Endmember Number Estimation Using Eigenvalue Thresholding

In hyperspectral image analysis, estimating the number of endmembers is a crucial step that affects the accuracy and efficiency of subsequent processing. This post introduces an eigenvalue thresholding approach for estimating the number of endmembers in hyperspectral data through noise whitening and eigenvalue analysis.

Noise Whitening of Hyperspectral Image

Let's consider a hyperspectral image \(\mathbf{X} \in \mathbb{R}^{M \times B}\), where \(M\) is the number of spatial pixels and \(B\) is the number of spectral bands. Each band can be represented as a column vector \(\mathbf{x}_i\) of \(\mathbf{X}\).

Noise whitening is the first step in preprocessing hyperspectral data, aiming to reduce noise and facilitate endmember identification. To achieve this, we first calculate the covariance matrix \(\mathbf{C}_x\) of the hyperspectral data, defined as:

\[ \mathbf{C}_x = \operatorname{Cov}(\mathbf{X}) = \left[\begin{array}{ccc} \sigma\left(\mathbf{x}_{1}, \mathbf{x}_{1}\right) & \cdots & \sigma\left(\mathbf{x}_{1}, \mathbf{x}_{B}\right) \\ \vdots & \ddots & \vdots \\ \sigma\left(\mathbf{x}_{B}, \mathbf{x}_{1}\right) & \cdots & \sigma\left(\mathbf{x}_{B}, \mathbf{x}_{B}\right) \end{array}\right] \in \mathbb{R}^{B \times B} \]

where \(\sigma\left(\mathbf{x}_{m}, \mathbf{x}_{k}\right)\) is the covariance between vectors \(\mathbf{x}_m\) and \(\mathbf{x}_k\), calculated as:

\[ \sigma\left(\mathbf{x}_{m}, \mathbf{x}_{k}\right) = \frac{1}{n-1} \sum_{i=1}^{n}\left(\mathbf{x}_{m i} - \bar{\mathbf{x}}_{m}\right)\left(\mathbf{x}_{k i} - \bar{\mathbf{x}}_{k}\right) \]

Next, we compute the noise covariance matrix \(\mathbf{N}\) from \(\mathbf{C}_x\):

\[ \mathbf{N} = \operatorname{diag}\left(\frac{1}{\operatorname{Getdiag}(\mathbf{C}_x^{-1})}\right) \in \mathbb{R}^{B \times B} \]

where \(\operatorname{Getdiag}(\cdot)\) extracts the diagonal elements of a matrix, and \(\operatorname{diag}(\cdot)\) converts a vector into a diagonal matrix.

With the noise covariance matrix \(\mathbf{N}\), we can calculate the whitened noise matrix \(\mathbf{Y}\):

\[ \mathbf{Y} = \mathbf{X}\left(\left(\sqrt{\mathbf{N}}\right)^{-1}\right)^\top \]

Eigenvalue Soft-Thresholding Algorithm

The whitened noise matrix \(\mathbf{Y}\) is used for endmember number estimation. We first compute the auto-covariance matrix \(\mathbf{C}_y\) and the auto-correlation matrix \(\mathbf{R}_y\) of \(\mathbf{Y}\):

\[ \mathbf{C}_y = \operatorname{Cov}(\mathbf{Y}) \in \mathbb{R}^{B \times B} \]

\[ \mathbf{R}_y = E\left(\mathbf{Y} \mathbf{Y}^{T}\right) \in \mathbb{R}^{B \times B} \]

where the elements \(\sigma\left(\mathbf{y}_{i}, \mathbf{y}_{j}\right)\) of \(\mathbf{C}_y\) and \(\rho(\mathbf{y}_i, \mathbf{y}_j)\) of \(\mathbf{R}_y\) are defined as:

\[ \rho(\mathbf{y}_i, \mathbf{y}_j) = \operatorname{corr}(\mathbf{y}_i, \mathbf{y}_j) = \frac{\operatorname{cov}(\mathbf{y}_i, \mathbf{y}_j)}{\sigma_Y \sigma_Y} \]

Next, we compute the eigenvalues of these two matrices:

\[ \mathbf{\Sigma}_C = \operatorname{eig}(\mathbf{C}_y) \]

\[ \mathbf{\Sigma}_R = \operatorname{eig}(\mathbf{R}_y) \]

The eigenvalue matrices \(\mathbf{\Sigma}_C\) and \(\mathbf{\Sigma}_R\) contain the eigenvalues of the auto-covariance and auto-correlation matrices, respectively. We then calculate the variance \(\mathbf{D}\) between these eigenvalues:

\[ \mathbf{D} = \mathbf{\Sigma}_R - \mathbf{\Sigma}_C \]

Subsequently, we estimate the noise variance \(\mathbf{V}\):

\[ \mathbf{V} = \frac{2}{M}\left(\mathbf{\Sigma}_R^2 + \mathbf{\Sigma}_C^2\right) \]

Finally, we compute the eigenvalue threshold gate based on the noise variance \(\mathbf{V}\):

\[ \operatorname{gate} = \sqrt{2 \times \mathbf{V}} \]

Using the eigenvalue soft-thresholding algorithm, we can obtain the estimated number of endmembers \(I\):

\[ I = \sum_{i = 1}^B (\mathbf{D}_{ii} > \operatorname{gate}) \]

This approach provides an automated method for estimating the number of endmembers in hyperspectral data, which is crucial for subsequent unmixing and analysis tasks.