5. POLARIMETRIC SAR DATA CLASSIFICATION

Transcription

1 Polarmetrc SAR data Classfcaton 5. POLARIMETRIC SAR DATA CLASSIFICATION 5.1 Classfcaton of polarmetrc scatterng mechansms - Drect nterpretaton of decomposton results - Cameron classfcaton - Lee classfcaton based on Freeman decomposton H-A-α classfcaton of scatterng mechansms Cloude and Potter proposed an algorthm to dentfy n an unsupervsed way polarmetrc scatterng mechansms n the H-α plane. The key dea s that entropy arses as a natural measure of the nherent reversblty of the scatterng data and that α can be used to dentfy the underlyng average scatterng mechansm. Fgure 1 Optcal mage (left, polarmetrc color coded mage (rghtof the Oberpfaffenhofen test ste. The H-α classfcaton plane s sub-dvded nto 8 basc zones characterstc of dfferent scatterng behavors. The basc scatterng mechansm of each pxel of a polarmetrc SAR mage can then be dentfed by comparng ts entropy and α parameters to fxed thresholds. The dfferent class boundares, n the H-α plane, have been determned so as to dscrmnate surface reflecton (SR, volume dffuson (VD and double bounce reflecton (DB along the α axs and low, medum and hgh degree of randomness along the entropy axs. Detaled explanatons, examples and comments concernng the dfferent classes can be found n the publcaton from Cloude and Potter. Fgure 2 shows the H-α plane and the occurrence of the studed polarmetrc data nto ths plane. 1

2 Polarmetrc SAR data Classfcaton α( H A Fgure 2 Polarmetrc decomposton man parameters: H, A and α. Grey zones n the H-α plane correspond to unfeasble areas. It can be seen, n Fgure 3, that the largest denstes n the occurrence plane correspond to volume dffuson and double bounce scatterng wth moderate and hgh randomness. Medum and low entropy surface scatterng mechansms are also frequently encountered n the scene under examnaton. 2

3 Polarmetrc SAR data Classfcaton α( 90 DR VD SR H Fgure 3 H-α scatterng mechansm dentfcaton plane (top. Polarmetrc data occurrence n the H- α plane (bottom. Data dstrbuton n the H-α plane does not show, for the consdered scene, dstnct natural clusters belongng to a sngle scatterng mechansm class. Therefore, dentfcaton results may hghly depend on segmentaton thresholds. Results of the unsupervsed dentfcaton procedure are presented n Fgure 4. It can be observed n Fgure 4 that the proposed segmentaton n the H-α plane permts to dentfy n a macroscopc way the type of scatterng mechansm. Agrcultural felds and bare sols are characterzed by surface scatterng. Scatterng over forested areas s domnated by volume dffuson whle urban areas are manly characterzed by double bounce scatterng. It may be noted that the dentfcaton process slghtly overestmates volume dffuson and double bounce scatterng over surfaces. 3

4 Polarmetrc SAR data Classfcaton Fgure 4 Unsupervsed scatterng mechansm dentfcaton n the H-α plane. 5.2 Polarmetrc data statstcal segmentaton Maxmum Lkelhood Supervsed segmentaton Bayes optmal decson rule The problem of optmal segmentaton over a SAR mage over a fxed number of clusters may be formulated as follows. How to assgn, n an optmal way, a SAR mage pxel, {, L, } 1 M, accordng to a SAR observable, x? p, to one of the M possble clusters The answer s gven by Bayes optmal decson rule usng the condtonal probablty of the dfferent clusters Decde p f P( x > P( x (1 A pxel s assgned to the most probable cluster condtonally to the observaton of x over the pxel under consderaton. The probablty of error related to ths decson s obtan from the a posteror probabltes of the unselected clusters by ( x P ( error x P P( x It s obvous, that assgnng a pxel to the cluster wth the hghest a posteror probablty mnmzes the condtonal error probablty. The total probablty s then computed usng (2 4

