ASR pixelsdisplay very low intensity. The phase of the low intensitypixels is effectively garbage, which manifests itself as thehuge phase standard deviation. However, the phase accuracyof the prominent points is relatively good (comparable tothe Fourier). Since the interferogram effectively weights eachpixel by its intensity, spatial smoothing of the interferogramspreads the phases of the good pixels spatially, replacing thephase of the bad pixels.

IV. CONCLUSIONS

This paper discussed the rationale for using modern 2-Dspectral estimation algorithms, rather than Fourier transforms,to form SAR imagery, and provided numerous examplesof their application to collected SAR data. Of the methodsdiscussed here, the adaptive methods that seem to offer themost immediate utility are MVM, Pisarenko, ASR, and SVA.MVM and Pisarenko offer improved resolution of prominentscatterers, reduced speckle, and imagery that is more “optical”in character than Fourier imagery. Pisarenko’s method pro-duces cleaner imagery than conventional ARLP, and obviatesthe need to choose a prediction element, since Pisarenkoaverages over all ARLP prediction elements. SVA offers sinc-like resolution without sidelobe artifacts; ASR offers slightlysharper resolution as well as TCR gain, albeit at the cost ofincreased speckle. The computational burden of SVA and ASRis trivial compared with that of MVM and Pisarenko. EVoffers great potential for enhancing resolution and contrastin SAR imagery; however, sophisticated new methods forestimating model order, based on spatial content, rather thansimply intensity, must be developed before EV can realizeits full potential. Methods such as PML and TKARLP offergreat promise for specialized applications in which accuratelocalization of point scatterers is of paramount importance,but offer little utility for general SAR imaging. MVM mayimprove the CFAR detectability of targets in clutter. Overhomogeneous regions, MVM intensity is distributed more log-normally than corresponding Fourier intensity.

We showed how MVM, ASR, and SVA methods can be ex-tended to estimate height interferometrically, and demonstratedthat these methods offer significant improvement over Fouriermethods for imaging interferometric scattering intensity andheight. MVM simultaneously improves resolution and reducesthe variance of both intensity and height imagery. ASR im-proves the spatial definition one sees in interferometric heightsignatures, although it also increases noise. SVA both improvesspatial definition of and reduces noise in interferometric heightsignatures slightly.

The ultimate utility of these adaptive imaging algorithmsfor various SAR applications has yet to be established. Forexample, it is unclear whether and/or to what degree thesemethods will improve automatic target detection and recogni-tion performance. Similarly, the ramifications of these methods


TABLE VIIACRONYMS

for systems-related issues such as area coverage rate, andrequirements on motion compensation accuracy have yet to beaddressed. Many other questions exist, as well. Which methodsare most appealing to SAR image analysts? Which methodsproduce imagery that is best suited to subsequent enhancement,exploitation, or compression methods? Is the computationalburden imposed by MVM, Pisarenko, and EV prohibitive, orjust large? This paper has demonstrated that these adaptivealgorithms can have a dramatic impact on SAR imagery, andthere is reason to believe that imaging techniques based onmodern 2-D spectral estimation methods may prove valuablein the SAR community.

APPENDIX

Table VII summarizes the definitions of the variousacronyms sprinkled liberally throughout the text.

ACKNOWLEDGMENT

The author wishes to thank Wright Laboratory for providingDCS SAR data of the ships near Toledo, OH, and DCS fullypolarimetric IFSAR data of the area around the University ofMichigan football stadium.

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Stuart R. DeGraaf (S’79–M’84) received the B.S.degree from Princeton University, Princeton, NJ,in 1979, and the M.S. and Ph.D. degrees fromRice University, Houston, TX, in 1982 and 1984,respectively, all in electrical engineering.

From 1985 to 1995, he was a Research En-gineer at the Environmental Research Institute ofMichigan, Ann Arbor, where he developed sig-nal processing and automatic target detection andrecognition algorithms for multichannel syntheticaperture radar and ISAR. Since 1995, he has been

an Advisory Engineer at Northrop Grumman Electronic Sensors and SystemsDivision, Baltimore, MD. His research interests are in the areas of adaptiveradar signal processing, automatic target recognition, and image processing.

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