An apparatus for spatially resolved, temperature dependent ...
Spatially resolved pair correlation functions for point cloud data
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Transcript of 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 Cu x +(1-x)
THE Material System Al-Cu solidification
x={.15,.2}
@ Eutectic Temperature
+5K
Holding Time
GOAL
TI
ME
in
S
EC
ON
DS
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 data 440954
9 datasets
θ
I X-CT
EXTRACTING CURVATURE
The flow of data to information.
Interface
Smoothing
Gaussian & Mean
curvature, Surface
Normals, & Nodal Area
Reconstruct
Time Steps 5 15 30 50 160 100
MEA
N C
UR
VATU
RE
This is a small subset of the actual data
CU
RVATU
RE
Time Steps 5 15 30 50 160 100
MEA
N C
UR
VATU
RE
A closed “pore” starts to form
CU
RVATU
RE
Time Steps 5 15 30 50 160 100
MEA
N C
UR
VATU
RE
“Pore” becomes isolated
CU
RVATU
RE
Time Steps 5 15 30 50 160 100
MEA
N C
UR
VATU
RE
CU
RVATU
RE
Time Steps 5 15 30 50 100 160
MEA
N C
UR
VATU
RE
CU
RVATU
RE
Time Steps 5 15 30 50 100 160
MEA
N C
UR
VATU
RE
CU
RVATU
RE
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
𝑓𝑟ℎℎ′ =
1
𝑆 𝑚𝑠
ℎ𝑚𝑠+𝑟ℎ′
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
Mate
rial In
form
atio
n
Sp
atia
l Co
rrela
tion
Objective
Comparison
𝑚𝑠ℎ 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
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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 = 𝑡𝑎𝑛𝜅1𝜅2
−𝜋
4
𝑟=
𝜅12+𝜅22
𝜅1
𝜅2
𝜅1 > 𝜅2
𝐻 𝜇𝑚−1 𝐾 𝜇𝑚
−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
2𝑛𝑛!
𝑑ℎ
𝑑𝑥ℎ𝑥2 − 1 ℎ
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
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I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
I will not make dumb coding mistakes I will not make dumb coding mistakes I will not make dumb coding mistakes
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.
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𝑚 𝑠′ℎ =
𝐴𝑖𝑚𝑖ℎ
𝑖∈𝑃 𝐴𝑖 𝑖∈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
Work flow
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
Correlation Function Visualization
hθ=3,hr=4
Correlation Function 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
EFFECT OF THE BASIS FUNCTION
Partition=50 &
Cutoff = 50
Partition= 5 &
Cutoff = 5
Partition= 20 &
Cutoff = 200
EFFECT OF THE BASIS FUNCTION
Partition=20 &
Cutoff = 200
Partition= 5 &
Cutoff = 5
Partition= 50 &
Cutoff = 50
20
30
10
20
30
20
10
Cutoff 5
Cutoff 100
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!