CLASSIFICATION AND REPRESENTATION OF MICROSTRUCTURES USING STATISTICAL LEARNING TECHNIQUES

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CLASSIFICATION AND REPRESENTATIONOF
MICROSTRUCTURES USINGSTATISTICAL LEARNING
TECHNIQUES
Veeraraghavan Sundararaghavan and Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
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RESEARCH SPONSORS
U.S. AIR FORCE PARTNERS Materials Process
Design Branch, AFRL Computational
Mathematics Program, AFOSR
ARMY RESEARCH OFFICE
Mechanical Behavior of Materials Program
NATIONAL SCIENCE FOUNDATION (NSF) Design
and Integration Engineering Program
CORNELL THEORY CENTER
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REPRESENTATION AT VARIOUS LENGTH SCALES
Average Properties Large-scale statistical
quantities
Lower order Descriptors (Grain sizes,ODF,OCF etc.)
Continuum
Lattice Positions, Interfacial energies etc.
Engineering
Micro- structural
Particle position/ momentum, potential
Atomistic
Materials
Electronic
Chemistry
Physics
Length Scales
nm
mm
mm
m
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THE PROBLEM STATEMENT
A Common Framework for Quantification of Diverse
Microstructure
Qualitative representation
Equiax grains Grain size small
Lower order descriptor approach
Grain size distribution
No. of grains
Grain size number
Equiaxial grain microstructure space
Quantitative approach
Microstructure represented by a set of numbers
Representation space of all possible polyhedral
microstructures
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LOWER ORDER DESCRIPTOR BASED RECONSTRUCTION
(Yeong Torquato, 1998)

• Non-uniqueness
• Computationally expensive
• Incomplete
• How many descriptors?
• Under constrained

Descriptor Two-point probability function and
lineal measure
New Descriptor P(3)( r,s,t ) (plotted as a
vector)
An under constrained case
Reconstructed
Actual
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REQUIREMENTS OF A REPRESENTATION SCHEME
A set of numbers which completely represents a
microstructure within its class
REPRESENTATION SPACE OF A PARTICULAR
MICROSTRUCTURE
Must differentiate other cases (must be
statistically representative)
Need for a technique that is autonomous,
applicable to a variety of microstructures,
computationally feasible and provides complete
representation
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APPROACH MICROSTRUCTURE LIBRARY
Input Image
Pre-processing
Identify and add new classes
Feature Detection
Employ lower-order features
Classifier
Quantification using incremental PCA
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DYNAMIC MICROSTRUCTURE LIBRARY CONCEPTS
Space of all possible microstructures
A class of microstructures (eg. Equiaxial grains)
New class partition
Hierarchical sub-classes (eg. Medium grains)
Expandable class partitions (retraining)
distance measures
New class
Dynamic Representation
New microstructure added
Axis for representation
Updated representation
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BENEFITS

• A data-abstraction layer for describing
  microstructural information.
• An unbiased representation for comparing
  simulations and experiments AND for evaluating
  correlation between microstructure and
  properties.
• An organized / self-organizing database of
  valuable microstructural information which can be
  associated with processes and properties.
• Data mining Process sequence selection for
  obtaining desired properties
• Identification of multiple process paths leading
  to the same microstructure
• Adaptive selection of basis for reduced order
  microstructural simulations.
• Hierarchical libraries for 3D microstructure
  reconstruction in real-time by matching multiple
  lower order features.
• Quality control Allows machine inspection and
  unambiguous quantitative specification of
  microstructures.

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PREPROCESSING
Inputs Microstructure Image (.bmp Format),
Magnification , Rotation (With respect to rolling
direction)

• DIGITIZATION
• Conversion of RGB format of .bmp file to a 2D
  image matrix
• PREPROCESSING
• Brings the image to the library format
• (RD x-axis, TD y-axis)
• Rotate and scale image
• Image enhancement steps
• Boundary detection for feature extraction

Preprocessing based on user inputs of
magnification and rotation
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ROSE OF INTERSECTIONS FEATURE ALGORITHM
(Saltykov, 1974)
Identify intercepts of lines with grain
boundaries plotted within a circular domain
Total number of intercepts of lines at each angle
is given as a polar plot called rose of
intersections
Count the number of intercepts over several lines
placed at various angles.
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GRAIN SHAPE FEATURE EXAMPLES
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GRAIN SIZE PARAMETER
Several lines are superimposed on the
microstructure and the intercept length of the
lines with the grain boundaries are recorded
(Vander Voort, 1993)
The intercept length (x-axis) versus number of
lines (y-axis) histogram is used as the measure
of grain size.
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GRAIN SIZE FEATURE EXAMPLES
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CLASSIFICATION BASED ON EXTRACTED FEATURES
y -1
Available data
y 1
xi 21.30,60.12
The problem
Class I Class II
Class Partition
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THE SVM ALGORITHM
w.xi b -1 if yi -1
r
Maximize the margin
w.xi b gt 1 if yi 1
s
Find w and b such that is
maximized and for all xi ,yi w . xi b 1 if
yi1 w . xi b -1 if yi -1
Maximal Margin Classifier The quadratic
optimization problem
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BINARY CLASSIFIER THE OPTIMIZATION PROBLEM
Maximal Margin Classifier The quadratic
optimization problem
Let w be of the form, ,
b yk- w . xk , k arg maxk ak

