2. +
How do we discuss the variety in materials science
information?
Materials are hierarchical and multi-physics.
3. +
Statistics are material descriptors
β-Titanium
REDUCED OUTPUT:
Grain size
Grain Faces
Number of Grains
Mean Curvature
Nearest Grain Analysis
4. +
First Order Statistics
n
Effective statistics the describe a material volume
n
n
n
Effective Statistics require:
n
n
n
Volume Fraction, Phase Distribution, Mean’s, Standard Deviation’s
Often times the value is a single feature parameters, but the
information in spatial materials data contains information about the
distribution.
n The distribution increases the number of variables in the system,
but adds to the fidelity of the material feature description.
Data processing
n Which could inject incorrect assumptions?
Limited return on the Time invested
How do we get more information out of spatial datasets & faster?
5. +
Goals of today: Advanced Spatial
Statistics and Signal Processing
n
Practical manipulation of multidimensional and multimodal
datasets.
n
New statistics tools to quantify material structures.
n
The variety of metadata and the uniformity of data.
n
Advanced methods for extracting structure-propertyprocessing connections.
n
To start thinking differently about the data you generate,
ingest, and manipulate.
6. +
Focus on Scalability
n
Datasets are getting larger, more channels can be extracted,
and the features are less understood.
n
Exploring the new space of data requires scalable
parametric and statistical material feature descriptors.
7. +
Types of Higher-Order Statistics
n
Moving Window Average – Code demo of image processing
filters
n
Neighborhood Connectivity – Code demo of Delaunay
tessellation and Voronoi Triangulation.
n
n
Shortest network path
GraphTehoryTest
n
Chord Length Distribution -Probably a chord of length d will
contiguously span a region containing some feature
n
Pair Correlation Functions – In depth
n
Vector-resolved spatial statistics – In depth
8. +
Spatial Statistics
n
Spatial statistics are a joint probability of material feature
domain with a posterior probability relating to a spatial
information.
Spatial statistics are the probability of finding <Feature A> and
<Feature B> separated by a <Vector,Distance> of <d-Tuple>"
n
Main Spatial Statistics to discuss
n
n
Pair Correlation Function
n Probability of two features two separated by a vector of magnitude
r
Vector resolved spatial statistics
n Probability of two features two separated by a vector t
n The pair correlation function is a reduced projection of the vector
resolved statistics
9. + The Breakdown
Index into features in the
spatial materials signal
• Direct or latent variables
• Basis function representation
Digital Signals i & j
• Gridded or Point Cloud
• Experimental or Simulated
• Periodic or non-periodic
• Any scale
Numerator is occurrence of true
conditions
• Summation only occurs when
s + t is a valid vector
Spatial Statistics
• Conditional, joint
probability
Joint Probability of two features i & j
• If i=j, autocorrelation
• otherwise, crosscorrelation
Index or vector into a spatial condition
Denominator :Number of tests on
the spatial condition
• Number of valid s+t vectors
10. +
Vector Resolved Spatial Correlation Function
of a Gridded Image
n
Computing this relationship directly is costly.
n
Since it is a convolution, we will use the Fourier transform
again.
n
Used to compute the numerator and denominator separately.
Code that Animates the
statistics
11. +
There is a Fourier Convolution
Property
n
Wikipedia
12. +
First Consideration: Signal pattern
n
The input signals must be on an
even grid to use DFT
methods.
Pattern
Point
Boundaries
n
Work around
n
Non-Uniform FFT’s ( Most accurate )
n
Binning point cloud data ( Introduces uncertainty )
Gridded
14. + Second Consideration: Periodicity Part 1
Source
Experiment
Simulation
Boundary Conditions
Boundary Conditions
Nonperiodic
Ø
Nonperiodic
Periodic
Ø
Ø
Group Discussion
If the denominator is the number of counts, how will it change with t?
15. +
The Denominator
n
If any dimensions are nonperiodic then the denominator
always varies with position. The number of times a variable
can be tested.
when
n
Convolution!
n
Needs to be computed less
frequently than the numerator.
n
Partial Periodicity is possible.
17. +
Pair Correlation Functions and
Spatial Statistics
n
Pair Correlation functions are a projection of the spatial
statistics. Either the magnitudes of the vectors or an average
of the vectors about their angle.
n
Group exercise : design a workflow to compute pair
correlation functions on periodic point cloud data.
