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Materials Science & Engineering Department Computational Materials Sci. Lab. 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

+,

,