omnibus.rst

Omnibus Test Change Detection

nd.change

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 ∈ {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

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

where ⟨ ⋅ ⟩ 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

$$\begin{aligned} {\langle C \rangle}_\text{dual} = \begin{bmatrix} {\langle S_{hh}S_{hh}^* \rangle} & {\langle S_{hh}S_{hv}^* \rangle} \\\ {\langle S_{hv}S_{hh}^* \rangle} & {\langle S_{hv}S_{hv}^* \rangle} \\\ \end{bmatrix} \end{aligned}$$

These covariance matrices follow a complex Wishart distribution as follows:

X_i ∼ W_C(p, n, Σ_i), i = 1, ..., k

where p is the rank of X_i = n⟨C_i⟩, E[X_i] = nΣ_i, and Σ_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 H₀ : Σ₁ = Σ₂ = ... = Σ_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(H₀). The hypothesis test is repeated iteratively over subsets of the time series in order to determine the actual time of change.

See Also:

• nd.change.OmnibusTest

References:

• Conradsen, K., Nielsen, A. A., & Skriver, H. (2016). Determining the Points of Change in Time Series of Polarimetric SAR Data. IEEE Transactions on Geoscience and Remote Sensing, 54(5), 3007–3024.