classification and representation of microstructures using statistical learning techniques

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

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




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




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.)




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




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




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




Input Image

Classifier

Feature Detection

APPROACH: MICROSTRUCTURE LIBRARYAPPROACH: MICROSTRUCTURE LIBRARY

Identify and add new classes

Employ lower-order features

Pre-processing




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




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.




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




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




GRAIN SHAPE FEATURE: EXAMPLESGRAIN SHAPE FEATURE: EXAMPLES




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







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




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




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




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




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




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




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




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




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




EIGEN VALUES AND RECONSTRUCTION OVER THE BASISEIGEN VALUES AND RECONSTRUCTION OVER THE BASIS

1.Reconstruction with 100% basis

2. Reconstruction with 80% basis



4 23 1

Reconstruction of microstructures over fractions of the basis

Significant eigen values capture most of the image features




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




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




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




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




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




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




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




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.

classification and representation of microstructures using statistical learning techniques

Documents

control laboratoryrequirements

microstructure libraryidentify

constrainednew descriptor

variety of microstructures

unbiased representation

aerospace engineering

set of numbers1

particle position momentum