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Spatial statistics employ the microstructure function to rapidly compute an objective description of the material information provided by model(s) within a similar sample volume. The spatial statistics are computed by the following relationship
f_t^(hh')=(∑_s▒〖m_s^h m_(s+t)^(h') 〗)/(S_t^(hhii'') )
where f_t^(hh') is the probability of finding local states h and h' separated by a vector t; h is a local state derived from signal i at the tail of t and h' is local state derived from signal i' is at the head . The complete set of statistics includes all the discrete set of statistics for all possible vectors within for the sampled region sampling pattern of the model. To better understand the definition above, it is useful to consider the numerator and denominator individually. The numerator is a cumulative sum of the positive outcomes where h and h' were observed to be separated by t. The denominator S_t^ii' S_t^ provides the total number of trials conducted with a vector t from the signal sources i corresponding to the local state indices h and h'. (Figure to illustrate statistics)
The spatial statistics are computed for all vectors in the sample volume, L, that satisfy the Nyquist criteria, |t_i |≤〖0.5 L〗_i for i=1…3.[ref] The correlation function of all vectors for states h and h' is defined as F_ ^(hh'). There are two types of correlations that are computed
Auto-correlation – occurs when h=h' and is represented as F_ ^(hh).
Cross-correlation – occurs when h≠h'.
Both correlations functions are smooth and differentiable; they are readily amenable to interpolation methods to extract correlations of arbitrary vectors t. Auto-correlation functions maintain C_2^ , 2-fold symmetry, about the origin of the statistics, or the 〈0,0,0〉 vector; slightly more than half of the t vectors are unique. The cross-correlation functions F_ ^(hh') and F_ ^(h'h) are respectively anti-symmetric. (IS THIS TRUE FOR COMPLEX BASIS?) The complete set of spatial statistics is defined for all combinations of local state indices of the microstructure function in the following anti-symmetric block matrix
F=[■(F^11&⋯&F^1H@⋮&⋱&⋮@F^H1&⋯&F^HH )]
with the auto-correlation functions on the diagonal and the cross-correlation functions on the upper and lower triangle. Previous literature reports the independent and dependent features of the complete statistics for an isolated, but common set of raw material information.[STEVEref]
2.1 The Numerator – An FFT Algorithm to Compute Joint Probabilities of Local Statesto Convolve Digitized Microstructure Signals
Both the numerator and denominator of the spatial statistics decompose into discrete convolutions of digital signals. The convolution has unique properties in the Fourier space that are often leveraged in algorithms to expedite computations. Fast Fourier Transforms (FFT) expedite the computation of the numerator. Fast Fourier Transforms (FFT)’s have unique convolution properties that drastically reduce the computational demands required to evaluate the expression; explicitly computing the numerator convolution has a complexity of S^2 whereas the FFT approach has Slog(S) complexity, S is the number of spatial points in the microstructure function.[ref] FFT methods extract the correlation function in a single computation; all vectors t are accounted for at once. In materials science applications to date, spatial statistics have only been computed using uniform FFT algorithms.[ref] This class of algorithms requires that the spatial domain of the microstructure function is on an evenly spaced cuboidal grid in three dimensions. Future work will include an application of non-uniform FFT methods to point cloud datasets.[ref]
Another caveat of the FFT methods is that they naturally impose periodicity on the volume being evaluated. This assumption is incorrect for a large variety of physics-based models. In fact, this assumption is erroneous for all experimental models. Figure rr explores the intrinsic problem of using a periodic assumption on an artificial non-periodic dataset using a single local state index, h, from the microstructure function. Figure rr(a) shows that if periodicity is assumed, a vector that goes beyond the boundary of the sample volume will enter in on the opposing side. When vectors transverse a boundary in a non-periodic sample volume then dubious counts are recorded into the numerator of the statistics; vectors with a dotted line identity these counts. Non-periodic statistics are computed using Fourier methods by padding the microstructure function with zeros to twice its length in each spatial dimension. Padding ignores vectors that transverse the boundaries because either m_s^h or m_(s+t)^(h') is zero thereby excluding the count from the sum of probabilities in the numerator. Pseudocode to compute the numerator is shown in Table YY. Periodic boundary conditions are discussed as a special case later in this paper; mixed periodic/non-periodic boundary conditions (i.e. slab boundary conditions) can exist when modeling surfaces/interfaces.[ref]
