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Novel Bayesian Multiscale MethodsforSpeckle Removal in SAR ImagesPanagiotis TsakalidesComputer Science DepartmentUniversity of CreteHeraklion, [email protected]

Presentation Outline Synthetic Aperture Radar (SAR) concept Heavy-tailed signals and non-Gaussian modeling Multiscale methods for SAR image processing A novel Bayesian processor for image denoising:The WIN-SAR algorithm SAR image denoising results

SAR Imaging ConceptMicrowave wavelenghts: 1 cm - 1 mFrequency ranges: 300 MHz-30 GHz1500 pulses/secPulse duration: 10-50 µsecsTypical Bandwidth: 10-200 MHzSAR produces a two-dimensional (2-D) image. The cross-track dimension inthe image is called range and is a measure of the "line-of-sight" distancefrom the radar to the target. The along-track dimension is called azimuthand is perpendicular to range.

SAR Imaging ConceptThe length of the radarantenna determines theresolution in the azimuth(along-track) direction ofthe image: the longer theantenna, the finer theresolution in this dimension.Each pixel in the imagerepresents the radarbackscatter for that areaon the ground: objectsapproximately the size ofthe wavelength (or larger)appearing bright (i.e. rough)and objects smaller thanthe wavelength appearingdark (i.e. smooth)

SAR QualitiesSynthetic aperture radar (SAR) systems take advantage of thelong-range propagation characteristics of radar signals and thecomplex information processing capability of modern digitalelectronics to provide high resolution imagery.SAR complements photographic and other optical imagingcapabilities because of the minimum constraints on time-of-day andatmospheric conditions and because of the unique responses ofterrain and cultural targets to radar frequencies.

SIR-C/X-SARImage of LA(space shuttleEndeavour, Oct. 1994) Area: 62x32 sq. miles L-band (24 cm) radarchannel Horizontal polarizarion Very dark grey: Pacificocean, LAX, freeway system. Dark grey: mountain slops Lighter grey: suburbanareas, low-density housing Bright white: high-risebuildings and housing allignedparallel to radar flight track Can be used to map fire scarsin areas prone to brush fires,such as Los Angeles

SAR Imaging ApplicationsCivilial ApplicationsHigh resolution remote sensing for mappingSurface surveillance for search and rescueTerrain structural information to geologists for mineralexplorationSea state and ice hazard maps to navigatorsEnvironmental monitoring: Oil spill boundaries on waterChanges in delicate ecosystemsAir pollution monitoring in urban areasAdministration of natural resourcesMilitary ApplicationsBattlefield intelligence: detection and identification ofpotential targets to infer enemy capabilities, tactics andstrategiesMine detectionAutomatic target recognition (ATR)

ProblemSAR images are inherently affected by multiplicative speckle noise, dueto the coherent nature of the scattering phenomenon:I ( x , y ) S ( x , y ) ηm( x , y )Speckle Noise(multiplicative):unit-mean, log-normaldistributed.Need to balancebetween specklesuppression andsignal detailpreservation!!!

Symmetric Alpha-Stable (SαS) Processes:A (fairly) New StatisticalSignal Processing Framework

Quotation“The tyranny of the normal distribution is that werun the world by attributing average levels ofcompetence to the whole population.What really matters is what we do with the tails ofthe distribution rather that the middle.”R. X. CringelyAccidental Empires, 1992It can also be said about least-squares in signalprocessing.

The Symmetric Alpha-Stable (SαS) ModelSαS Characteristic Function:φ (ω ) ejδω γ ω αa: characteristic exponent, 0 α 2 (determines thicknessof the distribution tails, α 2: Gaussian, α 1: Cauchy)δ: location parameter (determines the pdf’s point ofsymmetry)γ: dispersion parameter, γ 0 (determines the spread of thedistribution around its location parameter)for Gaussianγ 2 x variancefor Cauchyγ behaves like variance

SaS Probability Functions010α 1 (Cauchy)α 1.5 110P( X x)Pr( A a)α 1.8 210α 2, GaussianExponentiallydecaying tailsα 2.0 (Gaussian) 3For α 2,algebraic tails.α 1.5α 1.0 (Cauchy)α 0.510No 2nd-ordermoments exist: 4100123DataAmplitude,Amplitude,a x45infinitevarianceprocesses!!!

Properties of SαS LawsNaturally arise as limiting processes via the GeneralizedCentral Limit Theorem.Possess the stability property: The shape of a SαS r.v. ispreserved up to a scale and shift under addition.Contain Gaussian (a 2) and Cauchy (a 1) distributions asmembers.Have heavier tails than the Gaussian: Their tailMore likelyprobabilities are asymptotically power lawsto take values far away from the median (“Noah effect”):P( X x ) cα x-αas x

Properties of SαS LawsHave finite p-order moments only for p a:Ex p forpp pαDo not have finite second-order moments or variances:Ex 2Are self-similar processes: Exhibit long-range dependenceor long memory (“Joseph effect”).

