Closed-form Shannon entropies \(H = E\{-\log f(Z)\}\) for the models used in SAR (intensity) and PolSAR (matrix) image analysis. All functions are vectorised in the scalar parameters.
Usage
entropy_gamma(mu, L)
entropy_g0i(alpha, gamma, L)
entropy_ggamma(alpha, phi, mu)
entropy_wishart(Sigma, L)
entropy_g0p(alpha, Sigma, L)Arguments
- mu
Mean of the Gamma law, or inverse scale of the generalized Gamma law.
- L
Number of looks.
- alpha
Roughness (G0I, G0 polarimetric) or power (generalized Gamma) parameter.
- gamma
Scale parameter of the G0I law.
- phi
Shape parameter of the generalized Gamma law.
- Sigma
Hermitian positive definite covariance matrix.
Details
- Gamma
\(H = L - \log L + \log\mu + \log\Gamma(L) + (1 - L)\psi^{(0)}(L)\).
- G0I
(Ferreira and Nascimento, 2020) \(H = -\log\frac{L^L\Gamma(L-\alpha)}{\Gamma(-\alpha)\gamma^\alpha \Gamma(L)} + (1 - L)[\psi^{(0)}(L) - \log L + \log\gamma - \psi^{(0)}(-\alpha)] + (L - \alpha)[\psi^{(0)}(L - \alpha) + \log\gamma - \psi^{(0)}(-\alpha)]\).
- Generalized Gamma
\(H = -\log\alpha + \log\Gamma(\phi) + \phi - (\phi - 1/\alpha)\psi^{(0)}(\phi) - \log\mu\).
- Scaled complex Wishart
(Nascimento, Ferreira and Frery, 2023) \(H = m\log|\Sigma| - m^2\log L + \log\Gamma_m(L) + mL + (m - L)\psi^{(0)}_m(L)\), with \(\psi^{(0)}_m(L) = \sum_{i=0}^{m-1}\psi^{(0)}(L - i)\).
- G0 polarimetric
(Nascimento, Ferreira and Frery, 2023) obtained from \(E\log|Z| = m[\log(-\alpha-1) - \psi^{(0)}(-\alpha)] + \log|\Sigma| - m\log L + \psi^{(0)}_m(L)\) and \(E\log[L\,\mathrm{tr}(\Sigma^{-1}Z) - \alpha - 1] = \log(-\alpha-1) + \psi^{(0)}(mL - \alpha) - \psi^{(0)}(-\alpha)\).
References
Ferreira, J. A. and Nascimento, A. D. C. (2020). Shannon entropy for the G0I model: a new segmentation approach. IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 13, 2547–2553. doi:10.1109/JSTARS.2020.2997666
Nascimento, A. D. C., Ferreira, J. A. and Frery, A. C. (2023). Unsupervised segmentation of PolSAR data with complex Wishart and G0m distributions and Shannon entropy. Statistics and Computing, 33, 129. doi:10.1007/s11222-023-10298-0
Examples
entropy_gamma(mu = 1, L = 4)
#> [1] 0.6371121
entropy_g0i(alpha = -3, gamma = 2, L = 4)
#> [1] 0.8625083
entropy_ggamma(alpha = 1, phi = 4, mu = 4) # equals the Gamma entropy
#> [1] 0.6371121
S <- polsar_example_sigma("forest")
entropy_wishart(S, L = 4)
#> [1] -12.04464
entropy_g0p(alpha = -5, Sigma = S, L = 4)
#> [1] -12.48772