8.1. Omnibus Test Change Detection

Conradsen et al. (2016) present a change detection algorithm for time series of complex valued SAR data based on the complex Wishart distribution for the covariance matrices. \(S_{rt}\) denotes the complex scattering amplitude where \(r,t \in \{h,v\}\) are the receive and transmit polarization, respectively (horizontal or vertical). Reciprocity is assumed, i.e. \(S_{hv} = S_{vh}\). Then the backscatter at a single pixel is fully represented by the complex target vector

\[\boldsymbol s = \begin{bmatrix} S_{hh} & S_{hv} & S_{vv} \end{bmatrix}^T\]

For multi-looked SAR data, backscatter values are averaged over \(n\) pixels (to reduce speckle) and the backscatter may be represented appropriately by the (variance-)covariance matrix, which for fully polarimetric SAR data is given by

\[\begin{split}{\langle C \rangle}_\text{full} &= {\langle \boldsymbol s(i) \boldsymbol s(i)^H \rangle} = \begin{bmatrix} {\langle S_{hh}S_{hh}^* \rangle} & {\langle S_{hh}S_{hv}^* \rangle} & {\langle S_{hh}S_{vv}^* \rangle} \\ {\langle S_{hv}S_{hh}^* \rangle} & {\langle S_{hv}S_{hv}^* \rangle} & {\langle S_{hv}S_{vv}^* \rangle} \\ {\langle S_{vv}S_{hh}^* \rangle} & {\langle S_{vv}S_{hv}^* \rangle} & {\langle S_{vv}S_{vv}^* \rangle} \\ \end{bmatrix}\end{split}\]

where \({\langle \cdot \rangle}\) is the ensemble average, \(^*\) denotes complex conjugation, and \(^H\) is Hermitian conjugation. Often, only one polarization is transmitted (e.g. horizontal), giving rise to dual polarimetric SAR data. In this case the covariance matrix is

\[\begin{split}{\langle C \rangle}_\text{dual} = \begin{bmatrix} {\langle S_{hh}S_{hh}^* \rangle} & {\langle S_{hh}S_{hv}^* \rangle} \\ {\langle S_{hv}S_{hh}^* \rangle} & {\langle S_{hv}S_{hv}^* \rangle} \\ \end{bmatrix}\end{split}\]

These covariance matrices follow a complex Wishart distribution as follows:

\[\boldsymbol X_i \sim W_C(p,n,\boldsymbol\Sigma_i), \quad i = 1,...,k\]

where \(p\) is the rank of \(\boldsymbol X_i = n {\langle \boldsymbol C_i \rangle}\), \(E[\boldsymbol X_i] = n \boldsymbol\Sigma_i\), and \(\boldsymbol\Sigma_i\) is the expected value of the covariance matrix.

In the first instance, the change detection problem then becomes a test of the null hypothesis \(H_0 : \boldsymbol\Sigma_1 = \boldsymbol\Sigma_2 = ... = \boldsymbol\Sigma_k\), i.e. whether the expected value of the backscatter remains constant. This test is a so-called omnibus test.

A test statistic for the omnibus test can be derived as:

\[Q = k^{pnk} \frac{\prod_{i=1}^k \left| \boldsymbol X_i \right|^n }{\left| \boldsymbol X \right|^{nk}} = \left\{ k^{pk} \frac{\prod_{i=1}^k \left| \boldsymbol X_i \right| }{\left| \boldsymbol X \right|^k} \right\}^n\]

where \(\boldsymbol X = \sum_{i=1}^k \boldsymbol X_i \sim W_C(p,nk,\boldsymbol\Sigma)\). The test statistic can be translated into a probability \(p(H_0)\). The hypothesis test is repeated iteratively over subsets of the time series in order to determine the actual time of change.