Enhancement of Class Separability for Polarimetric TerraSAR-X Data and Its Application to Crop Classification in Leizhou Peninsula, Southern China

Abstract

In this paper, an enhanced class separability is proposed for multi-look fully polarimetric SAR classification. Instead of measuring the Wishart distance between two classes directly, we apply the deorientation procedure to eliminate the fluctuation of polarization orientation angle, which is an extrinsic property of targets and might result in larger inner class distance. Then the Barnes-Holm decomposition is used to factorize the deorientationed coherency matrix into a pure target and a distributed target, and the enhanced class separability, which is proved strictly, is measured by the sum of two Wishart distances based on pure targets and distributed targets respectively. The effectiveness of the proposed measure is demonstrated with the TerraSAR X-band PolSAR data in crop classification, in Leizhou Peninsula, southern China.

Appendix

Conjecture: Let ${\mathbf{T}}_{\mu }$ and ${\mathbf{T}}_{\nu }$ be the coherency matrix centers of two classes $\left\{ \mu \right\}$ and $\left\{ \nu \right\}$, respectively. Based on Barnes-Holm decomposition, $\left\{ \mu \right\}$ and $\left\{ \nu \right\}$ are expressed as $\left\{ {\mu_{0} } \right\} + \left\{ {\mu_{N} } \right\}$, $\left\{ {\nu_{0} } \right\} + \left\{ {\nu_{N} } \right\}$. Whether

$$ d\left( {\mu ,\nu } \right) \le d\left( {\mu_{0} ,\nu_{0} } \right) + d\left( {\mu_{N} ,\nu_{N} } \right) $$

(9)

Proof:

For an arbitrary sample $z \in \left\{ \mu \right\}$ with coherency matrix ${\mathbf{Z}}$, based on Barnes-Holm decomposition, $z$ is expressed as $z_{0} + z_{N}$, with coherency matrix ${\mathbf{Z}} = {\mathbf{Z}}_{0} + {\mathbf{Z}}_{N}$. According to Probability theory,

$$ \left( {z_{0} \in \left\{ {\nu_{0} } \right\}} \right) \cup \left( {z_{N} \in \left\{ {\nu_{N} } \right\}} \right) \subseteq \left( {z \in \left\{ \nu \right\}} \right) $$

(10)

which means that on the assumption of $z_{0}$ belongs to the class $\left\{ {\nu_{0} } \right\}$ and $z_{N}$ belongs to the class $\left\{ {\nu_{N} } \right\}$, it can be deduced that $z$ belongs to the class $\left\{ \nu \right\}$. In probability inequality, (10) can be expressed as

$$ p\left( {z_{0} \in \left\{ {\nu_{0} } \right\}} \right) \cdot p\left( {z_{N} \in \left\{ {\nu_{N} } \right\}} \right) \le p\left( {z \in \left\{ \nu \right\}} \right) $$

(11)

and thus

$$ p\left( {z_{0} |\nu_{0} } \right)P(\nu_{0} )*p\left( {z_{N} |\nu_{N} } \right)P(\nu_{N} ) \le p\left( {z|\nu } \right)P(\nu ) $$

(12)

where $P(\nu )$ is the priori probability of class $\left\{ \nu \right\}$.

For classes $\left\{ \nu \right\}$, $\left\{ {\nu_{0} } \right\}$, $\left\{ {\nu_{N} } \right\}$, their probability density functions follow Wishart distribution. According to the distance measure derived by Lee et al. [7], by taking the natural logarithm of (12) and changing its sign, we have

$$ d\left( {z_{0} ,\nu_{0} } \right) + d\left( {z_{N} ,\nu_{N} } \right) \ge d\left( {z,\nu } \right) $$

(13)

The average Wishart distance from class $\left\{ \mu \right\}$ to center of class $\left\{ \nu \right\}$ can be evaluated as

$$ d(\mu \left| \nu \right.) = \frac{1}{{N_{\mu } }}\sum\limits_{{z \in \left\{ \mu \right\}}} {d(z,\nu )} $$

(14)

where $N_{\mu }$ is the sample number of class $\left\{ \mu \right\}$. Combining Eqs. (13) and (14), we have

$$ d(\mu_{0} |\nu_{0} ) + d(\mu_{N} |\nu_{N} ) \ge d(\mu |\nu ) $$

(15)

and vice versa, that

$$ d(\nu_{0} |\mu_{0} ) + d(\nu_{N} |\mu_{N} ) \ge d(\nu |\mu ) $$

(16)

By substituting Eqs. (15) and (16) into

$$ d(\mu ,\nu ) = d(\mu |\nu ) + d(\nu |\mu ) $$

(17)

the proof of Eq. (9) is completed.

Equation (1) is a simplified expression by deleting the constant terms and it is assumed that $n > > q$, so it can be concluded that the Wishart distance based on Barnes-Holm decomposition enhance class separability in relative terms, as well as in absolute terms.