On the Convergence of JPEG Multiple Compression for Digital Forensics

Convergence means that the limit of the sequence will tend to a certain value. In some forensic scenarios, if the same manipulation is repeatedly performed on an image, the resulting image sequence tends to converge. Credible forensics can be achieved by utilizing this convergence tendency. Since convergence-based solutions for digital forensics are reliable and interpretable, it is widely applied in digital forensics, especially in image forensics such as identifying single JPEG compressed or double JPEG compressed images. But existing methods only focus on the convergence of the sequence without delving into its stability. To this end, we prove mathematically and completely an interesting property that, for a vector sequence \((W^{k})_{k\geq 0}\) defined by \(W^{k+1}=A^{-1}\left\{A\left\{W^{k}\right\}\right\}\) , where \(A\) is an orthogonal matrix and \(\{\cdot\}\) signifies the decimal part function, then for almost all matrices \(A\) (i.e., except for a measure-zero subset of all orthogonal matrices), the sequence \((W^{k})_{k\geq 0}\) ceases to change after a certain item. This property of sequence stability provides a solid theoretical foundation for double JPEG compression detection with the same quantization matrix. Extensive experiments demonstrate the above property, and experimental results show similar convergence and stability for truncation errors in JPEG compression.