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Tamper-evident Image using JPEG Fixed Points

Zhaofeng Si, Siwei Lyu

TL;DR

This work analyzes JPEG compression and decompression processes, revealing the existence of fixed points that can be reached within a few iterations, and develops a method to create a tamper-evident image from the original authentic image, which can expose tampering operations by showing deviations from the fixed point image.

Abstract

An intriguing phenomenon about JPEG compression has been observed since two decades ago- after repeating JPEG compression and decompression, it leads to a stable image that does not change anymore, which is a fixed point. In this work, we prove the existence of fixed points in the essential JPEG procedures. We analyze JPEG compression and decompression processes, revealing the existence of fixed points that can be reached within a few iterations. These fixed points are diverse and preserve the image's visual quality, ensuring minimal distortion. This result is used to develop a method to create a tamper-evident image from the original authentic image, which can expose tampering operations by showing deviations from the fixed point image.

Tamper-evident Image using JPEG Fixed Points

TL;DR

This work analyzes JPEG compression and decompression processes, revealing the existence of fixed points that can be reached within a few iterations, and develops a method to create a tamper-evident image from the original authentic image, which can expose tampering operations by showing deviations from the fixed point image.

Abstract

An intriguing phenomenon about JPEG compression has been observed since two decades ago- after repeating JPEG compression and decompression, it leads to a stable image that does not change anymore, which is a fixed point. In this work, we prove the existence of fixed points in the essential JPEG procedures. We analyze JPEG compression and decompression processes, revealing the existence of fixed points that can be reached within a few iterations. These fixed points are diverse and preserve the image's visual quality, ensuring minimal distortion. This result is used to develop a method to create a tamper-evident image from the original authentic image, which can expose tampering operations by showing deviations from the fixed point image.

Paper Structure

This paper contains 10 sections, 4 theorems, 1 equation, 6 figures.

Table of Contents

  1. Introduction
  2. Fixed Points for the JPEG Transform
  3. Overview of JPEG
  4. Basic Notations and Definitions
  5. DCT and IDCT Matrices
  6. JPEG Fixed Points of $8\times 8$ Blocks
  7. JPEG Fixed Points of Full-sized Images
  8. Tamper-evident JPEG Images
  9. Related Works
  10. Conclusion

Key Result

Theorem 1

$\Omega_t$ is finite and $\Omega_0 \supseteq \Omega_1 \supseteq \cdots \Omega_{t-1} \supseteq \Omega_{t}$.

Figures (6)

  • Figure 1: A demonstration of JPEG fixed points is presented. (Top): An example image approaches its JPEG fixed point through repeated JPEG transforms, along with the locations and numbers of pixels that change in each iteration. (Bottom): The pixel-wise $\ell_2$ distance between images from consecutive iterations of JPEG transforms is plotted for the example image in the top row under varying JPEG compression qualities.
  • Figure 2: The overall pipeline of JPEG compression and decompression.
  • Figure 3: The distribution of $\epsilon_t-\eta_{t+1}$ and $\epsilon_{t+1}-\eta_{t+1}$ from 1M $8\times 8$$8$-bit image blocks. See details in texts.
  • Figure 4: The $\ell_2$ difference between consecutive patches for 1M $8 \times 8$ patches with various compression quality. The starting point of the process of finding fixed point is single JPEG compressed patch.
  • Figure 5: The PSNR between fixed-point image and single JPEG image for different compression quality.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof