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CLASSIFICATION AND REPRESENTATION OF MICROSTRUCTURES USING STATISTICAL LEARNING TECHNIQUES

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

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CLASSIFICATION AND REPRESENTATION OF MICROSTRUCTURES USING STATISTICAL LEARNING TECHNIQUES

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  1. 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@cornell.edu URL: http://www.mae.cornell.edu/zabaras/ Materials Process Design and Control Laboratory

  2. 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 people Materials Process Design and Control Laboratory

  3. REPRESENTATION AT VARIOUS LENGTH SCALES Average Properties: Large-scale statistical quantities Lower order Descriptors (Grain sizes,ODF,OCF etc.) Lattice Positions, Interfacial energies etc. Continuum Engineering Micro- structural Particle position/ momentum, potential Atomistic Materials Electronic Chemistry Physics Length Scales nm mm mm m Materials Process Design and Control Laboratory

  4. 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 Materials Process Design and Control Laboratory

  5. LOWER ORDER DESCRIPTOR BASED RECONSTRUCTION Descriptor-1: P(2)( r ) Reconstructed Actual (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 Materials Process Design and Control Laboratory

  6. 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 Materials Process Design and Control Laboratory

  7. APPROACH: MICROSTRUCTURE LIBRARY Input Image Pre-processing Identify and add new classes Feature Detection Employ lower-order features Classifier Quantification using incremental PCA Materials Process Design and Control Laboratory

  8. 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 Materials Process Design and Control Laboratory

  9. 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. Materials Process Design and Control Laboratory

  10. 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 Materials Process Design and Control Laboratory

  11. 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. Materials Process Design and Control Laboratory

  12. GRAIN SHAPE FEATURE: EXAMPLES Materials Process Design and Control Laboratory

  13. 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. Materials Process Design and Control Laboratory

  14. GRAIN SIZE FEATURE: EXAMPLES Materials Process Design and Control Laboratory

  15. CLASSIFICATION BASED ON EXTRACTED FEATURES y = -1 Available data y = 1 xi = [21.30,60.12] The problem Class – I Class – II Class Partition Materials Process Design and Control Laboratory

  16. THE SVM ALGORITHM w.xi+ b ≤ -1 if yi= -1 r Maximize the margin w.xi + b > 1 if yi= 1 s 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 Maximal Margin Classifier – The quadratic optimization problem Materials Process Design and Control Laboratory

  17. 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 • Find α1…αN such that • is maximized and • = 0 (2) αi≥ 0 for all i Kernel function BINARY CLASSIFIER: THE OPTIMIZATION PROBLEM Maximal Margin Classifier – The quadratic optimization problem Let w be of the form, ,b= yk- w . xk, k = arg maxkαk Decision function Materials Process Design and Control Laboratory

  18. NON-LINEAR CLASSIFIERS Φ: x→φ(x) Relax constraints w . xi+ b≥ 1- if yi=1; w . xi+ b ≤ -1+ if yi= -1 Method: Map the non-separable data set to a higher dimensional space (using kernel functions) where it becomes linearly separable Non-separable case Minimize Materials Process Design and Control Laboratory

  19. SVM MULTI-CLASS CLASSIFICATION • One 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 p = 3 B Class-B C Class-A C A B A Class-C Materials Process Design and Control Laboratory

  20. SVM TRAINING FORMAT GRAIN FEATURES: GIVEN AS INPUT TO SVM TRAINING ALGORITHM Data point CLASSIFICATION SUCCESS % Materials Process Design and Control Laboratory

  21. 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 Materials Process Design and Control Laboratory

  22. 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 Materials Process Design and Control Laboratory

  23. 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 Materials Process Design and Control Laboratory

  24. EIGEN VALUES AND RECONSTRUCTION OVER THE BASIS Significant eigen values capture most of the image features 4 3 2 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 Materials Process Design and Control Laboratory

  25. 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. Materials Process Design and Control Laboratory

  26. 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 Materials Process Design and Control Laboratory

  27. REPRESENTATION FORMAT FOR MICROSTRUCTURE 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] 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 Materials Process Design and Control Laboratory

  28. EXTENSIONS: PCA REPRESENTATION OF 3D MICROSTRUCTURES Basis Components Reconstruct using two basis components X5.89 + X14.86 Project onto basis Pixel value round-off Representation using just 2 coefficients Materials Process Design and Control Laboratory

  29. 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 2D Imaging techniques Database Pattern recognition vision Microstructure Analysis Materials Process Design and Control Laboratory

  30. 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: <100> fiber Level 2: <110> fiber Level n: Values of ODF at the nodes Materials Process Design and Control Laboratory

  31. PROCESS DESIGN ALGORITHMS • 1. Exact methods • Heuristic methods 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 Materials Process Design and Control Laboratory

  32. 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. Materials Process Design and Control Laboratory

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