New Feature Presentation of
Transition Probability Matrix for
Image Tampering Detection
Luyi Chen1 Shilin Wang2 Shenghong Li1 Jianhua Li1
1Department
of Electrical Engineering, Shanghai Jiaotong
University
2School of Information Security, Shanghai Jiaotong University
1/18
Outline

Markov Transition Probability
 Second
order statistics and Feature
Extraction
 Dimension and correlation between variables

New Form of the feature
 Two
elements and three elements
Experiment Result
 Conclusion

2/18
Context


Inspired by applying Markov Transition
Probability Matrix to solve Image Tampering
Detection as a two-class classification (proposed
by Shi et al 07)
Current feature extraction method
 Every element from 2D matrix (huge dimension)
 Boosting selection or PCA for dimension reduction,
and the low dimensional features do not have
corresponding physical meaning

Goal: dimension reduction by decomposing
adjacent elements to be statistically uncorrelated
3/18
Second Order Statistical Modeling
of Image




Image transformed
with 8x8 BDCT
Horizontal difference
array
Modeled with
horizontal transition
probability
Can be applied to four
directions
Transition
Probability
Yij
Yi,j+1
Difference array
Xij : BDCT Coefficeints
4/18
Feature Extraction of Transition
Probability Matrix




Thresholding is applied to
difference array (with
threshold of T)
The transition probability
matrix is used as the
feature
Dimension of the feature
is (2T+1)2
If we consider four
directional transition, the
dimension needs to be
multiplied by 4.
 P4, 4

 P3, 4


 P3, 4
P
 4, 4
P4, 3
P3, 3
P4,3
P3,3
P3, 3
P4, 3
P3,3
P4,3
P4,4 

P3,4 


P3,4 
P4,4 
5/18
Example: Transition Probability
Matrix
0.8
0.6
0.4
0.2
0
-4
-3
-2
-1
0
1
2
3
4
1
2
3
4
5
6
7
8
9
6/18
Problem of Current Presentation of
the Feature
Dimension of the
feature is square
proportional to the
threshold
250
200
dimension of feature

150
100
50
0
2
3
4
5
threshold of difference element
6
7
7/18
Correlation Between Adjacent
Elements in Difference Array

Assume adjacent
BDCT coefficients are
uncorrelated, i.e.,
Transition
Probability
Yij
Yi,j+1
Difference array
E ( xij xi  k , j )  0 (k  1,2,...N  1)
 ( yij , yi  k , j ) 
0.5 (k=1)

(k>1)
0
E[( yij  y )( yi  k , j  y )]
 y2
Xij : BDCT Coefficeints
8/18
Correlation Calculated on Dataset
Figure . Correlation between adjacent elements on difference array of block DCT coefficients: (1) k=1; (2) k=2
9/18
PCA Transform of Two-component
Random Parameters

Correlation Matrix
1




 1 
 1 




2
2
 2  

1  
 1 
 1 





2

 2


1
Eigenvalues
 1  1  

2  1  
Eigenvectors

Uncorrelated new
random variables
 z1 
    1
 z2 
2 
T
 yij 


y
 i 1, j 
10/18
Decomposition of Second Order
Statistics into Marginal Ones

Marginal histograms
are output of two
linear filters
1
0.5
0
-4-3
-2-1
0
z  yij  yi 1, j
1
ij
 xij  xi  2, j
z  yij  yi 1, j
2
ij
 xij  2 xi 1, j  xi  2, j
sum histogram
0.4
0.3
0.3
0.2
0.2
0.1
0.1
-4 -3 -2 -1
0
1
2
34
9
78
56
4
3
12
difference histogram
0.4
0
12
3
4
0
-4 -3 -2 -1
0
1
2
3
4
11/18
Feature Dimension Linearly
Proportional to Threshold
250
Transition Probability Matrix
Our new form
Feature Dimension
200
150
100
50
0
2
3
4
5
6
7
Threshold
12/18
The Approach Can be Generalized
to Three Elements

Correlation Matrix
1


0



1

0
 
1 
Eigenvalues
 1  1

2  1  2 

3  1  2 
Eigenvectors

 1
 1 
 1 
2
 2 






 2 
1
1
  

1   0   2  
3


2
2






 1 
1
1
 




2





 2
 2 

Decomposed variables
 z1 
 
 z2    1
z 
 3
2
 yij 

T 
3   yi 1, j 
y

 i  2, j 
13/18
Dataset and Classifier




Columbia Splicing
Detection Evaluation
Dataset
921 authentic, 910
spliced
2/3 Training, 1/3 Test
LibSVM, Gaussian
RBF kernel
14/18
Single Feature Performance
Type of Joint Statistics Feature
Dimension
Accuracy
1st order Markov Transition
Probability
81
87.09 (1.39)
Our new form (Sec. 3.1)
46
87.97 (1.45)
2nd order Markov Transition
Probability
729
85.84 (0.92)
Our new form (Sec. 3.2)
77
85.54 (1.34)
2 elements
3 elements
15/18
Combined Features Performance
Feature
T
Dimension
Accuracy
Moment+Transition
Probability Matrix
3
266
89.86 (1.02)
3
220
89.62 (0.91)
4
236
89.78 (1.03)
5
252
89.78 (1.09)
Moment+New Form
16/18
Computation Complexity
Comparison
Feature Type
Computing Time (seconds)
Transition Probability Matrix
0.0516 (0.0054)
Marginal Distribution of two new
variables
0.0502 (0.0005)
On Core 2 Duo 1.6G, 3G Ram
17/18
Conclusion


Our new form has lower feature dimension,
faster computation, and almost as good
performance
Dimension Reduction is more obvious in higher
order, but further research is needed to improve
discrimination performance
18/18