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Computer Vision

This lecture focuses on classification techniques within the field of computer vision, specifically supervised learning methods like Support Vector Machines (SVM). It introduces the concepts of data labeling and feature extraction, highlighting the challenges of real-world applications like face and scene classification. The importance of training data, decision boundaries, and maximizing the margin for robust classifiers is discussed. Key methods such as Nearest Neighbor Classifiers and the SVM's kernel trick are explored, providing insights into optimizing classification performance.

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Computer Vision

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  1. Computer Vision Spring 2012 15-385,-685 Instructor: S. Narasimhan WH 5409 T-R 10:30am – 11:50am Lecture #23

  2. Classification and SVM Credits: Guru Krishnan and Shree Nayar Eric P. Xing

  3. Classification (Supervised Learning) Data: X = {X1, X2, … , Xn} Label: Y = {y1, y2, … , yn}, yi  {-1,1} Test: Xt -1 or 1? f: X  Y Classifier: X = Y = 1 1 1 -1 -1 -1 Face or not? Xt

  4. Classification and Computer Vision Handwritten characters Face Scene classification Object recognition

  5. Classification and Computer Vision Pedestrian detection Yes / No

  6. Procedure of Classification (1) Gather positive / negative training data (2) Feature Extraction Y = 1 1 1 -1 -1 -1 X = Very challenging in reality…  Rn

  7. Feature Extraction  Rn Binary codes are available ! Texturefilters SIFT Color histogram Haarfilter Pixel

  8. Feature Space Let the n-dimensional feature vector F be a point an n-D space Rn(Feature space) f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face (3) Learn classifier!

  9. Nearest Neighbor Classifier Nearest samples decide the result of the classifier. f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face Test Image

  10. Nearest Neighbor Classifier Nearest samples decide the result of the classifier. f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face Face

  11. Nearest Neighbor Classifier Nearest samples decide the result of the classifier. f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face Not Face

  12. Nearest Neighbor Classifier Larger the training set, more robust the NN classifier f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face False Positive

  13. Nearest Neighbor Classifier Larger the training set, slower the NN classifier f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face

  14. Decision Boundary A simple decision boundary separating the face and non-face classes will suffice. f2 ° ° ° ° ° ° f1 fN Training Dataof Face Training Dataof Non-Face

  15. Decision Boundary Find Decision Boundary in feature space Decision Boundary WTF+b=0 ° ° ° ° Faces ° ° ° WTF+b>0 ° ° ° ° Non-Faces WTF+b<0

  16. Decision Boundary How to find the optimal decision boundary? ° ° ° ° Face Class ° ° ° ° ° ° ° Non-Face Class

  17. Evaluating a Decision Boundary Margin ° ° ° ° ° ° ° ° ° ° ° Margin or Safe Zone: The width that the boundary could be increased by before hitting a data point.

  18. Evaluating a Decision Boundary Margin II Margin I + + ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° Decision I: Face Decision II: Non-Face Choose Decision Boundary with the Maximum Margin!

  19. Support Vector Machine (SVM) Classifier optimized to maximize Margin Margin ° ° ° ° ° ° ° ° ° ° ° Support Vectors: Closest data samples to the boundary. Decision Boundary and the Margin depend only on the Support Vectors.

  20. Finding Decision Boundary ρ -ρ Distance between a point x to a plan WTF+b=0 ° WTF+b=c ° ° ° ° ° wTxs + b 2c c ρ= ρ= ||w|| ||w|| ||w|| wTxs + b=c WTF+b=-c where WTF+b=0 wTx + b max (Hessian normal form) d= ||w|| w wTxi + b wTxi + b -c c s.t ≥ ≤ for all Xi with yi=1 for all Xi with yi=-1 ||w|| ||w|| ||w|| ||w||

  21. Founding Decision Boundary 2c 2c ||w|| ||w|| yi yi min max max ||w|| w w w Learning classifier Compute w by solving the QP wTxi + b wTxi + b wTxi + b -c c c s.t s.t s.t (wTxi + b) 1 ≥ ≥ ≥ ≤ for all Xi for all Xi for all Xi with yi=1 for all Xi with yi=-1 ||w|| ||w|| ||w|| ||w|| ||w|| ||w||

  22. Kernel Trick How about these points in 1-D? ° ° ° ° ° ° ° ° ° ° 0 0 ° No! ° Linearly separable Think about a quadratic mapping Φ(x)=x2. Yes!

  23. Kernel Trick Φ(x) ° ° ° ° ° ° ° ° ° ° ° ° ° ° Mathematically hard to describe! ° °

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