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\subsection{Model Sampling Patterns in the Spatial Domain}
A physics-based model will sample material information at a position x; sampling does not occur at infinitesimal points in space; rather, the position corresponds to a sampling within the precision of the simulation or the probe volume of an experimenta probe volume propagated by the model. Material information is sampled from a finite volume corresponding the resolution of the model and the resulting raw information is a probability density function of the local material states in that volume; concurrently, uncertainty is propagated by the precision of the local material states output by the model.[ref]
Two classifications of sampling patterns exist in the spatial domain: gridded and non-gridded.(Figure qq) Gridded sampling corresponds to information that is evenly spaced within the sample volume. Voxel, or 3-D pixel, based information is a gridded dataset because of this criteria; it is assumed that the voxel is the probe volume of the model. Non-gridded data is extracted from a sample volume of finite dimensions wherein x can take any value within the range. Models such as Atom Probe Microscopy [ref] and Molecular Dynamics [ref] will often exhibit non-gridded, or point cloud, sampling characteristics. The probe volume associated with point cloud datasets will relate to the resolution precision of the model from which the information is generated.
In prior work and this paper, indicator functions provide the basis to partition the spatial domain into non-overlapping, evenly spaced, cuboidal volumes [ref]; an investigation of other basis functions are currently underway (e.g. wavelets). Equation xx defines the basis for the spatial domain as χ_s (x). s is an index to a unique cuboidal volume ω_s in the spatial domain with the properties
EQUATION
and
EQUATION
Each underlying volume is identified spatially by its centroid and has a volume of l^d where d∈{1,2,3} is the number of spatial dimensions of the sample volume. Indicator functions can identify any convex or concave non-overlapping region in the sample volume. [ref]
The work presented in this paper is limited to microstructure functions that are expressed with an evenly gridded spatial basis function. This transformation is trivial for material information that is generated on an even grid (e.g. microscopy images). Raw pPoint cloud information can be transformed to uniform grid using the microstructure function by the appropriate basis function; it is acknowledged that some uncertainty will be propagated in this transformation. Alternate techniques are being developed to treat point cloud data on a non-uniform grid. A note: tree data structures may be most efficient to partition the spatial domain for point cloud datasets especially when the dataset is of a high-dimension.[ref] In applied problems, the microstructure function is a means to coarsen oversampled data or very large datasets; this will reduce computational demands incurred later.
