S. Mandayam/ ANN/ECE Dept./Rowan University Artificial Neural Networks ECE /ECE Fall 2006 Shreekanth Mandayam ECE Department Rowan University Lecture 7 October 30, 2006

S. Mandayam/ ANN/ECE Dept./Rowan UniversityPlan ANN Design Issues Input data processing Selection of training and test data - cross-validation Pre-processing: Feature Extraction Approximation Theory Universal approximation Final Project: Discussion Lab 3

S. Mandayam/ ANN/ECE Dept./Rowan University Feature Extraction Objective: Increase information content Decrease vector length Parametric invariance Invariance by structure Invariance by training Invariance by transformation

S. Mandayam/ ANN/ECE Dept./Rowan University Fourier Descriptors Object must be described as a function Function should be periodic Fourier Transform can be applied to analyze the frequencies Low Frequencies hold general shape information, while high frequencies carry more detail Effective for compression since reconstruction is possible with fewer values than the original

S. Mandayam/ ANN/ECE Dept./Rowan University Fourier Descriptors

S. Mandayam/ ANN/ECE Dept./Rowan University Fourier Descriptors Descriptors Near Zero Values

S. Mandayam/ ANN/ECE Dept./Rowan UniversityReconstruction Original Shape Reconstructed with 20 descriptors

S. Mandayam/ ANN/ECE Dept./Rowan UniversityMoments Statistical moments Mean, Variance, and higher order moments Give a well behaved statistical measure Moments of similar objects should share similar moment calculations 2-D moments evaluate the images without having to extract the boundary

S. Mandayam/ ANN/ECE Dept./Rowan University 2-D Central Moments Equation of 2-D moment is given as: Central moments:

S. Mandayam/ ANN/ECE Dept./Rowan University For a digital image the discrete equation becomes: Normalized Central Moments are defined as: Moments where,

S. Mandayam/ ANN/ECE Dept./Rowan University Invariant Moments

S. Mandayam/ ANN/ECE Dept./Rowan University Invariant Moments OriginalRotated and Resized

S. Mandayam/ ANN/ECE Dept./Rowan University Invariant Moments Image 1 Image 2 % Difference % % % % % % %

S. Mandayam/ ANN/ECE Dept./Rowan University Invariant Moments Ottawa #45 SandOttawa #20/70 Sand

S. Mandayam/ ANN/ECE Dept./Rowan University Invariant Moments Image 1 Image 2 % Difference % % % % % % %

S. Mandayam/ ANN/ECE Dept./Rowan University Approximation Theory: Distance Measures Supremum Norm Infimum Norm Mean Squared Norm

S. Mandayam/ ANN/ECE Dept./Rowan University Recall: Metric Space Reflexivity Positivity Symmetry Triangle Inequality

S. Mandayam/ ANN/ECE Dept./Rowan University Approximation Theory: Terminology Compactness Closure K F

S. Mandayam/ ANN/ECE Dept./Rowan University Approximation Theory: Terminology Best Approximation Existence Set E M u0u0 f min E M u0u0 ALL f min

S. Mandayam/ ANN/ECE Dept./Rowan University Approximation Theory: Terminology Denseness F f g 

S. Mandayam/ ANN/ECE Dept./Rowan University Fundamental Problem E M ? g min Theorem 1: Every compact set is an existence set (Cheney) Theorem 2: Every existence set is a closed set (Braess)

S. Mandayam/ ANN/ECE Dept./Rowan University Stone-Weierstrass Theorem Identity Separability Algebraic Closure F f g  x 1 x1x1 f(x 1 ) x2x2 f(x 2 ) F af+bg

S. Mandayam/ ANN/ECE Dept./Rowan University Final Project Discussion

S. Mandayam/ ANN/ECE Dept./Rowan University Lab 3: RBF Neural Nets /fall06/ann/lab3.htmlhttp://engineering.rowan.edu/~shreek /fall06/ann/lab3.html

S. Mandayam/ ANN/ECE Dept./Rowan UniversitySummary