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Recovering Sign Bits of DCT Coefficients in Digital Images as an Optimization Problem

Ruiyuan Lin, Sheng Liu, Jun Jiang, Shujun Li, Chengqing Li, C. -C. Jay Kuo

TL;DR

We address recovering unknown sign bits in DCT coefficients for digital images by formulating it as a mixed integer linear programming problem, which is NP-hard. To achieve practical performance, we propose two approximations: (i) LP relaxation that solves a continuous problem and derives signs from the solution, and (ii) a divide-and-conquer hierarchical MILP/LP approach that recovers regionwise signs and then globally aligns brightness. Applied to JPEG-encoded images, the methods accommodate encoding specifics such as DC level shifting, quantization, and DC differential encoding, and extensive experiments on 30 standard images show substantial improvements in PSNR and SSIM over naive methods. The results suggest these methods are applicable to selective encryption, error concealment, and efficient image restoration in error-prone environments, with potential extensions to color and video domains.

Abstract

Recovering unknown, missing, damaged, distorted, or lost information in DCT coefficients is a common task in multiple applications of digital image processing, including image compression, selective image encryption, and image communication. This paper investigates the recovery of sign bits in DCT coefficients of digital images, by proposing two different approximation methods to solve a mixed integer linear programming (MILP) problem, which is NP-hard in general. One method is a relaxation of the MILP problem to a linear programming (LP) problem, and the other splits the original MILP problem into some smaller MILP problems and an LP problem. We considered how the proposed methods can be applied to JPEG-encoded images and conducted extensive experiments to validate their performances. The experimental results showed that the proposed methods outperformed other existing methods by a substantial margin, both according to objective quality metrics and our subjective evaluation.

Recovering Sign Bits of DCT Coefficients in Digital Images as an Optimization Problem

TL;DR

We address recovering unknown sign bits in DCT coefficients for digital images by formulating it as a mixed integer linear programming problem, which is NP-hard. To achieve practical performance, we propose two approximations: (i) LP relaxation that solves a continuous problem and derives signs from the solution, and (ii) a divide-and-conquer hierarchical MILP/LP approach that recovers regionwise signs and then globally aligns brightness. Applied to JPEG-encoded images, the methods accommodate encoding specifics such as DC level shifting, quantization, and DC differential encoding, and extensive experiments on 30 standard images show substantial improvements in PSNR and SSIM over naive methods. The results suggest these methods are applicable to selective encryption, error concealment, and efficient image restoration in error-prone environments, with potential extensions to color and video domains.

Abstract

Recovering unknown, missing, damaged, distorted, or lost information in DCT coefficients is a common task in multiple applications of digital image processing, including image compression, selective image encryption, and image communication. This paper investigates the recovery of sign bits in DCT coefficients of digital images, by proposing two different approximation methods to solve a mixed integer linear programming (MILP) problem, which is NP-hard in general. One method is a relaxation of the MILP problem to a linear programming (LP) problem, and the other splits the original MILP problem into some smaller MILP problems and an LP problem. We considered how the proposed methods can be applied to JPEG-encoded images and conducted extensive experiments to validate their performances. The experimental results showed that the proposed methods outperformed other existing methods by a substantial margin, both according to objective quality metrics and our subjective evaluation.
Paper Structure (29 sections, 6 equations, 8 figures, 14 tables)

This paper contains 29 sections, 6 equations, 8 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Sign bit encryption for image and video encryption and privacy protection
  4. Simple image and video recovery against transmission errors and encryption
  5. Optimization-based DCT coefficient recovery
  6. Sign bit recovery of DCT coefficients
  7. Proposed Methods
  8. The model
  9. Two approximation methods
  10. Method 1--LP with relaxation
  11. Method 2--Hierarchical MILP or Hybrid MILP and LP
  12. Extension to different image encoding methods and settings
  13. Experimental Results
  14. Performance of the relaxed LP method
  15. Accelerating computation by ignoring small DCT coefficients
  16. ...and 14 more sections

Figures (8)

  • Figure 1: Distribution of the difference between neighboring pixel values of the standard test image "Lenna" of size $512\times 512$.
  • Figure 2: Time consumption under different values of threshold.
  • Figure 3: Quality of the recovery result under different values of threshold $T$: a) PSNR; b) SSIM.
  • Figure 4: Recovered images under different DC prediction modes ($U=2$): a) mode 0; b) mode 1; c) mode 2; d) mode 3.
  • Figure 5: Performance comparison under the different number of missing coefficients: a) PSNR; b) SSIM.
  • ...and 3 more figures