An Adaptiv Clustring MAP Algorithm to Filtr Spckl in Multilook SAR Imags FÁTIMA N. S. MEDEIROS 1,3 NELSON D. A. MASCARENHAS LUCIANO DA F. COSTA 1 1 Cybrntic Vision Group IFSC -Univrsity of São Paulo Caia Postal, 369, SP,13560-970,Brasil -mails: {luciano,fsombra}@ultra3000.ifsc.sc.usp.br UFSCar - DC Via Washington Luiz Km 35, Caia Postal 676 13565-905 - São Carlos, SP, Brasil nlson@dc.ufscar.br 3 UFC- DEE Campus do PICI Bloco 705 60455-760 - Fortalza,CE, Brasil fsombra@d.ufc.br Abstract. SAR imags ar gnrally affctd by a multiplicativ nois, calld spckl, which dgrads th quality of ths imags. Using this modl w prsnt an algorithm basd on th Maimum a Postriori (MAP) critrion to rduc spckl nois of multilook amplitud data. Th spckl in ths imags is approimatly dscribd by th Squar Root of Gamma distribution, which is usd to dvlop MAP filtrs using diffrnt a priori distributions. W also suggst th combind us of MAP and th k- mans clustring algorithm as a formal way to choos th bst window siz to updat noisy pils. W conclud this work by calculating th cofficint of variation, dfind as th ratio of th standard dviation to man, of MAP filtrd imags and of th original imag to masur th rduction of th spckl in homognous aras. Kywords: Multilook imag, MAP stimator, k-mans classifir, spckl nois, filtring.
I. Introduction Spckl nois is on of th main charactristics prsnt in imags obtaind by cohrnt imagry systms such as synthtic aprtur radar (SAR), lasrs, and ultrasound imags. This kind of signal-dpndnt nois limits th visual intrprtation of ths imags bcaus it obscurs th scn contnt. It is rcommndd to ovrcom this difficulty prior to classification procdurs, for ampl. In such cass, filtring algorithms usd as "a prior" stp would improv th classification prformanc. In th litratur, classical filtrs, lik L (L, 1980), Kuan, (Kuan t al. 1985), Sigma (L, 1983), Frost (Frost t al., 198), Mdian (Castlman, 1996) amongst othrs aim to rduc th nois spckl. Th idal filtr must smooth th nois without liminating radiomtric and ttural information that ar fundamntal for dtail prsrvation (Lops t al., 1988). It has bn primntally vrifid in svral works that for SAR imags ovr homognous aras, th standard dviation of th signal is proportional to its man L (1981). This fact suggsts th us of th multiplicativ modl for th spckl and it was usd by Kuan t al. (1987) to propos an adaptiv non-linar pointwis filtr that satisfis th MAP critrion for singl look, quadratic dtction and Gaussian a priori dnsity. Th varianc ratio of th original and noisy imag is usd as a masur of local proprtis by th adaptiv filtrs to control filtr window siz (Li, 1988). By combining th MAP filtr and th k-mans clustring algorithm ovr Changl Li s varianc ratio (Li, 1988) it is possibl to classify th noisy imag in rgions of homognous statistics. In ordr to adapt th MAP filtring to th local statistics, th thrsholds on th varianc ratio ar chosn to dtrmin th window sizs for paramtr stimation. In practical applications th nois is oftn rducd by multilook procssing, which can b don by avraging indpndnt sampls of svral imags. With an incrasing numbr of avragd sampls, th Rayligh distribution of a signal approimats a Gaussian distribution (Hagg t al., 1996). Although improving th signal to nois ratio by N, whr N is th numbr of looks usd to gnrat th imag, this tchniqu also diminishs th spatial rsolution. In this work th spckl distribution ovr multilook amplitud data is modlld by a Squar Root of Gamma and w us it in th proposd MAP filtr. Th statistical paramtrs in th filtring algorithm ar calculatd by using a fid window siz (55) around ach pil or choosing a window siz according to th dgr of roughnss of th non-noisy signal around th pil. Th clustring of th cofficints of variation dtrmins th suitabl filtring window. It will b shown in this articl that this fact lads to a bttr filtring rsult. In sction II w prsnt th multiplicativ modl for th spckl and drivd from this modl in sction III th MAP stimator is formulatd using th a priori Gaussian, Gamma, Chisquar, Eponntial and Rayligh dnsitis. Thr is a brif discussion about th implmntation rsults in sction IV. Sction V summarizs th conclusions and sction VI outlins possibilitis for futur work.