Decision function
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NON-LINEAR CLASSIFIERS
Method Map the non-separable data set to a
higher dimensional space (using kernel functions)
where it becomes linearly separable
Non-separable case Minimize
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SVM MULTI-CLASS CLASSIFICATION
p 3
B
Class-B
C
Class-A
C
A
B
A
Class-C
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SVM TRAINING FORMAT
GRAIN FEATURES GIVEN AS INPUT TO SVM TRAINING
ALGORITHM
Data point
CLASSIFICATION SUCCESS
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CLASS HIERARCHY


Level 1 Grain shapes
Class 2
Class 1
Level 2 Subclasses based on grain sizes

Class 1(a)
Class 1(b)
Class 1(c)
Class 2(a)
Class 2(b)
Class 2(c)
New classes Distance of image feature from the
average feature vector of a class
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Microstructure Representation PRINCIPAL
COMPONENT ANALYSIS

• Let be n images.
• Vectorize input images
• Create an average image
• Generate training images

• Create correlation matrix (Lmn)
• Find eigen basis (vi) of the correlation matrix
• Eigen faces (ui) are generated from the basis
  (vi) as
• Any new face image ( ) can be transformed to
  eigen face components through n coefficients
  (wk) as,

Reduced basis
Data Points
Representation coefficients
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PCA REPRESENTATION OF MICROSTRUCTURE AN EXAMPLE
Input Microstructures
Eigen-microstructures
Representation coefficients (x 0.001)
Basis 1
Image-1 quantified by 5 coefficients over the
eigen-microstructures
Basis 5
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EIGEN VALUES AND RECONSTRUCTION OVER THE BASIS
Significant eigen values capture most of the
image features
4
2
3
1
Reconstruction of microstructures over fractions
of the basis
1.Reconstruction with 100 basis
3. Reconstruction with 60 basis
2. Reconstruction with 80 basis
4. Reconstruction with 40 basis
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INCREMENTAL PCA METHOD

• For updating the representation basis when new
  microstructures are added in real-time.
• Basis update is based on an error measure of the
  reconstructed microstructure over the existing
  basis and the original microstructure

Newly added data point
Updated Basis
IPCA Given the Eigen basis for 9
microstructures, the update in the basis for the
10th microstructure is based on a PCA of 10 x 1
coefficient vectors instead of a 16384 x 1 size
microstructures.
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IPCA QUANTIFICATION WITHIN CLASSES
Class-j Microstructures (Equiaxial grains,
medium grain size)
Class-i Microstructures (Elongated 45 degrees,
small grain size)
The Library Quantification and image
representation
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REPRESENTATION FORMAT FOR MICROSTRUCTURE
Date 1/12 0223PM, Basis updated Shape Class 3,
(Oriented 40 degrees, elongated) Size Class 1,
(Large grains) Coefficients in the basis2.42,
12.35, -4.14, 1.95, 1.96, -1.25
Improvement of microstructure representation due
to classification
Reconstruction with 6 coefficients (24 basis) A
class with 25 images
Improvement in reconstruction 6 coefficients (10
of basis) Class of 60 images
Original image
Reconstruction over 15 coefficients
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EXTENSIONS PCA REPRESENTATION OF 3D
MICROSTRUCTURES
Basis Components
Reconstruct using two basis components
X 5.89

X 14.86
Project onto basis
Pixel value round-off
Representation using just 2 coefficients
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EXTENSIONS REAL-TIME 3D RECONSTRUCTION

• Real-time reconstruction of 3D microstructures
  from planar image features using statistical
  learning

Experimental image AA3002 Al alloy
3D reconstruction through statistical learning
Database
2D Imaging techniques
Pattern recognition
vision
Microstructure Analysis
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EXTENSIONS TEXTURE CLASSIFICATION

• Statistical learning for recognition of
  crystallographic textures (Orientation
  distribution functions).
• Adaptive selection of reduced-order basis for
  control of microstructure sensitive properties
• Data-mining for process selection, identifying
  multiple process paths for obtaining desired
  properties

Level 1 lt100gt fiber
Level 2 lt110gt fiber
Level n Values of ODF at the nodes
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CONCEPT OF A STATISTICAL LEARNING TOOLBOX
Training samples
NUMERICAL SIMULATION OF MATERIAL RESPONSE
Update data In the library

• Multi-length
• scale analysis
• Polycrystalline
• plasticity

Adaptive Selection of reduced order basis
STATISTICAL LEARNING TOOLBOX
Image

• Functions
• Feature extraction/ Classification
• Identify new classes

ODF
Initial guesses
Associate data with a class update classes
Process controller
Pole figures
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MAIN REFERENCES

• V. Sundararaghavan and N.Zabaras, Acta
  Materialia, in press.
• C.L.Y. Yeong and S. Torquato, Physical Review E.,
  57(1),495-506 (1998).
• M. Turk and A. Pentland, J Cognitive Neurosci,
  3(1) ,71-86 (1991).
• D. Skocaj and A. Leonardis, "Incremental
  approach to robust learning of eigenspaces" in
  26th Workshop of the Austrian Association for
  Pattern Recognition (ÖAGM/AAPR), edited by F.
  Leberl and F. Fraundorfer, Graz (Austria), 2002,
  pp. 71-78.
• G.F. Vander Voort, "Examination of some grain
  size measurement problems" in Metallography
  Past, Present and Future (75th Anniversary
  Volume), edited by G.F. Vander Voort et al.,
  ASTM STP1165, Philadelphia, 1993, pp. 266-273.
• S.A. Saltykov, Stereometrische metallographie,
  DeutscherVerlag fur Grundstoffindustrie, Leipzig,
  1974.
• V. Sundararaghavan and N.Zabaras, Comp. Materials
  Sci, submitted.

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