A Graph-Matching Kernel for Object Categorization

Olivier Duchenne ， ArmandJoulin ， Jean Ponce

Willow Lab ， ICCV2011

Many applications:

1. Object recognition2. Text categorization3. time-series prediction4. Gene expression profile analysis ......

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.

Kernel Method

:| ( )

NR Fx x

1,...,N

nx x R

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.

Kernel Method

1 1( ( ), ),..., ( ( ), )n nx y x y F Y

Idea: The nonlinear problem in a lower space can be solved by a linear method in a higher space.

Example:

Kernel Method

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

Kernel Method

, ( ), ( )x y x y

: | ( )x x F

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

Kernel Method

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]

Kernel Method--Kernel function

Linear Kernel

Polynomial Kernel

RBF (Gaussian) Kernel

Inverse multiquadric Kernel

Common Kernel functions

( , ) ,x z x z ( , ) ( , 1)rx z x z

2

2

|| ||( , ) exp( ), 02x zx z

2 2

1( , )|| ||

x zx z c

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)

Kernel Method

1

( ) , 'n

i ii

g x w x w x w x

1( ' ) 'w x x x y

Dual form

where K is the kernel matrix.

Kernel Method

1 2( ' ) ' ' ( ' ) ' 'w x x x y x x x x x y x

' 'x xw x y ' ' 'x xx x y ' ' 'xx xx xx y 1K y

11

( , )( ) '( ) ' ...

( , )

test

test

l test

x xg x y K

x x

Kernel Method

1 1 1 2 1

2 1 2 2 2

1 2

( , ) ( , ) ... ( , )( , ) ( , ) ... ( , )

( , ) ( , ) ... ( , )

l

l

l l l l

k x x k x x k x xk x x k x x k x x

K

k x x k x x k x x

Kernel Method

Input: hundreds of thousands of Images

Output (goal): Object Categorization

CAT

DINOSAUR

PANDA

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.

Motivations

We need to design a good image similarity measure:

Camparing images

≈?

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-wiseInformation•State-of-the-art performance

An image I = 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....

Image Representation

Matching two images

,( , )

( ) ( ) ( , )n n m n m nn V m n E

E d U d B d d

Matching two iamges is realized by maximizing the energy function:

Matching two images

, 1

,

( , ) || ||

[ ] 1,( , ) [ ] , 1

0

m n m n m n

n m n m n m

m n m n n m n m n m

u d d d d

dx dx x x y yv d d dy dy x x y y

others

, , ,( , ) ( , ) ( , )m n m n m n m n m n m nB d d u d d v d d