II. Multiplicativ Modl and Spckl Statistics Th modl usd to dscrib th spckl is givn in trms of a multiplicativ nois givn by quation (1), whr z A dscribs th amplitud SAR noisy imag, is th original signal and n A is th nois with unitary man and standard dviation n. Th multiplicativ modl is a good modl ovr homognous aras bcaus th standard dviation is proportional to th man. Th spckl nois and th original imag ar assumd to b dcorrlatd. z = A. n A (1) Equation rprsnts th β ind which is th ratio of th standard dviation to th man usd to masur th strngth of th spckl in this kind of imag and N is th numbr of looks. µ = 057. N () Th spckl for 1 look amplitud SAR imag obys a Rayligh distibution as in quation 3. An N look intnsity spckl imag is obtaind by avraging N intnsity singl look imags and is modlld as a Gamma distribution (quation 3.a). Th multilook amplitud spckl can b obtaind by avraging th N amplitud singl look imags or by avraging th N intnsity imags and thn taking th squar root (Frry t al., 1997). Th lattr follows a Squar Root of Gamma distribution (L t al.,1994) as dscribs th quation 3.b. Th formr is dscribd by th convolution of N Rayligh distributions and for N= thr is a closd form for it. Sinc thr is no closd form for th distribution for N 3 it is costumary to mak an approimation (Yanass t al., 1995) and dscrib it by th Squar Root of Gamma distribution. g f ( g) = (3) g whr g is th random variabl with paramtr. f ( g) = ( g) Γ( λ) λ 1 g, g > 0 whr Γ(λ) is a valu of th gamma function and g is th random variabl with paramtrs and λ. For λ=1 th Gamma distribution is idntical to th Eponntial distribution. For λ=n/ (n>0) and =1/ th Gamma distribution is quivalnt to a Chi-squar distribution. (3.a) f ( g) Ng N N N 1 ( g), g, N > 0 N Γ( N) = (3.b) III. MAP Estimator Th MAP stimator of is obtaind by maimizing th a postriori probability dnsity function f ( z), which can b rlatd to th a priori distribution f ( ) through quation (4). To simplify th notation, th inds (A) in th following quations ar droppd out. Th
conditional distribution f(z ) which dscribs th modl follows a Squar Root of Gamma distribution is givn by th quation (5). f ( z) = f ( z ) f ( ) f ( z) (4) ln f ( z ) ln f ( ) + = ˆ MAP = 0 (4.b) N N N N 1 ( z ) = ( ) N f (5) Γ( N) whr N is th numbr of looks. This follows from th multiplicativ modl (quation 1) sinc givn th signal, th conditional probability dnsity of z is a Squar Root of Gamma dnsity with man valu (th man valu of n is on). ln N z Γ ( N + 1 / ) [ f ( z )] = + 3 Γ ( N ) (6) W formulatd svral MAP filtrs using diffrnt a priori dnsitis. Ths MAP quations ar prsntd in th following. III.1 Givn th Gaussian a priori dnsity f ( ) = 1 π 1 µ ( ) Th Gaussian MAP filtr is givn by th solution of th quation (8). This quation was obtaind using th "a priori" knowldg from (7) combind with (6) in quation (5). 4 Γ ( N) 3 Γ ( N ) µ + Γ ( N) N z Γ ( N + 1/ ) = 0 (8) Th stimators for µ and ar obtaind by th following prssions: µ ˆ ˆ w 1 = µ ˆ z = m = z w ˆ R = z z µ ˆ z. = 1+ n n i= 1 i whr w is th siz of window around th filtrd pil and n is th nois varianc, which is a constant dtrmind by th numbr of looks and th typ of dtction. R (Changl Li s (7) (9)
paramtr) is th local ratio of original and noisy imag varianc. Th st of prssions in quation (9) aros from th multiplicativ modl with unitary man for th spckl nois. Bfor filtring th noisy imag, w calculat th local R (Li s ratio) paramtr for all pils using 33 windows. Th on dimnsional k-mans algorithm ovr Li s ratio classifis pils with similar statistics. Pils assignd to th sam clustr ar filtrd with th sam window siz for paramtr stimation (33 or 55). Th ral and positiv roots of th MAP quations whos valus ar btwn th man and th obsrvd pil ar takn as th filtrd pil valus. III. Givn th Gamma a priori dnsity f ( ) = ( Γ( λ) ) λ 1 Th MAP stimator is givn by th solution of th quation 3 Γ ( N ) + Γ ( N )( N + 1 λ) z Γ ( N + 1/ ) = 0 (11) (10) ˆ µ ˆ λ = sˆ (1) whr th paramtrs and λ ar stimatd by th sampl man and th sampl varianc through th mthod of momnts, using th multiplicativ modl. Th stimatd paramtr s ˆ is th varianc of th original signal calculatd from th noisy signal through quation 9. III.3 Givn th Chi-squar a priori dnsity 1 f ( ) = n/ Γ( n / ) n/ 1 / whr n dnots a Chi-squar distribution with n dgrs of frdom. Th MAP quation is givn by (13) 3 Γ ( N ) + Γ ( N )[ + 4N n] 4z Γ ( N + 1/ ) = 0 n) 1 = µ ) = µ ) = w w z z k k = 1 (14) III.4 Givn th Eponntial a priori dnsity f ( ) = Th noisy pil is updatd with th solution of th MAP quation (15)
3 ˆ Γ ( N) 1 = µ ˆ + Γ ( N)N z Γ ( N + 1/ ) = 0 (16) III.5 Givn th Rayligh a priori dnsity f ( ) = (17) Th MAP stimator is givn by th solution of th quation 4 Γ ( N) + Γ ( N) (N 1) z Γ ( N + 1/ ) = 0 (18) ˆ = µ ˆ π IV. Eprimntal Rsults Th original imag in Figur 1.a is a pic of 481481 pils imag of th National Forst of Tapajós, Pará, Brazil, takn on Jun, 6, 1993 by th JERS-1 satllit. It is a thr looks, amplitud dtctd imag. Th prsntd filtrs wr applid and thir prformanc was valuatd in trms of th spckl rduction ind, β, which is th ratio btwn th standard dviation and th man ovr homognous aras. In Tabl 1, th stimatd β inds in a 4141 pils pic of th original imag with initial coordinats (51,376) ovr a forst rgion ar shown. Th last row ar th thortical and practical valus of β inds ovr this rgion. Th closnss of th thortical and practical β cofficints implis that this forst ara can b considrd homognous. In th first column ar th stimatd β inds in th MAP filtrd ara without k-mans and in th scond column ar th β inds in th MAP filtrd ara with th k-mans algorithm.
