markov random field representation of microstructures and

Michigan Engineering Multiscale Structural Simulations Laboratory

Markov Random Field Representation of Microstructures and its Applications

Prof. Veera Sundararaghavan University of Michigan-Ann Arbor

MURI: Mosaic of Microstructures

Contributors: Abhishek Kumar,

Lily Nguyen, Marc DeGraef Tony Fast, David Turner, Surya Kalidindi


Motivation Develop mathematical models that can be used

to explain the notion that: • Different windows of a microstructure ‘look-

alike’: Notion of a stationary probability distribution/ Markov Random Fields

• Use data to develop a model for microstructures in the absence of a physical simulation – DATA is the MODEL

• Apply the ideas for

– Computational sampling of microstructures – 3D reconstruction from 2D orthogonal

sections – Understand location specific microstructures

in macro-specimens – Verify methods by comparing statistical

features and physical properties


Highlights


Probabilistic models

Markov Random Fields

Bayesian Networks


1D Markov Chain Ordered Markov property is the assumption that a node only depends on its immediate parents, not on all predecessors in the ordering


1D Markov chain (second order)


Markov Random Fields • Undirected graph

• Symmetric and more natural for 2D, 3D image data • Pairwise, local and global Markov properties


MRFs for 2D data

Ising model

Higher order interactions

Microstructures = Markov random fields: a set of n neighbor pixels completely determine the probability of the next pixel.


Inference Problems

Find unknown pixel X8 given the rest of the image. • Need to compute P(X8| X3, X7, X9, X13) • Building explicit probability tables from

experimental images intractable, especially for higher order interactions

A harder problem: Sample a new image given an experimental image • Start from a small seed image and

‘grow’ image based on its neighbors • P(p | known neighbor pixels) = ? • This is related to the previous problem,

but only a part of the neighborhood is known


Sampling approach

- Sample patches in the experimental image that look similar to the neighborhood. All patches within 10% of the closest match are stored in a list. The center pixel values of patches in the list give a histogram for the unknown pixel

- The similarity measure for patches is based on a least square distance with Gaussian weighting. The distance is normalized by the number of known pixels.

- Tradeoff: PDF may not stay valid as neighbor pixels are filled in: but works well nevertheless


Two-phase microstructure – Effect of window size

Microstructure is a silver-tungsten composite from Umekawa et al (1965)

Abhishek Kumar, PhD thesis, University of Michigan Ann Arbor, 2014. (recipient of Ivor McIvor award for outstanding research) Paper in MSMSE Journal (2015)


Synthesis movie

Input image that is sampled to identify pixel probabilities in the Markov chain

Window size = 11


Computational estimation of microstructure field

Experimental sample

SAMPLING

Example: Sampling polycrystals Test stereological properties of the reconstructed image

Heyn intercept histogram (top) and rose of intersections (bottom)


Two-phase microstructure

MRF

• Average Stress Value for synthesized image = 7.19 GPa • Average Stress Value for sample image = 7.36 GPa

MRF Sample Autocorrelation function

Stress distribution (in GPa) under y-axis tensile test

Stress histogram (% of pixels with a given stress, stress value is indicated on x axis, number above the bar reflects percentage of each stress value)

r g

g’

Sampling code from Kalidindi group


Crystal orientations

AA3002 alloy, Wittridge, Knutsen 1999 Color blot indicating crystal orientations in experimental image (left) and the synthesized image (right)


Microstructure properties from crystal plasticity simulations

0 1 2 3

x 10-4

5

10

15

20

25

30

35

40

45

50

Equi

vale

nt S

tress

(Mpa

)

Equivalent Strain

SampleSynthesis 1Synthesis 2

Plastic response

Elastic modulus

Sample Synthesized MRF Sample

Stress histogram (% of pixels with a given stress, stress value is indicated along x)


• Problem: Generate 3D microstructures from 2D orthogonal image sections

• State-of-the-art methods based on matching statistical features, restricted to simple (eg. two phase) microstructures

• Direct extension of MRF sampling is computational expensive • A multiscale optimization approach is proposed: Low resolution 3D

images are first constructed (16x16x16) and then progressively refined.

