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

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 = 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
























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 =

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


hθ=3,hr=4


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

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!

Technology

Spatially resolved pair correlation functions for point cloud data