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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
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
Highlights
Michigan Engineering Multiscale Structural Simulations Laboratory
Probabilistic models
Markov Random Fields
Bayesian Networks
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
1D Markov chain (second order)
Michigan Engineering Multiscale Structural Simulations Laboratory
Markov Random Fields • Undirected graph
• Symmetric and more natural for 2D, 3D image data • Pairwise, local and global Markov properties
Michigan Engineering Multiscale Structural Simulations Laboratory
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.
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
Synthesis movie
Input image that is sampled to identify pixel probabilities in the Markov chain
Window size = 11
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
Crystal orientations
AA3002 alloy, Wittridge, Knutsen 1999 Color blot indicating crystal orientations in experimental image (left) and the synthesized image (right)
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
• 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
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
(a) (b)
(c)
Experiment
Synthesized
Example 1: Anisotropic microstructure
Michigan Engineering Multiscale Structural Simulations Laboratory
3D Interconnected structures Experimental
Final Reconstruction
Michigan Engineering Multiscale Structural Simulations Laboratory
Example 2: random disks Experiment Synthesized
0 10 20 30 400.2
0.3
0.4
0.5
0.6
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
0.6
0.7
0.8
r (µm)
Two
poin
t cor
rela
tion
func
tion
(S 2(r))
Experimental (2D) image3D reconstruction
Michigan Engineering Multiscale Structural Simulations Laboratory
Example 3: W-Ag composite experimental image
Umekawa et al, JMPS (1965)
Michigan Engineering Multiscale Structural Simulations Laboratory
Example 4: Polycrystal
Reconstruction iterations
Input microstructure
Michigan Engineering Multiscale Structural Simulations Laboratory
(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
Michigan Engineering Multiscale Structural Simulations Laboratory
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.
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
3D to 3D sampling
AL6XN microstructure from serial sectioning, a 643 RVE (Alexis Lewis, NRL)
Synthesized microstructure using MRFs showing cut section (right)
Michigan Engineering Multiscale Structural Simulations Laboratory
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.
Michigan Engineering Multiscale Structural Simulations Laboratory
Full Field Microstructural Data
AL6XN microstructure from serial sectioning, a 643 RVE (Alexis Lewis, NRL)
Synthesized geometry
Michigan Engineering Multiscale Structural Simulations Laboratory
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
Michigan Engineering Multiscale Structural Simulations Laboratory
Controlled axial orientations
Grain structure from MC simulation
Synthesized image
Cross section
Michigan Engineering Multiscale Structural Simulations Laboratory
Controlled grain size
AL6XN microstructure from serial sectioning, a 643 RVE (Alexis Lewis, NRL)
Synthesized with controlled RVE sizes
Michigan Engineering Multiscale Structural Simulations Laboratory
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)
Michigan Engineering Multiscale Structural Simulations Laboratory
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
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