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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