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\subsection{The Microstructure Function}  The microstructure function expresses spatially resolved material structure information from physics-based models as digital signals. A physics-based model extracts material structure-response behaviors with either simulated simulation or empirical experimental techniques; materials information generated by either technique, or source, are uniquely identified by a set of boundary conditions and control parametersset of modeling conditions (i.e. control parameters). In a model,A model includes spatially resolved raw model information is outputabout the material structure, β(x), $\beta(x)$,  as either an input or output with local material states, β, at disparate positions, x, $x$,  in the sample volume, L. $L$.  The sample volume has dimensions L_1×⋯×L_d $L_1×⋯×L_d$  where d $d$  is the dimensionality of the dataset; the dimensionality corresponds to the number of independent axes in the spatial domain and will have the property, d≤3. $d$≤3.  The local material state is an ordered set of salient material features such as phase, classification, grain orientation, volume fraction, spin, curvature, etc. The generality of the local material state definition enables a framework to describe most material information from most sources. The microstructure function digitizes the raw model information by the following equation EQUATION  where m_s^h $m_{s}^{h}$  is a digitized coefficients of the raw material information corresponding to any normalized basis functions $\chi $  χ_s (x) and χ^h (β(x)) applied to represent the spatial domain and local state of the material, respectively. The spatial domain and local state space are orthogonal to one another and will require different basis functions. Each basis function is normalized requiring that m_s^h is bounded between zero and one . The selection of the basis functions is an extremely important matter and will be discussed extensively throughout this paper. From this point on, m_s^h will be referred to as the microstructure function where h is a local state index corresponding to the hth function in the local material state basis; similarly, s is an index pertaining to a volume contained within the sample volume prescribed by the spatial basis function .