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CLASSIFICATION AND REPRESENTATION OF MICROSTRUCTURES USING STATISTICAL LEARNING TECHNIQUES. Veeraraghavan Sundararaghavan and Nicholas Zabaras. Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University - PowerPoint PPT Presentation
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
CLASSIFICATION AND REPRESENTATIONOF MICROSTRUCTURES USING
STATISTICAL LEARNING TECHNIQUES
Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering
188 Frank H. T. Rhodes HallCornell University
Ithaca, NY 14853-3801
Email: [email protected]: http://www.mae.cornell.edu/zabaras/
Veeraraghavan Sundararaghavan and Nicholas Zabaras
peoplepeople
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
RESEARCH SPONSORS
U.S. AIR FORCE PARTNERS
Materials Process Design Branch, AFRL
Computational Mathematics Program, AFOSR
CORNELL THEORY CENTER
ARMY RESEARCH OFFICE
Mechanical Behavior of Materials Program
NATIONAL SCIENCE FOUNDATION (NSF)
Design and Integration Engineering Program
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
REPRESENTATION AT VARIOUS LENGTH SCALESREPRESENTATION AT VARIOUS LENGTH SCALES
Length Scales
Electronic
Atomistic
Ph
ysic
sC
hem
istr
yM
ater
ials
En
gin
eeri
ng
Micro-
structural
Continuum
Average Properties: Large-scale
statistical quantities
Lattice Positions, Interfacial
energies etc.Particle position/
momentum, potential
nm m mm m
Lower order Descriptors (Grain
sizes,ODF,OCF etc.)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
THE PROBLEM STATEMENT
A Common Framework for Quantification of Diverse Microstructure
Representation space of all possible polyhedral microstructures
Equiaxial grain microstructure space
Qualitative representation
Lower order descriptor approach
Equiax grains
Grain size: small
Grain size distribution
Grain size number
No.
of
grai
ns
Quantitative approach
1.41.4 2.62.6 4.04.0 0.90.9 ……....
Microstructure represented by a set of numbers
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
LOWER ORDER DESCRIPTOR BASED RECONSTRUCTIONLOWER ORDER DESCRIPTOR BASED RECONSTRUCTION
(Yeong & Torquato, 1998)
Descriptor: Two-point probability function and lineal measure
1. Non-uniqueness
2. Computationally expensive
3. Incomplete
• How many descriptors?
• Under constrained
Descriptor-1: P(2)( r )
Reconstructed
Actual
New Descriptor: P(3)( r,s,t )
(plotted as a vector)Reconstructed
Actual
An under constrained
case
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
REQUIREMENTS OF A REPRESENTATION SCHEMEREQUIREMENTS OF A REPRESENTATION SCHEME
REPRESENTATION SPACE OF A PARTICULAR MICROSTRUCTURE
Need for a technique that is autonomous, applicable to a variety of microstructures, computationally feasible and provides complete
representation
A set of numbers which completely represents a microstructure within its class
2.72.7 3.63.6 1.21.2 0.10.1 ……....
8.48.4 2.12.1 5.75.7 1.91.9 ……....
Must differentiate other cases: (must be statistically representative)
Quantification using incremental PCA
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Input Image
Classifier
Feature Detection
APPROACH: MICROSTRUCTURE LIBRARYAPPROACH: MICROSTRUCTURE LIBRARY
Identify and add new classes
Employ lower-order features
Pre-processing
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
DYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTSDYNAMIC MICROSTRUCTURE LIBRARY: CONCEPTS
Space of all possible microstructures
New class
New class: partition
Expandable class partitions
(retraining)
Hierarchical sub-classes (eg. Medium grains)
A class of microstructures (eg. Equiaxial grains)
Dynamic Representation:
Axis for representation
New microstructure
added
Updated representation
distance measures
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
BENEFITSBENEFITS
1. A data-abstraction layer for describing microstructural information.
2. An unbiased representation for comparing simulations and experiments AND for evaluating correlation between microstructure and properties.
3. 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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
DIGITIZATIONDIGITIZATION
Conversion of RGB format of Conversion of RGB format of *.bmp file to a 2D image matrix*.bmp file to a 2D image matrix
PREPROCESSINGPREPROCESSING
Brings the image to the library Brings the image to the library formatformat
(RD : x-axis, TD : y-axis)(RD : x-axis, TD : y-axis)– Rotate and scale imageRotate and scale image– Image enhancement stepsImage enhancement steps– Boundary detection for Boundary detection for
feature extractionfeature extraction
Inputs: Microstructure Image (*.bmp Format), Magnification , Rotation (With respect to rolling direction)
Preprocessing based on user inputs of magnification and rotation
PREPROCESSINGPREPROCESSING
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, ROSE OF INTERSECTIONS FEATURE – ALGORITHM (Saltykov, 1974)1974)
Identify intercepts of lines with grain boundaries plotted within a circular domain
Count the number of intercepts over several lines placed at various angles.
