# 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.