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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 University2School of Information Security, Shanghai Jiaotong University
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Outline
Markov Transition ProbabilitySecond order statistics and Feature
ExtractionDimension and correlation between variables
New Form of the featureTwo elements and three elements
Experiment Result Conclusion
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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
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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
Xij : BDCT Coefficeints
Yij Yi,j+1 Difference array
Transition Probability
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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.
4, 4 4, 3 4,3 4,4
3, 4 3, 3 3,3 3,4
3, 4 3, 3 3,3 3,4
4, 4 4, 3 4,3 4,4
P P P P
P P P P
P P P P
P P P P
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Example: Transition Probability Matrix
12
34
56
78
9
-4-3
-2-1
01
23
4
0
0.2
0.4
0.6
0.8
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Problem of Current Presentation of the Feature Dimension of the
feature is square proportional to the threshold
2 3 4 5 6 70
50
100
150
200
250
threshold of difference element
dim
ensi
on o
f fe
atur
e
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Correlation Between Adjacent Elements in Difference Array Assume adjacent
BDCT coefficients are uncorrelated, i.e.,
Xij : BDCT Coefficeints
Yij Yi,j+1 Difference array
Transition Probability
,, 2
[( )( )]( , )
0.5 (k=1)
0 (k>1)
ij i k jij i k j
y
E y y y yy y
,( ) 0 ( 1,2,... 1)ij i k jE x x k N
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Correlation Calculated on Dataset
Figure . Correlation between adjacent elements on difference array of block DCT coefficients: (1) k=1; (2) k=2
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PCA Transform of Two-component Random Parameters Correlation Matrix Eigenvectors
1
1
1
2
1
1
1 2
1 1
2 2
1 1
2 2
Uncorrelated new random variables
11 2
1,2
T ij
i j
yz
yz
Eigenvalues
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Decomposition of Second Order Statistics into Marginal Ones Marginal histograms
are output of two linear filters
1 2 3 4 5 6 7 8 9-4-3-2-101234
0
0.5
1
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4sum histogram
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4difference histogram
11,
2,
21,
1, 2,
2
ij ij i j
ij i j
ij ij i j
ij i j i j
z y y
x x
z y y
x x x
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Feature Dimension Linearly Proportional to Threshold
2 3 4 5 6 70
50
100
150
200
250
Threshold
Fea
ture
Dim
ensi
on
Transition Probability Matrix
Our new form
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The Approach Can be Generalized to Three Elements Correlation Matrix Eigenvectors
1 0
1
0 1
Eigenvalues
1
2
3
1
1 2
1 2
1 2 3
1 112 2
21 1
0 2 2
11 1
22 2
Decomposed variables
1
2 1 2 3 1,
3 2,
ijT
i j
i j
z y
z y
z y
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Dataset and Classifier
Columbia Splicing Detection Evaluation Dataset
921 authentic, 910 spliced
2/3 Training, 1/3 Test LibSVM, Gaussian
RBF kernel
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Single Feature Performance
Type of Joint Statistics Feature Dimension Accuracy
2 elements
1st order Markov Transition Probability
81 87.09 (1.39)
Our new form (Sec. 3.1) 46 87.97 (1.45)
3 elements
2nd order Markov Transition Probability
729 85.84 (0.92)
Our new form (Sec. 3.2) 77 85.54 (1.34)
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Combined Features Performance
Feature T Dimension Accuracy
Moment+Transition Probability Matrix
3 266 89.86 (1.02)
Moment+New Form
3 220 89.62 (0.91)
4 236 89.78 (1.03)
5 252 89.78 (1.09)
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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
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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