Total number of intercepts of lines at each angle is given as a polar plot called rose of intersections
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
GRAIN SHAPE FEATURE: EXAMPLESGRAIN SHAPE FEATURE: EXAMPLES
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
GRAIN SIZE PARAMETERGRAIN 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.
GRAIN SIZE FEATURE: EXAMPLESGRAIN SIZE FEATURE: EXAMPLES
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
CLASSIFICATION BASED ON EXTRACTED FEATURESCLASSIFICATION BASED ON EXTRACTED FEATURES
Class – I Class – II
y = 1
y = -1
xi = [21.30,60.12]
ClassClass
labellabelFeature vectors
(x data)(x data)
00 24.0224.02 52.1552.15
11 20.1020.10 58.2058.20
11 23.3223.32 54.1254.12
00 24.1224.12 52.6552.65
???? 20.1020.10 63.1263.12
The problem
Available data
Class Partition
THE SVM ALGORITHMTHE SVM ALGORITHM
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
2s
ws
w.xi + b ≤ -1 if yi = -1
w.xi + b > 1if yi = 1
r
T br
w x
w
Find w and b such that
is maximized and for all {xi ,yi}
w . xi + b ≥ 1 if yi=1; w . xi + b ≤ -1 if yi = -1
2s
w
Maximal Margin Classifier – The quadratic optimization problem
Maximize the margin
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
BINARY CLASSIFIER: THE OPTIMIZATION PROBLEMBINARY CLASSIFIER: THE OPTIMIZATION PROBLEM
Find w and b such that
is maximized and for all (xi ,yi)
w . xi + b ≥ 1 if yi=1; w . xi + b ≤ -1 if yi = -1
2s
w
Maximal Margin Classifier –
The quadratic optimization problem
Decision function
Find α1…αN such that
is maximized and
(1) = 0 (2) αi ≥ 0 for all iKernel function
1 , 1
1( ) ( . )
2
n n
i i j i j i ji i j
Q y y
α α α α x x
, 1
n
i ii j
y
α
, 1
( ) sgn( ( . ) )n
i i ii j
f x y b
α x x
Let w be of the form, , b= yk- w . xk , k = arg maxk αk1
n
i i ii
y
w α x
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
NON-LINEAR CLASSIFIERSNON-LINEAR CLASSIFIERS
Method:
Map the non-separable data set to a higher dimensional space (using kernel functions) where it becomes linearly separable
Φ: x → φ(x)
Non-separable case
Minimize 2
1
1( , )
2
n
jj
J w w C
Relax constraints
w . xi + b ≥ 1- if yi=1; w . xi + b ≤ -1+ if yi = -1i i i
j
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
SVM MULTI-CLASS CLASSIFICATIONSVM MULTI-CLASS CLASSIFICATION
Class-AClass-B
Class-CA
CB
AB
C
p = 3One Against One Method:
•Step 1: Pair-wise classification, for a p class problem
•Step 2: Given a data point, select class with maximum votes out of ( 1)
2
p p
( 1)
2
p p
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CCOORRNNEELLLL U N I V E R S I T Y
SVM TRAINING FORMAT
CLASSIFICATION SUCCESS %
Total Total imagesimages
Number of Number of classesclasses
Number of Number of Training imagesTraining images
Highest Highest success ratesuccess rate
Average Average success ratesuccess rate
375375 1111 4040 95.8295.82 92.5392.53
375375 1111 100100 98.5498.54 95.8095.80
ClassClass Feature Feature numbernumber
Feature Feature valuevalue
Feature Feature numbernumber
Feature Feature valuevalue
11 11 23.3223.32 22 21.5221.52
22 11 24.1224.12 22 31.5231.52
Data point
GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM
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CCOORRNNEELLLL U N I V E R S I T Y
CLASS HIERARCHYCLASS HIERARCHY
Class –2Class –1
Class 1(a) Class 1(b) Class 1(c) Class 2(a) Class 2(b) Class 2(c)
Level 1 : Grain shapes
Level 2 : Subclasses based on grain sizes
New classes:
Distance of image feature from the average feature vector of a class
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Microstructure Representation: PRINCIPAL COMPONENT ANALYSISMicrostructure Representation: PRINCIPAL COMPONENT ANALYSIS
Let be n images.
1. Vectorize input images2. Create an average image
3. Generate training images
1 2 n, ,.....
