1. 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

2. 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)

3. 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)

4. 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:

5. 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)

6. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 6 Linear Decision Functions (1/5)  2 categories:  Μ categories:

7. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 7  Μ=3  Shaded areas: Many d i (x) positive  Can’t decide category X2X2 X1X1 C1C1 C2C2 C3C3 d 1 (x) d 3 (x) d 2 (x) Linear Decision Functions (2/5)

8. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 8 Linear Decision Functions (3/5)  2 nd methodology  Μ categories: Mutual per two separable  decision functions (Μ per 2) of the form:  x belongs to the category C i  Shaded area: equation doesn’t apply If Μ high, Many d ij (x) Linear Decision Functions (3/5)

9. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 9  3 rd methodology  Μ categories: M decision functions  x belongs to category C i  Subcase of 2 nd methodology Linear Decision Functions (4/5)

10. 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)

11. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 11 Generalized Decision Functions (1/4) C1αC1α C 2β C1βC1β C 2α Sebestyen’s Problem

12. 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)

13. 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)

14. Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Intelligent Systems Laboratory 14 C1αC1α C 2β C1βC1β C 2α Generalized Decision Functions (4/4) Whereconstants Curse of dimensionality

15. 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

16. 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)

17. 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, 1982. (2nd edition 2000).  Fukunaga: Introduction to Statistical Pattern Recognition. Academic Press, 1990.  Bishop: Neural Networks for Pattern Recognition. Claredon Press, Oxford, 1997.  Schlesinger, Hlaváč: Ten lectures on statistical and structural pattern recognition. Kluwer Academic Publisher, 2002.  Satosi Watanabe Pattern Recognition: Human and Mechanical, Wiley, 1985  E. Gose, R. Johnsonbaught, S. Jost, Pattern recognition and image analysis, Prentice Hall, 1996.  Sergios Thodoridis, Kostantinos Koutroumbas, Pattern recognition, Academiv Press, 1998. References