deletions | additions       diff --git a/Sampling Spatial.tex b/Sampling Spatial.tex  index e6a3aa8..becacba 100644  --- a/Sampling Spatial.tex  +++ b/Sampling Spatial.tex 
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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)/*{Tony qq)\*{Tony  needs to add a figure} 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 ~\ref{??} defines the basis for the spatial domain as $\chi_{s}(x)$. $s$ is an index to a unique cuboidal volume $\omega_{s}$ in the spatial domain with the properties