Spatially resolved pair correlation functions for point cloud data

Peter Voorhees
John Gibbs Surya Kalidindi
Tony Fast
MURI Annual Review Meeting
Chicago, IL
Spatially Resolved Pair Correlation
Functions for Point Cloud Data

Al Cux +(1-x)
THE Material System
Al-Cu solidification
x={.15,.2}
@
Eutectic Temperature
+5K
Holding Time
GOAL

TIMEinSECONDS
5
10
15
30
50
100
160
1,007,923
826,898
697,839
617311
525364
786,212
732,051
685,239
20% 15%V O L U M E F R A C T I O N
440954
amount of data440954
9 datasets

θ
IX-CT
EXTRACTING CURVATURE
The flow of data to information.
Interface
Smoothing
Gaussian & Mean
curvature, Surface
Normals, & Nodal Area
Reconstruct

Time Steps5 15 30 50 160100
MEANCURVATURE
This is a small subset
of the actual data
CURVATURE

Time Steps5 15 30 50 160100
MEANCURVATURE A closed “pore” starts to form
CURVATURE

Time Steps5 15 30 50 160100
MEANCURVATURE
“Pore” becomes isolated
CURVATURE

Time Steps5 15 30 50 160100
MEANCURVATURE
CURVATURE

Time Steps5 15 30 50 100 160
MEANCURVATURE
CURVATURE

μInformatics is material and hierarchy independent statistical framework
aimed to distill rich physical data into tractable forms that facilitate
structural taxonomies and bi-directional structure-property/processing
homogenization and localization relationships. It provides a foundation
for rigorous microstructure sensitive materials design.
3 Statistical Modules
5 Value Assessment
4
Data-Mining Modules
2
μS Signal Processing Modules
Experiment &
Simulation
Objective &
Subjective μS
metrics
DSP and image
segmentation
“HUGE influence on μI”
1
Physical Models
DSP
Spatial
Statistics
MKS Dimension
Reduction
MICROSTRUCTURE
INFORMATICS (μI)

Hey, I don’t
know what
direction to
hold this
microscope
image so I’m
going home!
MATERIAL / population RVE / sample
Materials science Statistics
?
? ?
Difference Between
Direct comparison of microstructures is most often
impractical thereby demanding statistical
interpretations.
Statistically speaking,
you probably never
will, so stay here and
use some statistics!

reveal
𝑓𝑟
ℎℎ′
=
1
𝑆
𝑚 𝑠
ℎ 𝑚 𝑠+𝑟
ℎ′
𝑆
𝑠=1
Statistical correlations between random points in space/time which reveal systematic patterns
in the microstructure. Contains the original μS within a translation & inversion.
Difference
Between
MaterialInformation
SpatialCorrelation Objective
Comparison
𝑚 𝑠
ℎ A digital signal of the microstructure at a position maybe voxel in the volume, s,
of S total positions for a channel, h, of H total channels. The channels describe
material features (e.g. phase, angle, curvature) using a prescribed basis function.

Evenly Gridded
Spatial Domain
&
Build a kd-tree & partition the spatial domain
Build: O(N) & Search: O(log(N))
Evenly gridded data allows for FFT methods
Outside Cell
Inside Cell
k-d tree range to
find point indices in
each partition
8
47
22
Grid in the Spatial Domain or the Fourier Domain
That is the question!
An Algorithm for Point
Cloud Spatial Statistics
Provides a look-up table
for material features

𝜃 = 𝑡𝑎𝑛
𝜅1
𝜅2
−
𝜋
4
𝑟=𝜅1
2+𝜅2
2
𝜅1
𝜅2
𝜅1 > 𝜅2
𝐻 𝜇𝑚−1
𝐾𝜇𝑚−2
99.99% of Original Data 
𝜅𝑖 = 𝐻 ± 𝑚𝑎𝑥 0, 𝐻2 − 𝐾

References
Legendre Polynomial Basis Functions
Legendre basis functions provide a compact representation of continuous local state
features. They provide a richer description than the primitive basis, but don’t be
deceived because there may be better ones. It’s an open problem, but let’s start here. r
vs. θ is an ideal space to define the polynomials after normalizing the LSS to [-1,1] in
each dimension. r is normalized with an affine mapping and theta by cos( θ ).
𝑃ℎ 𝑥 =
1
2 𝑛 𝑛!
𝑑ℎ
𝑑𝑥ℎ
𝑥2 − 1 ℎ

ot make dumb coding mistakes I will not make dumb coding mistakes I will not
t make dumb coding mistakes I will not make dumb coding mistakes I will not
Combining Domains - The μS Function
𝑚 𝑠′
ℎ
is the average of the weighted average of Legendre Polynomials of the
processed digital signal in each partition.
8
47
22
𝑚 𝑠′
ℎ =
𝐴𝑖 𝑚𝑖
ℎ
𝑖∈𝑃
𝐴𝑖𝑖∈𝑃
Position of the Partition(𝑠′
)
𝑑𝑥
Note to self: Parametric studies of the informatics variables are preferable in
gridded spatial domain, NFFT’s need to be recomputed too often.

PCA Distance
Visualization# of
Polynomials
Cutoff of
Stats Size of Partition
Microstructure
Function of Partitions
Legendre
k-d tree
Partition cells
H vs. K
Kappa1 vs. kappa2
R vs. theta
Correlation Functions via Fast
Fourier Transform
Embedding & Analytics
Raw Data
(Next) Results
Normalize
kd Range Search for
Look up table
WORKFLOW

hθ=1,hr=1
Correlation Function
Visualization

hθ=2,hr=3
Visualization

hθ=3,hr=4
Visualization

Principal Components Analysis – Reduces D variables to d variables. Each axis
corresponds to the i-th greatest direction of variance.
15% Vf
20% Vf
Each point corresponds to
the statistics of the digital signal

EFFECT OF THE BASIS FUNCTION
Partition=5 &
Cutoff = 5
Partition= 50 &
Cutoff = 50
Partition= 20 &
Cutoff = 200

Partition=50 &
Cutoff = 50
Partition= 5 &
Cutoff = 5
Partition= 20 &
Cutoff = 200

Partition=20 &
Cutoff = 200
Partition= 5 &
Cutoff = 5
Partition= 50 &
Cutoff = 50

Improved metrics for comparison
Hellinger, KL Divergence, other information gain metrics
Embed more data into the μI process
The current amount of data is inconclusive
Try NFFT to see if they are faster
Are there other spatial transforms, Wavelets anyone?
Achievements: Algorithms exist to analyze this data!

Spatially resolved pair correlation functions for point cloud data

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