A Graph-Matching Kernel for Object Categorization

1. A Graph-Matching Kernel for Object Categorization Olivier Duchenne， Armand Joulin， Jean Ponce Willow Lab， ICCV2011

2. Kernel Method • Many applications: • Object recognition • Text categorization • time-series prediction • Gene expression profile analysis ......

3. Kernel Method • Given a set of data (x1, y1), (x2, y2), ..., (xn, yn), the Kernel Method maps them into a potentially much higher dimensional feature space F.

4. Kernel Method • For a given learning problem one now considers the same algorithm in instead of RN, one works with the sample • The kernel method seeks a pattern among the data in the feature sapce.

5. Kernel Method • Idea: The nonlinear problem in a lower space can be solved by a linear method in a higher space. Example:

6. Kernel Method

7. Kernel Method • 【Kernel function 】A kernel function is a function k that for all x, z∈X satisfies where is a mapping from X to an (inner product) feature space F

8. Kernel Method • The computation of a scalar product between two feature space vectors, can be readily reformulated in terms of a kernel function k

9. Kernel Method--Kernel function • Is necessary? Not necessary • What kind of k can be used? symmetric positive semi-definite ( kernel matrix ) • Given a feature mapping, caan we compute the inner product in feature space? Yes • Given a kernel function k, whether a feature mapping is existence? Yes [Mercer’s theorem]

10. Common Kernel functions • Linear Kernel • Polynomial Kernel • RBF (Gaussian) Kernel • Inverse multiquadric Kernel

11. Kernel Method • Kernel matrix • Consider the problem of finding a real-valued linear function that best intopolates a given training set S = {(x1, y1), (x2, y2), ..., (xl, yl)} (least square)

12. Kernel Method • Dual form where K is the kernel matrix.

13. Kernel Method

14. Kernel Method

15. Input: hundreds of thousands of Images

16. Output (goal): Object Categorization DINOSAUR PANDA CAT

17. Motivations • Feature correspondences can be used to construct an image comparison kernel that is appropriate for SVM-based classification, and often outperforms BOFs. • Image representations that enforce some degree of spatial consistency usually perform better in image classification tasks than pure bags of features that discard all spatial information.

18. Camparing images • We need to design a goodimage similarity measure: ≈ ?

19. Graph-matching Method in this paper • Sparse Features • NN Classifier • Slow • Use pair-wise Information • Lower performance • As Dense • SVM Classifier • Fast enough • Use pair-wise • Information • State-of-the-art performance

20. Image Representation • An imageI = a graph G = Nodes + Edges A node n=dn(xn,yn) represent a region of I, • Each region is represented by a image Feature vector Fn ,e.g. SIFT....

21. Matching two images Matching two iamges is realized by maximizing the energy function:

22. Matching two images