Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 1 Prof. George Papadourakis, Ph.D. PATTERN CLASSIFICATION WITH DECISION FUNCTIONS Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 2 The main function of a pattern recognition system is the pattern classification into categories. Use of decision functions w 1, w 2, w 3 : parameters (position, line inclination) Simple Linear Decision Functions (1/2) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 3 For any pattern if: then x belongs C 2 then x belongs C 1 then indefinite status Simple Linear Decision Functions (2/2) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 4 Simple Linear Decision Functions (1/2) Decision Function can have any form Depends: General form of d(x): geometrical set properties Coefficients of d(x): after transformations, the problem is reduced to find Linear Decision Functions The form of a n-dimensional linear function is: Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 5 n=2: straight line, separates two-dimensional space n=3: level, separates three-dimensional space n>3: superlevel n-1, separates n-dimensional space Simplified form: w: Parameters Vector Simple Linear Decision Functions (1/2) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 6 Linear Decision Functions (1/5) 2 categories: categories: Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 7 =3 Shaded areas: Many d i (x) positive Cant decide category X2X2 X1X1 C1C1 C2C2 C3C3 d 1 (x) d 3 (x) d 2 (x) Linear Decision Functions (2/5) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 8 Linear Decision Functions (3/5) 2 nd methodology categories: Mutual per two separable decision functions ( per 2) of the form: x belongs to the category C i Shaded area: equation doesnt apply If high, Many d ij (x) Linear Decision Functions (3/5) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 9 3 rd methodology categories: M decision functions x belongs to category C i Subcase of 2 nd methodology Linear Decision Functions (4/5) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 10 Linearly separable categories. Coefficients need to be calculated. X2X2 X1X1 C3C3 C2C2 C1C1 d 1 (x)- d 2 (x) d 1 (x)- d 3 (x) d 2 (x)- d 3 (x) Linear Decision Functions (5/5) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 11 Generalized Decision Functions (1/4) C1C1 C 2 C1C1 C 2 Sebestyens Problem Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 12 2 categories, 2 subcategories each Non-linearly separable categories General form of decision functions: {f i (x)}: basis functions Indefinite variation of decision functions Generalized Decision Functions (2/4) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 13 Simple form: linear decision function: In this case we have: Secondary decision function: Simple two-dimensional : Generalized Decision Functions (3/4) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 14 C1C1 C 2 C1C1 C 2 Generalized Decision Functions (4/4) Whereconstants Curse of dimensionality Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 15 Segmental Linear Separation (1/2) Segmental Linear Separation in Sebestyen Problem Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 16 d 1, d 2 : separate all the subcategories d 1, d 2 give binary decision +(1) -(0) Category Separation with logical equations: Knowledge of the space topology necessary Segmental Linear Separation (2/2) Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 17 Duda, Heart: Pattern Classification and Scene Analysis. J. Wiley & Sons, New York, (2nd edition 2000). Fukunaga: Introduction to Statistical Pattern Recognition. Academic Press, Bishop: Neural Networks for Pattern Recognition. Claredon Press, Oxford, Schlesinger, Hlav: Ten lectures on statistical and structural pattern recognition. Kluwer Academic Publisher, Satosi Watanabe Pattern Recognition: Human and Mechanical, Wiley, 1985 E. Gose, R. Johnsonbaught, S. Jost, Pattern recognition and image analysis, Prentice Hall, Sergios Thodoridis, Kostantinos Koutroumbas, Pattern recognition, Academiv Press, References