5 Polarmetrc SAR data Classfcaton P ( error x P( error x P( x dx (3 The set of a posteror probabltes P( x express such quanttes usng Bayes rule P ( x P x P( x P( s generally uneasy to derve. One prefers then to ( (4 Insertng ths expresson n (1, the expresson of the optmal decson rule becomes Decde p f P( x P( > P( x P( (5 Where P( x s the lkelhood of the observaton x, gven and probablty of. P( s the pror Addtonally, f the pror probabltes are supposed to be equal, the optmal decson rule may wrtten usng only cluster lkelhood as Decde p f Arg max P( x ( Supervsed segmentaton scheme A supervsed segmentaton scheme may be represented as a three-step process descrbed n the followng. Frst tranng data, separated nto M dstnct groups or classes, are suppled to the segmentaton algorthm. Tranng nputs may be : - user suppled under the correct format - sampled from the mage under study - sampled from SAR mages acqured over other scenes Durng the second phase, the segmentaton algorthm learns from the tranng data sets how to dscrmnate the M classes and establshes decson rules. In the case of supervsed segmentaton based on Bayes optmal decson rule, the segmentaton algorthm learns the dfferent statstcal quanttes useful to compute the expresson mentoned n (6. Once the learnng phase has converged, the algorthm assgns each pxel of the SAR mage under study to one of the M clusters provded n the frst phase, accordng to decson procedure derved durng the second step Supervsed ML segmentaton of [S matrx mages In the case of [S matrx mages, the SAR observaton x s replaced by a 3 element complex T target vector k S, 2S, S. [ It has been verfed that when the radar llumnates an area of random surface of many elementary scatterers, k can be modeled as havng a multvarate complex crcular gaussan probablty densty functon N C ( 0,[ Σ of the form : 5

6 Polarmetrc SAR data Classfcaton exp( k [ Σ P( k q π [ Σ 1 k (7 where q stands for the number of elements of k, equal to three n the monostatc case, represents the determnant, and [ Σ E( kk s the global (3 3 covarance matrx of k. The lkelhood of a target vector k gven a cluster s then gven by ( P k exp( k [ Σ q π [ Σ 1 k (8 Wth [ Σ practce the actual value of the covarance matrx of cluster [ Σ ts maxmum lkelhood estmate Wth n computed durng the learnng phase. In remans unknown and the covarance matrx s replaced by Σˆ defned as [ 1 [ Σˆ kk (9 n p the number of pxels belongng to the tranng cluster From a computatonal pont of vew, t s generally preferable to deal wth log-lkelhood functon, L( k nstead of the expresson gven n (8 Where ( ˆ 1 tr [ kk L ( ln[ ˆ ([ ˆ 1 k Σ tr Σ kk q lnπ. 1 Σ stands for the trace of ˆ 1 [ Σ kk and equals k [ ˆ Σ k. The logarthm functon beng strctly ncreasng wth ts argument, the optmal decson rule s transformed, n the case of [S matrx segmentaton, to Decde p f ( [ ˆ Arg mn d k Σ wth ˆ 1 ln[ ˆ [ ˆ (11 d k [ Σ + Σ + tr Σ kk ( ( Where the varable ( ˆ (10 d k [ Σ may be assmlated to a statstcal dstance, derved from the opposte of log-lkelhood functon wthout constant terms. Durng the segmentaton process a pxel s then assgned to the mnmum dstance cluster, havng the closest statstcs. The supervsed segmentaton algorthm may be summarzed as follows 6

7 Polarmetrc SAR data Classfcaton Intalze pxel dstrbuton over M custers fro m tranng data sets For each cluster 1 [ Σˆ k k N For each pxel, k m f d ( k, < d ( k / 1, L, M Fgure 5 Supervsed ML segmentaton scheme Supervsed ML segmentaton of [T or [C matrx mages It has been shown that assumng that target vectors have a N C ( 0,[ Σ dstrbuton, a sample n-look covarance matrx degrees of freedom, Where W C [ C 1/ n kk, follows a complex Wshart dstrbuton wth n ( n,[ Σ, gven by n p([ T wth qn n [ T n q K( n, q π Γ(. represents the Gamma functon. exp( tr( n[ Σ K( n, q[ Σ q( q 1 / 2 q 1 n 1 [ T Γ( n + 1 A development smlar to the one presented n the case of scatterng matrces leads to the followng decson rule Decde p f ( [ ˆ Arg max L [ T Σ wth [ ˆ ln[ ˆ [ ˆ 1 (13 L [ T Σ n Σ ntr Σ [ T + qnln n + ( n qln[ T ln K( n, q Wth ( ( Σˆ, the maxmum lkelhood estmate of the coherency matrx defned as [ 1 [ Σˆ [ T (14 n p Takng the opposte of the lower expresson of relaton (13 and removng terms that do not depend on the cluster under test, the optmal decson rule becomes Decde p f ( [ ˆ Arg mn d [ T Σ wth [ ˆ ln[ ˆ [ ˆ (15 d [ T Σ + Σ + tr Σ 1 [ T ( ( 7

