A Graph-Matching Kernel for Object Categorization Olivier Duchenne , Armand Joulin , Jean Ponce Willow Lab , ICCV2011
Mar 23, 2016
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