Key Question!Since the variance if associated with the concept ofpower, are infinite variance distributions inappropriatefor signal modeling and processing?No!! Variance is only one measure of spread! Whatreally matters is an accurate description of the shapeof the distribution. Particularly true when outliersappear in the data.Note that bounded data are routinely modeled by theGaussian distribution, which has infinite support.

Real Data ModelingReal sea clutter @ nominal sea condition:sea state 3X-band radar8o look-down anglespatial resolution of 1.52 m (5 ft)sampled at 40 HzClutter probability density modelingSαS with α 1.75Excellent tail fitThe impulsive nature of the clutter data isobvious.Exponential densities

Real Data ModelingRunning estimates of α file: 63131133; 25 24Time [sec]Running estimates of α file: 63131133; 25 e [sec]20251

SαS ApplicationsAstronomy (Holtzmark, 1919)Economic Time Series (Mandelbrot 60’s, McCulloch 90’s)Statistics (Zolotarev, Cambanis, Taqqu, Koutrouvelis, 70’s-90’s)Modeling of Signals and Noise:Radar clutter (Tsakalides and Nikias, 1995)Underwater Noise (Tsakalides and Pierce, 1997)Communications Applications: Telephone line noise (Stuck and Kleiner, 1974) Fading in mobile systems (Hatzinakos and Llow, 1997) Traffic modeling over comm. nets (Taqqu, 1996 – Petropulu, 2002)Multimedia Applications: Modeling, compression, watermarking, and image restoration in theDCT and Wavelet transform domains (Tsakalides et al., 1999-2002)

Multiscale methods for SAR image processing:The Wavelet-based Image-DenoisingNonlinear SAR (WIN-SAR) Processor

Wavelets for Image mModifiedCoefficientsDenoisedImage

2-D Dyadic Wavelet TransformReconstruction of animage from itsapproximation anddetailsDecomposition of animage into anapproximation and 3detail subbandsExpand a signal using a set of basisfunctions obtained from a singleprototype: the “mother wavelet.”Result: A sequence of signalapproximations a successively coarserresolutions.

Multiresolution decomposition – 1st levelLow-resolution approximationThree spatiallyoriented waveletdetailsThe 2-D wavelettransform is appliedalong both thehorizontal andvertical directions,decomposing theimage into fourregions referred asimage subbands.

Multiresolution decomposition – 2nd levelThe LL subbandcontains the low-passinformation and itrepresents a lowresolution version ofthe original image.

Multiresolution decomposition – 3rd levelThe HL (LH) subbandscontain high (low) passinformationhorizontally and low(high) pass informationvertically. The HHsubbands contain highpass information inboth directions.

Previous Work in Wavelet-based Image DenoisingDonoho’s pioneering work: “Denoising by softthresholding” IEEE Trans. Inf. Theory 1995Simoncelli’s “Noise removal via Bayesian waveletcoding,” 1996Gagnon & Jouan’s wavelet coefficient shrinkage(WCS) filter, 1997Simoncelli’s work on texture synthesis, 1999Sadler’s multiscale point-wise product technique,1999Achim’s work on heavy-tailed modeling, 2001Pizurica’s work on inter & intra-scale statisticalmodeling, 2002

Wavelet Shrinkage MethodsSoft ThresholdingTs sgn( s )( s t ),( s) 0, s ts tHard ThresholdingThards s ,(s) 0 ,s tProcessor Outputsofts tInput Coeff.

The WIN-SAR ProcessorWIN-SAR fundamentals:1. Wavelet transform the speckle SAR image.2. SaS modeling of signal wavelet coefficients.3. Bayesian processing of the coefficients in every level ofdecomposition.

Wavelet Coefficients Modeling (1)Empirical pdfdoes not followthe straightGaussian lineEmpirical pdfaccuratelyfollows theSaS line, α 1.3Normal and SαS probability plots of the vertical subband at the first level of decompositionof the image HB06158 from the MSTAR* collection.* http://www.mvlab.wpafb.af.mil/public/sdms/

Wavelet Coefficients Modeling (2)Empirical APDLaplacian APD, p 0.43SαS APD, a 1.3Amplitude Probability Density (APD) plot for the data ofthe previous slide: The SaS provides an excellent fit toboth the mode and the tails of the empirical distribution.

SaS Modeling of Wavelet Subband CoefficientsImage 1.30221.4181.1251.29531.2861.0191.380The tabulated key parameter α defines the degree ofnon-Gaussianity as deviations from the value α 2.

