Unsupervised Image Classification using the Entropy/Alpha/Anisotropy Method in Radar Polarimetry Shane R Cloude*, Eric Pottier**, Wolfgang M Boerner *** *AEL Consultants, 26 Westfield Av., Cupar, Fife, KY15 5AA,Scotland, UK Tel/Fax: +44 (1334) 653958 e-mail: [email protected], web-site: http://www.aelc.demon.co.uk ** Institut dElectronique et de Tlcommunications de Rennes, UMR — CNRS, Universit de Rennes 1, Ple Micro-Ondes Radar, Campus de Beaulieu — Bat. 11D, 35042 Rennes Cedex France e-mail:[email protected] ***UIC-ECE Communications, Sensing & Navigation Laboratory 900 W. Taylor St., SEL (607) W-4210, M/C 154, CHICAGO IL/USA-60607-7018 T&F: +[1] (312) 996-5480 e-mail: [email protected] Abstract In this paper we re-examine the entropy alpha approach to radar polarimetry and show how the basic method may be augmented by the addition of two new polarizing parameters, the propagation and helicity phase angles and three depolarizing parameters, the anisotropy A and two depolarizing eigenvector angles. We apply the technique to Polarimetric AIRSAR data for land, ice and forestry applications. 1. Introduction In this paper we describe a method for unsupervised classification of polarimetric SAR imagery based on an eigenvector analysis of the coherency matrix. In previous publications [1,2,3], the entropy/alpha plane was introduced as a convenient means of displaying these eigenvector properties. It was further shown how, by using simple physical models, up to 8 important classes of terrain cover can be classified using this technique [1]. This method has also recently been used as a starting condition for a more sophisticated iterative classification method employing multi-variate Wishart statistics [5,6,7]. These studies have all highlighted the importance of optimising the number and diversity of classes to be used as input to the classifier. For this reason it is of interest to reconsider the details of the method and assess any possible extensions into new class types. Given the widespread availability of high quality calibrated POLSAR data and the imminent launch of space based polarimetric imaging radars, it is of timely interest to consider such an extension. To do this, we first up-date the classification boundaries using recent developments in radar polarimetry [3,8,9] and interferometry [10,11,12] and then highlight the potential of using several new parameters for further refinements in the classification procedure. We concentrate on two new sets of polarimetric parameters, namely those arising from a polarising/depolarising decomposition of the coherency matrix [13]. The former are derived from the maximum eigenvector and offer two invariant phase angles, which so far have not been employed for classification. The latter arise from the depolarising subspace alone. One of these, the scattering anisotropy A, has already been suggested as a new feature to distinguish depolarising mechanisms in surface and volume scattering [3,5]. For example, A can be used to distinguish rough from smooth surfaces and to classify different types of vegetation cover. However, the depolarization subspace also offers two new parameters, which so far have not been fully exploited. We first derive the key features of this classification method and then present some examples of its application to POLSAR data for land, sea ice and forestry applications. 2. Navigating the Entropy/Alpha Plane The basic observable in radar polarimetry is the 2 x 2 complex scattering matrix [S]. For backscatter problems, the reciprocity theorem forces HV=VH and so this matrix is symmetric. When such SLC data is available then the coherency matrix can be directly derived by vectorising [S] into k using the Pauli spin matrices and then averaging products of the complex elements to generate a 3 x 3 positive semi-definite Hermitian matrix [T], as shown in equation 1. However, often the user is provided not with SLC data but with multi-look Stokes matrix data [M]. It is important to realise that this format is entirely equivalent to the coherency matrix, although some care is required in transforming between matrices. The problem arises that, because of coding or measurement errors, small negative eigenvalues are obtained in converting from [M] to [T]. To cope with this case, matrix-filtering techniques have been developed [4]. In equation 1 we also show for reference the explicit mapping between the real symmetric Stokes matrix [M] and the coherency matrix [T]. Using this relationship [T] can be easily derived from Stokes matrix format data such as provided from the AIRSAR system. ¡ ¤ ¦ £ m1 m2 m3 m4 £ ¦ ¦ £ m m m m § [M] 2 5 6 7 £ ¦ m3 m6 m8 m9 £ ¦ ¢ ¥ m4 m7 m9 m10 ! 