1. CHAPTER 13 Pattern Recognition and Classification CLASSIFICATION A. Dermanis

2. Examples of pattern recognition applications: Recognition of letters(scanning of texts – artificial reading) Speech recognition(artificial hearing) Voice recognition(identification of a person by his speech) Medical diagnosis(identification of decease) Classification of image pixels (Remote Sensing) Examples of pattern recognition applications: Recognition of letters(scanning of texts – artificial reading) Speech recognition(artificial hearing) Voice recognition(identification of a person by his speech) Medical diagnosis(identification of decease) Classification of image pixels (Remote Sensing) Pattern Recognition: Identification of a particular object within a class of similar objects by exploitation of object-related measurements and a-priori information (patterns) on all class elements Pattern Recognition: Identification of a particular object within a class of similar objects by exploitation of object-related measurements and a-priori information (patterns) on all class elements Pattern Recognition = A collection of mathematical-statistical tools and algorithms independent from particular applications Pattern Recognition = A collection of mathematical-statistical tools and algorithms independent from particular applications A. Dermanis

3. d ij (x)  d i (x) – d j (x) = w i T x + w i 0 – w j T x – w j 0 = (w i T – w j T ) x + (w i 0 – w j 0 )   w ij T x + w ij 0 = 0 Boundary between any 2 classes ω i and ω j : w T x + w 0 = 0 d i (x) = w i 1 x 1 + w i 2 x 2 + … + w i B x B + w i 0 = w i T x + w i 0 d j (x) = d k (x) x   j : d j (x) > d k (x),  k  j d i (x) = d i (x 1, x 2, …, x B ) Decision functions Linear decision functions A decision function for each class ω 1, ω 2, …, ω Κ : Assign each pixel to the class with maximum value of decision function : Spectral space separated into class regions with boundaries : simplified to : A. Dermanis

4. n = w = w |w| w T w 1 d = – w 0 / | w | d(x) = w T x + w 0 = 0 Examples of boundaries for linear decision functions A. Dermanis

5. d i (x) = C –  i (x),d i (x) = C – [  i (x)] 2,d i (x) = C /  i (x) x   j :  j (x) >  k (x),  k  j d i (x) > d i (x)   i (x) >  i (x) Replace d i (x) = max with ρ i (x) = ρ i (m i,x) = min : Classification by distance functions Decision functions derived from distance functions : m i = center of class ω i (usually mean of sample of class pixels) Assign each pixel to the class with minimum value of distance function : A. Dermanis

6.  (m i, x) = || x – m i || = (x – m i ) T (x – m i ) Euclidean distance function : Classification by distance functions Boundary between classes ω i and ω j = = hyperplane perpendicular to the middle of segment joining m i and m j Boundary between classes ω i and ω j = = hyperplane perpendicular to the middle of segment joining m i and m j  (m i, x) =  (m j, x)   (m j – m i ) T [x –  ½ (m i + m j )] = 0  (m i, x) =  (m j, x)   (m j – m i ) T [x –  ½ (m i + m j )] = 0 A. Dermanis

7. Non-linear decision functions Example of non-linear decision functions: 2 classes, 1 band Example of non-linear decision functions: 2 classes, 1 band Example of quadratic (non-linear) decision functions: 2 classes, 2 bands Example of quadratic (non-linear) decision functions: 2 classes, 2 bands A. Dermanis