This repository is an implementation of the method in
"Color image zero-watermarking based on Fast Quaternion Generic Polar Complex Exponential Transform", Signal Processing: Image Communication, 2020.
Code implemented by Shuren Qi ( i@srqi.email ). All rights reserved.
In 2011, the Generic Polar Complex Exponential Transform (GPCET) was proposed by Hoang et al. [1]. As a generalization of various harmonic function-based moments, GPCET has numerous beneficial mathematical properties such as orthogonality, completeness and rotation-invariance. The main distinctive property of GPCET is that the computed coefficients can put emphasis on certain portions of an image by changing a parameter, which is similar to the time-frequency analysis tools. Later, in [2], Hoang et al. extended and improved the research of [1]; in [3], a fast computation strategy for GPCET based on the recurrence relations and geometrical symmetry was proposed. However, these methods basically rely on zero-order approximation (ZOA) in Cartesian coordinate system, namely direct calculation. Three defects limit its practical application: computationally expensive, numerically unstable and inaccurate.
When a large number of coefficients are needed or large-sized images are used, the complexity of direct calculation may become excessively high. Since these requirements are common in practical applications, we propose a fast Fourier transform (FFT)-based calculation method in polar coordinate system, namely Fast Generic Polar Complex Exponential Transform (FGPCET). Compared with the direct calculation method, the proposed FGPCET has the following three advantages
• Low Complexity. Suppose K order GPCET coefficients need to be calculated and the up-sampling parameter M is proportional to the image size N. If only multiplication is considered, complexity of the direct calculation is O(N^2·K^2) . In contrast, the FGPCET has a lower complexity O(N^2·logN) due to the use of the 2D-FFT.
• Numerical Stability. For the direct calculation, it is numerically unstable due to the unboundedness (very high values near the origin) of the radial kernels. For the FGPCET, the numerical instability is eliminated.
• High Precision. Since the up-sampling is used, i.e., M>>N , numerical error and geometric error of the FGPCET are much smaller than those of the direct calculation scheme.
[1] T.-V. Hoang, S. Tabbone, Generic polar harmonic transforms for invariant image description, in: Proceedings of the 18th IEEE International Conference on Image Processing, 2011, pp. 829–832.
[2] T.-V. Hoang, S. Tabbone, Generic polar harmonic transforms for invariant image representation, Image Vis. Comput. 32 (8) (2014) 497-509.
[3] T.-V. Hoang, S. Tabbone, Fast generic polar harmonic transforms, IEEE Trans. Image Process. 23 (7) (2014) 2961-2971.