1 # 2 m1 m5 m8 m10 m2 im9 m3 im7 ¨ © # 1 1 # T m2 im9 2 m1 m5 m8 m10 m6 im4 - 1) " 2 1 m3 im7 m6 im4 2 m1 m5 m8 m10 D G D G (*) +,.-0/132 ' ' I F 2 * * I Shh Svv Shh Svv Shh Svv 2 Shh Svv SHV F I F a z1 z2 F I $ % & 5.6 7839;: <.=: I F * 2 * J 4 I T S S S S S S 2 S S S F z b z I F hh vv hh vv hh vv hh vv HV 1 p 3 E H ?3@ B.CA F I > * * 2 H E zzp z3 p c 2 Shh Svv SHV 2 Shh Svv SHV 4 Shv If we write [T] in the explicit form shown in equation 1 then the 3 real non-negative eigenvalues of [T] can then be derived analytically as 1 1 3 3 K LNMOM2 S2 1 S3 1 1 Tr(T) 1 3 3 3 3.S3 3.2 1 P M 3 K (1 i 3)S 1 (1 i 3)S P MOP L 2 3 - 2) 2 2 1 Tr(T) 1 3 3 3 3 3.2 .S3 6.2 1 M P 3 K (1 i 3)S 1 (1 i 3)S P MOP L 2 3 3 2 1 Tr(T) 1 3 3 3 3 3.2 .S3 6.2 where Tr(T) = (a+b+c) and the secondary parameters S2 and S 3 can be calculated from the following relationships QR RSTSNS S1 ab ac bc z1z1 p z2z2 p z3z3 p QUSRUSVSVRURWR3R2 2 2 S2 a ab b ac bc c 3z1z1 p 3z2z2 p 3z3z3 p QXSYSZR[R\S - 3) det(T) abc cz1z1 p bz2z2 p z1z2 p z3 z1 p z2z3 p az3z3 p c d b 2 ^_^a`0^ Q]S 3 3 3 S3 27det(T) 9S1Tr(T) 2Tr(T) 27det(T) 9S1Tr(T) 2Tr(T) 4S2 The eigenvectors of [T] can also be calculated as the columns of a 3 x 3 matrix [U3] = [e1 e2 e3] where j m g p p f fhg l o i c (( i c)z1 p z2 p z3 )z3 p l o p fif o l z ((b )z z z )z 2 p i 2g p 1 p 3 p 2 p p f o l ( c)z z z e i 1 p 2 p 3 - 4) l o ei p fWf o l (b i )z2 p z1 p z3 p o l 1 k n The coherency matrix [T] also has an associated Hermitian form qi, q 0, which can be used to generate the level of scattered power into mechanism wi as [14] q rtsvuw*T T w i - 5) The eigenvalues of [T] therefore have direct physical significance in terms of the components of scattered power into a set of orthogonal unitary scattering mechanisms given by the eigenvectors of [T], which for radar backscatter themselves form the columns of a 3 x 3 unitary matrix. Hence we can write an arbitrary coherency matrix in the form } 1 0 0 w x y z|{ U *T i T U 0 0 U 0 0 P 1 - 6) | . 3 2 3 1 2 3 i ~ 0 0 3 where the P i may be interpreted as probabilities for a Bernoulli model of scattering from random media [1] as a weighted sum of coherent scattering mechanisms given by the 3 eigenvectors or columns of [U 3] as shown in equation 7 1 0 0 || *T T U3 0 2 0 U3 0 0 3 - 7) cos 1 cos 2 cos 3 h i 1 i 2 i 3 U 3 sin 1cos 1e sin 2cos 2e sin 3cos 3e i 1 i 2 i 3 sin 1 sin 1e sin 2 sin 2e sin 3 sin 3e While the eigenvalues and eigenvectors are the primary variables of interest, several secondary parameters can be defined as functions of the components of the eigen-decomposition. There are four of these of interest in radar, two from the eigenvalues, namely the entropy and anisotropy, and two from the eigenvectors, the and angles, as shown in figure 1. The parameters and can be interpreted as generalised rotations of the mechanism w as shown in figure 2. In fact the parameter is just the physical orientation of the object about the line of sight. However, the parameter is an indicator of the type of scattering and is called the scattering mechanism. ¢ £¤£¥£ ¦§¦ H P1 log3 P1 P2 log3 P2 P3 log3 P3 0 H 1 ¨ £ Eigenvalue Parameters ¦§¦ ¢ P P A 2 3 0 A 1 P2 P3 ¿ ©ª©N©«©¬©© k ¯±°«°]²²³¯ ´µ o 1 0 ¡ ¶ · P1 1 P2 2 P3 3 0 90 i cos µ k · ¼ * Eigenvector Parameters ­§­ ­®­ ¿ ´ 2Re(k k ) ²§¸¹¯»º ¯ o 1 1 µ¾½1 2 ¶ 1 0 360 i 2 tan * * · k1k1 k2k2 Figure 1 : Secondary Scattering Parameters Derived from the Eigenvalues and Eigenvectors of [T] If the pair H/ À are plotted on a plane then they are confined to a finite zone as shown in figure 3.