17 Featurization, The microstructure function m(x,l) • For each image, the microstructure function !(#, %) is defined as a wide-sense stationary stochastic process in which ℎ is random variable associated with the probability of finding a specific local state at spatial position #. – ℎ ∈ ), complete set of local states of interest – # is an index, similar to time for a signal • The two-point correlation function is then defined as: *+,+, -. , -/ = 1 ! -. , % ! -/ , %′ , • But stationarity of ! allows us to reduce *+,+3 to a function of one spatial variable 4 = -/ − -. : *+,+, 4 = 1 ! -, % ! - + 4, %′ . • Microstructure function discretizing and periodic boundary conditions and a primitive basis gives: *+,+, 4 = 1 : ; < =< + =<>4 +, ,