8 Polarmetrc SAR data Classfcaton It may be noted that, snce covarance and coherence matrces are related by a smlarty transformaton, the form of the statstcal dstance expressed n (15 s nvarant under a change from coherency matrx form to covarance matrx form and (15 s adapted to both [T and [C representatons. The supervsed segmentaton algorthm may be summarzed as follows Intalze pxel dstrbuton over M custers fro m tranng data sets For each cluster 1 [ Σˆ [ T N For each pxel, [ T m f d ([ T, < d ([ T / 1, L, M Fgure 6 Supervsed ML segmentaton scheme Supervsed segmentaton performance The performance of a supervsed segmentaton process s generally estmated va a confuson matrx computed from the tranng data. Each element of the tranng data set s processed through the supervsed classfer. The confuson matrx summarzes the percentage of pxels belongng to an orgnal cluster,, effectvely affected to one of the M possble clusters, matrx of an deal, error-free, segmentaton s dagonal., wth 1L M. The confuson User-defned tranng data sets are generally orented towards the dscrmnaton of dfferent knds of medum and may not represent all scatterng behavors encountered over a natural scene. Therefore, some parts of a SAR mage may have statstcs that do not ft wth any of the proposed cluster ones. In ths case, the assgnment of the correspondng pxels may lead to rrelevant nterpretaton of the segmentaton results. One possble soluton to ths problem conssts n creatng a reect class. A pxel s assgned to ths reect class f t s too much dstant, from a statstcal pont of vew, from all the learnng clusters. A possble metrc may be defned from the statstcal dstances defned n (11 and (15. A pxel belongs to the reect class f ts dstance to the closest cluster exceeds a gven factor of the wthn-class dstance standard devaton. Ths decson process may be summarzed as follows Decde p re d [ T [ Σ ˆ > σ d Arg mn d [ T [ Σˆ f ( r wth (. (16 8

9 Polarmetrc SAR data Classfcaton and ( ( σ d [ T [ ˆ d [ T [ ˆ d Σ N p N p Where the varable r represents a user-defned reecton factor. Σ Maxmum Lkelhood unsupervsed segmentaton It has been showed, n former paragraphs that a pxel can be assgned to a cluster n an optmal way, accordng to the followng decson rule. Decde p f Arg max P( x (17 In the case of an unsupervsed segmentaton scheme, the lkelhood of a cluster, ( x P, cannot be estmated snce ts calculaton requres to compute the ML estmate of the cluster coherency matrx from the samples contaned n the cluster. An optmal soluton to the unsupervsed segmentaton problem conssts n dstrbutng the pxels of an mage over the set of M clusters so as to maxmze the global lkelhood defned as the product of all ndvdual lkelhood functons. A rgorous resoluton requres to test all the possble combnatons and to select the one correspondng to the maxmum ont lkelhood value. Ths optmal soluton cannot be appled due to the unrealstc computatonal load t nvolves. Alternatve solutons based on sub-optmal teratve optmzaton procedures are generally preferred Prncple of statstcal K-means clusterng The K-means procedure s an teratve optmzaton algorthm descrbed by the followng synopss : 9

10 Polarmetrc SAR data Classfcaton Intalze pxel dstrbuton over M clusters Compute each cluster center Assgn each pxel to the closests cluster No Convergence? Yes Fgure 7 Synopss of the K-means clusterng algorthm. The algorthms begns wth the ntalzaton of the mage pxel dstrbuton over the M clusters. Ths dstrbuton may be done n a random way or accordng to user specfcatons. Once all pxels are affected, the dfferent cluster centers are computed accordng to the processed data type. Each pxel s then affected to the closest cluster accordng to a dstance measure. The convergence of the algorthm s then tested usng stablty metrcs. If a termnaton crteron s met, the segmentaton stops, otherwse a new teraton starts over from the class center computaton step. The K-means algorthm ams to optmze a global functon by teratvely optmzng local expressons. It s known ths type of technques may get stuck nto locally stable states and fal to determne the optmal pxel dstrbuton. The ntalzaton of the pxel dstrbuton nto N classes s then a crtcal stage of the K-mean clusterng algorthm. An adequate ntalzaton permts a fast convergence and provdes correctly segmented clusters. The convergence of the algorthm s evaluated by testng a condton of termnaton. Such a crteron may be defned from the estmaton of the classfcaton qualty, or consst n a maxmum number of teratons or n a suffcently low number of pxels that are dfferently classfed from one teraton to the other Unsupervsed ML segmentaton of [T or [C matrx mages The K-means clusterng algorthm may be adapted to the segmentaton of [T or [C polarmetrc representatons as shown on the followng synopss 10

