Estimation of geometric transformation matrices using grid-shaped pilot signals

TL;DR

This work tackles watermark robustness to cropping by introducing a grid-shaped pilot signal embedded in a color channel and recovering the geometric transformation from its distorted Radon-domain representation. The method embeds the pilot using QIM, separates vertical and horizontal grid lines, and uses the Radon transform to extract detection angles and intervals, which are then mapped to the transformation matrix. Extensive simulations demonstrate accurate estimation for single and composite geometric attacks on high-resolution images, with some limitations for low-resolution data where grid interval tuning is critical. When integrated with an existing watermarking approach, the technique enables watermark recovery with low BER (often <0.1) after distortion and cropping, while incurring modest PSNR loss. Overall, the grid-pilot Radon framework provides a practical path to robust synchronization and watermark extraction under cropping and geometric manipulation.

Abstract

Digital watermarking techniques are essential to prevent unauthorized use of images. Since pirated images are often geometrically distorted by operations such as scaling and cropping, accurate synchronization - detecting the embedding position of the watermark - is critical for proper extraction. In particular, cropping changes the origin of the image, making synchronization difficult. However, few existing methods are robust against cropping. To address this issue, we propose a watermarking method that estimates geometric transformations applied to a stego image using a pilot signal, allowing synchronization even after cropping. A grid-shaped pilot signal with distinct horizontal and vertical values is embedded in the image. When the image is transformed, the grid is also distorted. By analyzing this distortion, the transformation matrix can be estimated. Applying the Radon transform to the distorted image allows estimation of the grid angles and intervals. In addition, since the horizontal and vertical grid lines are encoded differently, the grid orientation can be determined, which reduces ambiguity. To validate our method, we performed simulations with anisotropic scaling, rotation, shearing, and cropping. The results show that the proposed method accurately estimates transformation matrices with low error under both single and composite attacks.

Figures (29)

• Figure 1: Embedded area of the pilot signal: Values $-1$, $0$, and $+1$ are embedded in the blue, pink, and green areas as a pilot signal.
• Figure 2: Example of extracted signal: black, gray, and white pixels represent the $-1$, $0$, and $+1$ values extracted by the QIM extractor.
• Figure 3: Radon coefficient for the extracted signal: horizontal and vertical axes represent the projection angle and position, respectively.
• Figure 4: Schematic drawing of Radon coefficient: detection angles and detection intervals are shown.
• Figure 5: Original