Maximum Likelihood / Bayesian (4) — full decision rule.

The pixel is assigned to the class for which D is the lowest:

\[D \;=\; \log_e(a_c) \;-\; \tfrac{1}{2}\log_e(|V_c|) \;-\; \tfrac{1}{2}(X - M_c)^{\!T}\,V_c^{-1}(X - M_c)\]

• X — measurement vector of the candidate pixel.
• M_c — mean vector of the data in class c.
• V_c — covariance matrix of class c.

• **

  V_c

  ** — determinant of V_c.

• V_c⁻¹ — inverse of V_c.
• T — transposition.

• a_c — prior probability that class c occurs in the image (equal for all classes, or user-entered from a priori knowledge).

• The third term is the Mahalanobis distance squared — distance from the pixel to the class mean, scaled by the class’s covariance. Mahalanobis alone (without the first two log terms) is its own decision rule.