The WIN-SAR MAE Bayesian Estimator After applying the DWT:d ij , k s ij , k ξ ji , k The Bayes risk estimator of s minimizes the conditionalrisk, i.e., the loss function averaged over the conditionaldistribution of s given the measured wavelet coeffs:))s (d ) arg) min s s (d ) Ps d (s d ) dss (d ) The mean absolute error (MAE) estimator is theconditional median of s, given d, which coincides with theconditional mean (due to the symmetry of the distributions):Pd / s (d / s )P(s )s ds) s (d ) s Ps d (s d ) ds Pd / s (d / s )P(s ) ds

The WIN-SAR MAE Bayesian EstimatorPξ (d s )P(s )s ds Pξ (ξ )P(s )s ds) s (d ) Pξ (d s )P(s ) ds Pξ (ξ )P(s ) ds Signal Parameter Estimation - by means of a LSfitting in the characteristic function domain:2n{aˆ s , γˆ s , σˆ } argmin [Φ d (ω i ) Φ d e (ω i )]14243aˆ s ,γˆ s ,σˆwhere:Φdi(ω ) exp( γ s ωαsσ22) exp( ω )2

WIN-SAR MAE Processor I/O CurvesBayesian Processing:Only for a 2 (Gaussian signal), theprocessing is a simple linearrescaling of the measurement:σ s2sˆ ( d ) 2d2σs σFor a given ratio γ/σ, the amount ofshrinkage decreases as α decreases:The smaller the value of α, theheavier the tails of the signal PDF andthe greater the probability that themeasured value is due to the signal.Processor OutputPξ (ξ )P(s )s ds) s (d ) Pξ (ξ )P(s) dsThe WIN-SAR MAEnonlinear “coring”operation preserveslarge-amplitudeobservations andsuppresses smallamplitude values in astatistically optimalfashion.Input Coeff.

Real SAR Imagery Results MSE16.2β0.54s/m0.35

Real SAR ImageryResults (2)WIN-SARShoftThresholdingUrban scene(dense set of largecross-sectiontargets w.intermingled treeshadows

Real SAR ImageryResults (3)WIN-SARShoftThresholdingRural scene

Conclusions1.Introduced a new statistical representationfor wavelet coefficients of SAR images.2.Designed and tested Bayesian processors andfound them more effective than traditionalwavelet shrinkage methods, both in terms ofspeckle reduction and signal detail preservation.3.Proposed processors based on solid statisticaltheory: do not depend on the use of any ad hocthresholding parameter.4. Future work: Analyze multiscale products forstep detection and estimation.

Related Publications1.2.3.4.5.A. Achim, A. Bezerianos, and P. Tsakalides, “SAR Image Denoising viaBayesian Wavelet Shrinkage based on Heavy-Tailed Modeling,” IEEETransactions on Geoscience and Remote Sensing, submitted for publicationconsideration, July 2002.P. Tsakalides and C. L. Nikias, “High Resolution Autofocus Techniques forSAR Imaging based on Fractional Lower-Order Statistics,” IEEProceedings - Radar, Sonar and Navigation, vol. 148, no. 5, pp. 267-276,October 2001.A. Achim, A. Bezerianos, and P. Tsakalides, “Novel Bayesian MultiscaleMethods for Speckle Removal in Medical Ultrasound Images,” IEEETransactions on Medical Imaging, vol. 20, no. 8, pp. 772-783, August 2001.P. Tsakalides, R. Raspanti, and C. L. Nikias, “Angle/Doppler Estimation inHeavy-Tailed Clutter Backgrounds,” IEEE Transactions on Aerospace andElectronic Systems, vol. 35, no. 2, pp. 419-436, April 1999.P. Tsakalides and C. L. Nikias, “Robust Space-Time Adaptive Processing(STAP) in Non-Gaussian Clutter Environments,” IEE Proceedings - Radar,Sonar and Navigation, vol. 146, no. 2, pp. 84-94, April 1999.

The WIN-SAR MAP Bayesian Estimator The MAP estimator is the Bayes risk estimatorunder an uniform cost function:)s d arg maxs d (s d ) arg maxd s (d s) P(s) PP))ss arg maxPξ (d s) Ps (s) arg maxPξ (ξ ) Ps (s)))ss Parameter estimation method: After estimating thelevel of noise σ we find the parameters αs and γs byregressing22 2[(y log log Φ d (ω ) σ ωon w log ωin the model:)]yk µ α wk ε kwhere: µ log(2γ), εk – error term, and (ω κ , κ 1,., Κ ) R

Processor OutputWIN-SAR MAP processor I/O curvesInput Coeff.The plots illustrate the processordependency on the parameter α ofthe signal prior PDF. For a givenratio γ/σ, the amount of shrinkagedecreases as α decreases.Theintuitiveexplanationforthisbehavior is that the smaller the valueof α, the heavier the tails of thesignal PDF and the greater theprobability that the measured valueis due to the signal.