1
1=
n
iin
i i 4. Create correlation matrix (Lmn)
5. Find eigen basis (vi) of the correlation matrix
6. Eigen faces (ui) are generated from the basis (vi) as
7. Any new face image ( ) can be transformed to eigen face components through ‘n’ coefficients (wk) as,
Tmn m nL
i i iLv v
i ij ju v
( )Tk ku
Representation coefficients
Reduced basis
Data Points
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
PCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLEPCA REPRESENTATION OF MICROSTRUCTURE – AN EXAMPLE
Eigen-microstructures
Input Microstructures
Representation coefficients (x 0.001)
Image-1 quantified by 5 coefficients over the eigen-microstructures
0.0125 1.3142 -4.23 4.5429 -1.6396
-0.8406 0.8463 -3.0232 0.3424 2.6752
3.943 -4.2162 -0.6817 -9718 1.9268
1.17961.1796 -1.3354-1.3354 -2.8401-2.8401 6.20646.2064 -3.2106-3.2106
5.82945.8294 5.22875.2287 -3.7972-3.7972 -3.6095-3.6095 -3.6515-3.6515Basis 5
Basis 1
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EIGEN VALUES AND RECONSTRUCTION OVER THE BASISEIGEN VALUES AND RECONSTRUCTION OVER THE BASIS
1.Reconstruction with 100% basis
2. Reconstruction with 80% basis
3. Reconstruction with 60% basis
4. Reconstruction with 40% basis
4 23 1
Reconstruction of microstructures over fractions of the basis
Significant eigen values capture most of the image features
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
INCREMENTAL PCA METHODINCREMENTAL 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
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.
Updated BasisNewly added data point
IPCA QUANTIFICATION WITHIN CLASSESIPCA QUANTIFICATION WITHIN CLASSES
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Class-j Microstructures (Equiaxial grains, medium grain size)
Class-i Microstructures (Elongated 45 degrees, small grain size)
Representation Matrix
Image -1 Image-2 Image-3…
Component in basis vector 1
123 23 38
2 91 54 -85
3 -54 90 12
Average Image
21 23 24…
Eigen Basis
0.9 0.84 0.23..
0.54 0.21 0.74..
The Library – Quantification and image representation
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
REPRESENTATION FORMAT FOR MICROSTRUCTUREREPRESENTATION FORMAT FOR MICROSTRUCTURE
Improvement of microstructure representation due to classificationImprovement of microstructure representation due to classification
Date: 1/12 02:23PM, Basis updated
Shape Class: 3, (Oriented 40 degrees, elongated)
Size Class : 1, (Large grains)
Coefficients in the basis:[2.42, 12.35, -4.14, 1.95, 1.96, -1.25]
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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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXTENSIONS: PCA REPRESENTATION OF 3D MICROSTRUCTURESEXTENSIONS: PCA REPRESENTATION OF 3D MICROSTRUCTURES
Pixel value round-off
Basis Components
X 5.89
X 14.86
+
Project
onto basis
Reconstruct using two basis components
Representation using just 2 coefficients
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXTENSIONS: REAL-TIME 3D RECONSTRUCTIONEXTENSIONS: REAL-TIME 3D RECONSTRUCTION
• Real-time reconstruction of 3D Real-time reconstruction of 3D microstructures from planar microstructures from planar image features using statistical image features using statistical learninglearning
vision
Database
Pattern recognition
MicrostructureAnalysis
2D Imaging techniques
Experimental image AA3002 Al alloy
3D reconstruction through statistical learning
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
EXTENSIONS : TEXTURE CLASSIFICATIONEXTENSIONS : TEXTURE CLASSIFICATION
•Statistical learning for recognition Statistical learning for recognition of crystallographic textures of crystallographic textures (Orientation distribution (Orientation distribution functions). functions).
•Adaptive selection of reduced-Adaptive selection of reduced-order basis for control of order basis for control of microstructure sensitive microstructure sensitive propertiesproperties
•Data-mining for process Data-mining for process selection, identifying multiple selection, identifying multiple process paths for obtaining process paths for obtaining desired propertiesdesired properties
Level 1: <100> fiber
Level n: Values of ODF at the nodes
Level 2: <110> fiber
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Training samples
ODF
Image
Pole figures
STATISTICALLEARNING TOOLBOX
Functions:1. Feature
extraction/ Classification
2. Identify new classes
NUMERICAL SIMULATION OF
MATERIAL RESPONSE
1. Multi-length scale analysis
2. Polycrystalline plasticity
PROCESS DESIGN
ALGORITHMS
1. Exact methods
2. Heuristic methods
Update data
In the library
Associate datawith a class;
update classesProcesscontroller
CONCEPT OF A STATISTICAL LEARNING TOOLBOX
Adaptive Selection of
reduced order basis
Initial guesses
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
MAIN REFERENCESMAIN REFERENCES
1. V. Sundararaghavan and N.Zabaras, Acta Materialia, in press.
2. C.L.Y. Yeong and S. Torquato, Physical Review E., 57(1),495-506 (1998).
3. M. Turk and A. Pentland, J Cognitive Neurosci, 3(1) ,71-86 (1991).
4. 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.
5. 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.
6. S.A. Saltykov, Stereometrische metallographie, DeutscherVerlag fur Grundstoffindustrie, Leipzig, 1974.
7. V. Sundararaghavan and N.Zabaras, Comp. Materials Sci, submitted.