Camera Source Identification Game with Incomplete Information

Abstract

Image forensics with the presence of an adversary has raised more and more attention recently. A typical case is the interplay between the sensor-based camera source identification and fingerprint-copy attack. This paper gives a game theory analysis in such an adversarial environment. We use a counter anti-forensic method based on noise level estimation to detect the possible forgery (forgery test). Next, we introduce a game theory model to evaluate the ultimate performance when both the investigator and the forger have complete information. Finally, for a more practical scenario that one of the parties has incomplete information, a Bayesian game is introduced and the ultimate performance is compared with that of complete information game.

Appendix

Proposition 1.

For selected patches in noise level estimation [13], the fingerprint-copy attack (1) can be considered as a procedure of Gaussian noise addition.

Proof.

For each selected weak textured patch J _p , the procedure of fingerprint-copy attack can be denoted as:

$$ \begin{gathered} \varvec{J}_{p}^{\prime} = \varvec{J}_{p} (1 + \beta \hat{\varvec{K}}_{p} )\\ \text{ } = \varvec{J}_{p} + \varvec{J}_{p} \beta \hat{\varvec{K}}_{p}\\ \text{ } \approx \varvec{J}_{p} + j_{p} \beta \hat{\varvec{K}}_{p}\\ \end{gathered} $$

(15)

where $\hat{\varvec{K}}_{p}$ is the fingerprint $\hat{\varvec{K}}$ of the corresponding location. The last approximately equal sign is because only weak textured patch is selected in the noise level estimation [13]. That is, J _p ≈ j _p is approximately constant on the selected patch. j _p β $\hat{\varvec{K}}_{p}$ can be modelled as Gaussian noise due to $\hat{\varvec{K}}_{p}$ is modelled as Gaussian noise. Hence, the fingerprint-copy attack is a procedure of Gaussian noise addition from (15). QED.