3D reconstruction from 2D samples


Voxel,

- Match voxel neighborhood of 3D slices to a neighborhood in the 2D micrograph

- Run optimizer to mesh together the voxels from three orthogonal directions.

3D Markov Random Fields: Optimization problem


Optimization algorithm for MRFs Expectation step: Minimize the cost function with respect to Vv

Minimization step: Minimize cost function with respect to the set of input neighborhoods Sv keeping Vv fixed at the value estimated in the E-step.

Iterate until convergence (ie. until Sv unchanged between iterations)


(a) (b)

(c)

Experiment

Synthesized

Example 1: Anisotropic microstructure


3D Interconnected structures Experimental

Final Reconstruction


Example 2: random disks Experiment Synthesized

0 10 20 30 400.2

0.3

0.4

0.5

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0.7

0.8

0.9

1

Feature index

Thre

e po

int c

orre

latio

n (S

3(r,s,

t))

Experimental (2D) image3D reconstruction

- Comparison of rotationally invariant correlation functions (2 point and 3 point) for a random disk microstructure reconstruction

- Marked Point Process (MPP) for improving reconstructed shapes (Comer, Purdue)

0 2 4 6 8 100.3

0.4

0.5

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r (µm)

Two

poin

t cor

rela

tion

func

tion

(S 2(r))

Experimental (2D) image3D reconstruction


Example 3: W-Ag composite experimental image

Umekawa et al, JMPS (1965)


Example 4: Polycrystal

Reconstruction iterations

Input microstructure


(a) (b)

(c)

Example 4: Polycrystalline case

Experiment Synthesized

V. Sundararaghavan, Reconstruction of three-dimensional anisotropic microstructures from two-dimensional micrographs imaged on orthogonal planes, Integrating Materials and Manufacturing Innovation, 3:19, p.1-11, 2014


Oblique sections

• Georgia Tech group has been working on using additional oblique section information to further improve reconstruction.

• The new algorithm development is being supported by ONR.


3D to 3D sampling Approach: 1) Select elements in geometry space 2) Map geometry space to microstructure space: Associate nodes in the

geometry to nodes in the microstructure mesh 3) Patches are merged through optimization*

Tensor field (F= RU, James/Minnesota)


3D to 3D sampling

AL6XN microstructure from serial sectioning, a 643 RVE (Alexis Lewis, NRL)

Synthesized microstructure using MRFs showing cut section (right)


Improved 3D to 3D sampling

Approach: 1) User defines the scale and direction of patches – to control grain size 2) Warp microstructure according to geometry – to control anisotropy directions

• Same approach can be extended towards any arbitrary geometry.


Full Field Microstructural Data


Synthesized geometry


Controlled anisotropy

Sundararaghavan V, Synthesis of location specific 3D microstructures using Markov random fields, Book chapter, in preparation, 2015

Grain structure from MC simulation

3D sampling with user controlled anisotropy directions

Elongated

Equiax


Controlled axial orientations

Grain structure from MC simulation

Synthesized image

Cross section


Controlled grain size


Synthesized with controlled RVE sizes


MURI interactions •Verification using shape moment invariants and autocorrelation functions (DeGraef/CMU and Kalidindi/Georgia Tech) •Collaboration: Fiber composites (U Washington, Oak Ridge National Lab) •3D reconstruction using oblique sections (Kalidindi/Georgia Tech) •Model deformation using Ising representation (papers in Acta Materialia (2015) and IJSS (2014)) Future •Feature constraints (Surface microstructure): Collaboration with DoE PRISMS program to interpret DIC 2D strain data •Grain shape constraints (MPP, Comer/Purdue) and 4D simulations (model microstructure evolution using temporal data, Voorhees)


Conclusions

Experiment (2D) Reconstruct (3D)

Full Field

Pixel based Markov Random Fields

Patch-based

Validate: Compare metallurgical features, shape moments, properties Control scale and anisotropy to generate

location specific microstructures

- New methods based on MRFs - Open source codes - Testing methodologies

markov random field representation of microstructures and

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