11 Polarmetrc SAR data Classfcaton Intalze pxel dstrbuton over M custers For each cluster 1 [ Σˆ [ T N For each pxel, [ T m f d ([ T, < d ([ T / 1, L, M No Convergence? Yes Fgure 8 Unsupervsed Maxmum Lkelhood segmentaton scheme. The class center ntalzaton step conssts n computng each cluster coherency matrx ML estmate. Pxels are then affected to a cluster accordng to the ML dstance measure derved n ( Unsupervsed Wshart H-α segmentaton The partcularty of the dentfcaton procedure based on H and α, ntroduced n a former paragraph, resdes n the estmaton of the type of observed meda from a physcal nterpretaton of canoncal scatterng mechansms usng robust ndcators. Nevertheless, the analyss of natural scenes usng ths unsupervsed approach may reach some lmtatons: - The arbtrarly fxed lnear decson boundares n the H-α plane may not ft data dstrbuton. A natural cluster correspondng to smlar targets may le across a fronter n the decson plane. In ths case, pxels wth very smlar characterstcs may be assgned, n an almost random way, to dfferent classes due slghtly dfferent locatons n the H-α plane. Ths effect can be observed n Fgure 4 where the varablty n natural meda polarmetrc features lead to nosy classfcaton results. - Even f the computaton of H and α requres fully polarmetrc data, these two parameters do not represent the whole polarmetrc nformaton. The use of other ndcators such as the span or specfc correlatons coeffcents may mprove the classfcaton results n a sgnfcant way. Segmentaton procedures based on the whole coherency matrx statstcs permt to overcome the lmtatons mentoned above. Nevertheless, t s shown n the followng, that the physcal nterpretaton of the scatterng phenomenon permts to enhance n a sgnfcant way the performance of statstcal segmentaton schemes. 11

12 Polarmetrc SAR data Classfcaton The unsupervsed segmentaton scheme mentoned above s ntalzed n an effcent way wth the results of the unsupervsed dentfcaton of a scatterng mechansm, usng H, A and α. Ths ntalzaton provdes 8 stable clusters relatng to the underlyng physcal scatterng mechansm. Intalze pxel dstrbuton over 8 clusters For each cluster compute Σ ˆ [ Segmented H- α plane Assgn each pxel to a cluster No Convergence? Yes Fgure 9 Unsupervsed Wshart H-α segmentaton scheme. Fgure 10 Wshart H-α segmentaton results. C1 C2 C3 C4 C5 C6 C7 C8 An mportant mprovement n the segmentaton accuracy can be observed n the mage presented n Fgure 10. The man knds of natural meda are clearly dscrmnated by the Wshart H-α segmentaton scheme. Ths unsupervsed classfcaton algorthm modfes the 12

13 Polarmetrc SAR data Classfcaton decson boundares n an adaptve way to better ft the natural dstrbuton of the scatterng mechansms and takes nto account nformaton related to the back-scattered power Unsupervsed Wshart H-A-α segmentaton The ML Wshart segmentaton may be further mproved by explctly ncludng the ansotropy nformaton durng the segmentaton procedure. As mentoned prevously, the ansotropy ndcates the relatve mportance of secondary mechansms obtaned from the expanson of a coherency matrx. Ths polarmetrc ndcator s partcularly useful to dscrmnate scatterng mechansms wth dfferent egenvalue dstrbutons but wth smlar ntermedate entropy values. In such cases, a hgh ansotropy value ndcates two domnant scatterng mechansms wth equal probablty and a less sgnfcant thrd mechansm, whle a low ansotropy value corresponds to a domnant frst scatterng mechansm and two non-neglgble secondary mechansms wth equal mportance. Among the dfferent approaches tested, the best way to ntroduce the ansotropy nformaton n the classfcaton algorthm conssts n mplementng two successve segmentaton procedures as shown n Fgure 11. Intalze pxel dstrbuton over 8 clusters Segmented H- α plane A > 0.5 Χ m Χ m clusters Ansotropy Fgure 11 Unsupervsed Wshart H-A-α segmentaton scheme. Polarmetrc data are frst segmented accordng to the algorthm presented n the former paragraph. Once ths procedure has converged, the 8 resultng clusters are splt nto 16 ones 13

14 Polarmetrc SAR data Classfcaton by comparng the ansotropy of each pxel to a threshold fxed to 0.5. The 16 segments are then used to ntalze a second Wshart ML segmentaton procedure. C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 Fgure 12 Wshart H-A-α segmentaton results. The segmentaton results presented n Fgure 12 show an enhanced descrpton of the Oberpfaffenhofen scene. The ntroducton of the ansotropy n the clusterng process permts to splt large segments nto smaller clusters dscrmnatng small dspartes n a refned way. Several knds of agrcultural felds are separated. The runway and other low ntensty targets are dstngushed from other surfaces. Buldngs are dscrmnated from other types of scatterers present n urban areas. The Wshart H-A-α classfcaton scheme gathers nto segments pxels wth smlar statstcal propertes, but does not provde any nformaton concernng the nature of the scatterng mechansm assocated to each cluster. 14