MAP FILTER β= z/µ z β= z/µ z GAUSSIAN 0.113 0.017 GAMMA 0.137 0.019 CHI-SQUARE 0.16 0.071 EXPONENTIAL 0.00 0.18 V. Conclusions RAYLEIGH 0.19 0.16 HOMOGENEOUS REGION (41X41 PIXELS) 3 LOOKS THEORETICAL 0.941 Tabl 1-Estimatd β inds PRACTICAL Th nonlinar, adaptiv algorithms basd on th MAP critrion proposd in this papr, bsids dcrasing substantially th standard dviation to th man ratio improvd th discrimination of th prdominant classs (rgnration and forst) as shown by th histograms. Th smoothing of th spckl in th Gaussian, Gamma, Chi-squar, Eponntial and Rayligh MAP filtrd imags has bn valuatd by th β ind in Tabl 1 and from th histograms. Th inds wr calculatd ovr an homognous ara of forst (4141 pils). Som spckl rduction can b discrnd in Figurs.a, 3.a, 4.a and 5.a which prsntd th bst β inds and in Figurs 3.b and 5.b th discrimination of classs has bn improvd by th us of th k-mans algorithm. In th Chi-squar MAP filtrd imag histogram, Figur 9.b, th classs ar bttr discriminatd than for th othr distributions. Th β inds for th Eponntial and Rayligh MAP filtrs wr th lowst, and vn whn using k-mans th classs discrimination is not as vidnt as in Figurs 6.b and 11.b. Basd on ths rsults, th improvmnt obtaind through th us of th k- mans algorithm bcom clar. VI. Furthr Dvlopmnts Futur dvlopmnts will us rgion growing tchniqus to dtrmin windows with adaptiv siz and shap (not ncssarily squar) to stimat th local paramtrs of th MAP filtrs. Acknowldgmnts Mrs. Fátima N. S. Mdiros was partially supportd by a PICD-CAPES scholarship. Th authors wish to thank Sidni J. S. Sant Anna, Pdro R. Viira, Antônio M. V. Montiro Aljandro C. Frry and Corina C. F. Yanass for thir assistanc with tst data st. Rfrncs L, J.S. Digital Imag Enhancmnt and Filtring by Us of Local Statistics, IEEE Trans. on Pattrn Analysis and Machin Intllignc, Vol., No., pp. 165-168, 1980. 0.309
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(.a) GAUSSIAN MAP FILTERED (.b) GAUSSIAN MAP FILTERED HISTOGRAM (3.a) GAUSSIAN MAP FILTERED IMAGE (3.b) GAUSSIAN MAP FILTERED IMAGE HISTOGRAM (4.a) GAMMA MAP FILTERED IMAGE (4.b) GAMMA MAP FILTERED IMAGE HISTOGRAM
(5.a) GAMMA MAP FILTERED IMAGE (5.b) GAMMA MAP FILTERED IMAGE HISTOGRAM (6.a) EXPONENTIAL MAP FILTERED IMAGE (6.b) EXPONENTIAL MAP FILTERED IMAGE HISTOGRAM (7.a) EXPONENTIAL MAP FILTERED IMAGE (7.b) EXPONENTIAL MAP FILTERED IMAGE HISTOGRAM
(8.a) CHI-SQUARE MAP FILTERED IMAGE (8.b) CHI-SQUARE MAP FILTERED IMAGE HISTOGRAM (9.a) CHI-SQUARE MAP FILTERED IMAGE (9.b) CHI-SQUARE MAP FILTERED IMAGE HISTOGRAM (10.a) RAYLEIGH MAP FILTERED IMAGE (10.b) RAYLEIGH MAP FILTERED IMAGE HISTOGRAM
(11.a) RAYLEIGH MAP FILTERED IMAGE (11.b) RAYLEIGH MAP FILTERED IMAGE HISTOGRAM