Robust color image watermarking using invariant quaternion Legendre-Fourier moments

Abstract

In this paper, a geometrically invariant color image watermarking method using Quaternion Legendre-Fourier moments (QLFMs) is presented. A highly accurate, fast and numerically stable method is proposed to compute the QLFMs in polar coordinates. The proposed watermarking method consists of three main steps. First, the Arnold scrambling algorithm is applied to a binary watermark image. Second, the QLFMs of the original host color image are computed. Third, the binary digital watermark is embedding by performing the quantization of selected QLFMs. Two different groups of attacks are considered. The first group includes geometric attacks such as rotation, scaling and translation while the second group includes the common signal processing attacks such as image compression and noise. Experiments are performed where the performance of proposed method is compared with the existing moment-based watermarking methods. The proposed method is superior over all existing quaternion moment-based watermarking in terms of visual imperceptibility capability and robustness to different attacks.

Appendices

Appendix 1

A counter-clockwise rotation with an angle α, the transformed image intensity function is defined as:

$$ {\mathrm{f}}^{\mathrm{rot}}\left(\mathrm{r},\uptheta \right)=\mathrm{f}\left(\mathrm{r},\uptheta +\upalpha \right) $$

(49)

Assume $\widehat{\uptheta}=\uptheta +\upalpha$, then $\uptheta =\widehat{\uptheta}-\upalpha$, $\mathrm{d}\uptheta =\mathrm{d}\widehat{\uptheta}$. The QLFMs of the two images f^rot and f have the following relations:

$$ {\displaystyle \begin{array}{l}{\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{rot}}\right)=\frac{2\mathrm{p}+1}{\uppi}{\int}_0^{2\uppi}{\int}_0^1{\mathrm{f}}^{\mathrm{rot}}\left(\mathrm{r},\uptheta \right){\left[{\mathrm{L}}_{\mathrm{p}\mathrm{q}}\left(\mathrm{r},\uptheta \right)\right]}^{\ast}\mathrm{rdrd}\uptheta\ \\ {}\kern4em =\frac{2\mathrm{p}+1}{\uppi}{\int}_0^{2\uppi}{\int}_0^1\mathrm{f}\left(\mathrm{r},\uptheta +\upalpha \right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{r}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}\mathrm{rdrd}\uptheta \\ {}\kern4em =\frac{2\mathrm{p}+1}{\uppi}{\int}_0^{2\uppi}{\int}_0^1\mathrm{f}\left(\mathrm{r},\widehat{\uptheta}\right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{r}\right){\mathrm{e}}^{-\upmu \mathrm{q}\left(\widehat{\uptheta}-\upalpha \right)}\ \mathrm{rdrd}\widehat{\uptheta}\\ {}\kern4em =\frac{2\mathrm{p}+1}{\uppi}{\int}_0^{2\uppi}{\int}_0^1\mathrm{f}\left(\mathrm{r},\widehat{\uptheta}\right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{r}\right){\mathrm{e}}^{-\upmu \mathrm{q}\widehat{\uptheta}}{\mathrm{e}}^{\upmu \mathrm{q}\upalpha}\mathrm{rdrd}\widehat{\uptheta}\\ {}\kern4em ={\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left(\mathrm{f}\right){\mathrm{e}}^{\upmu \mathrm{q}\upalpha}\end{array}} $$

(50)

Where ${\mathrm{M}}_{\mathrm{pq}}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{rot}}\right)$ and ${\mathrm{M}}_{\mathrm{pq}}^{\mathrm{R}}\left(\mathrm{f}\right)$ are the QLFM of f^rot and f respectively.

Appendix 2

Let f^S be the scaled version of the image f. Let f^S(r, θ) = $\mathrm{f}\left(\ \frac{\mathrm{r}}{\mathrm{a}},\uptheta \right)$ be the color image expanded by the scale factor a, and $\widehat{\mathrm{r}}=\frac{\mathrm{r}}{\mathrm{a}}$, then $\mathrm{dr}=\mathrm{ad}\widehat{\mathrm{r}},\kern0.5em$we could express the scaling QLFMs using the following equation:

$$ {\displaystyle \begin{array}{l}{\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{S}}\right)=\frac{2\mathrm{p}+1}{\uppi}\underset{0}{\overset{2\uppi}{\int }}\underset{0}{\overset{1}{\int }}{\mathrm{f}}^{\mathrm{s}}\left(\mathrm{r},\uptheta \right){\left[{\mathrm{L}}_{\mathrm{p}\mathrm{q}}\left(\mathrm{r},\uptheta \right)\right]}^{\ast}\mathrm{rdrd}\uptheta \\ {}\kern3.5em =\frac{2\mathrm{p}+1}{\uppi}\underset{0}{\overset{2\uppi}{\int }}\underset{0}{\overset{1}{\int }}{\mathrm{f}}^{\mathrm{S}}\left(\mathrm{r},\uptheta \right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{r}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}\mathrm{rdrd}\uptheta \\ {}\kern3.5em =\frac{2\mathrm{p}+1}{\uppi}\underset{0}{\overset{2\uppi}{\int }}\underset{0}{\overset{1}{\int }}\mathrm{f}\left(\frac{\mathrm{r}}{\mathrm{a}},\uptheta \right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{r}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}\mathrm{rdrd}\uptheta\ \\ {}\kern3.5em =\frac{2\mathrm{p}+1}{\uppi}\underset{0}{\overset{2\uppi}{\int }}\underset{0}{\overset{1}{\int }}\mathrm{f}\left(\widehat{\mathrm{r}},\uptheta \right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{a}\widehat{\mathrm{r}}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}{\mathrm{a}}^2\widehat{\mathrm{r}}\mathrm{d}\widehat{\mathrm{r}}\mathrm{d}\uptheta \\ {}\kern3.5em ={\mathrm{a}}^2\frac{2\mathrm{p}+1}{\uppi}\underset{0}{\overset{2\uppi}{\int }}\underset{0}{\overset{1}{\int }}\mathrm{f}\left(\widehat{\mathrm{r}},\uptheta \right){\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{a}\widehat{\mathrm{r}}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}\widehat{\mathrm{r}}\mathrm{d}\widehat{\mathrm{r}}\mathrm{d}\uptheta \end{array}} $$

(51)

Where ${\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{a}\widehat{\mathrm{r}}\right)$ is scaled version of ${\overline{\mathrm{P}}}_{\mathrm{p}}\left(\widehat{\mathrm{r}}\right)$. The scaled substituted shifted Legendre polynomials ${\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{a}\widehat{\mathrm{r}}\right)$ could be expressed in terms of the substituted shifted Legendre polynomials ${\overline{\mathrm{P}}}_{\mathrm{p}}\left(\widehat{\mathrm{r}}\right)$ as follows:

$$ {\overline{\mathrm{P}}}_{\mathrm{p}}\left(\mathrm{a}\widehat{\mathrm{r}}\right)=\sum \limits_{\mathrm{k}=0}^{\mathrm{p}}\left(\sum \limits_{\mathrm{i}=\mathrm{k}}^{\mathrm{p}}{\mathrm{a}}^{2\mathrm{i}}{\mathrm{C}}_{\mathrm{p}\mathrm{i}}{\mathrm{d}}_{\mathrm{i}\mathrm{k}}\right){\overline{\mathrm{P}}}_{\mathrm{k}}\left(\widehat{\mathrm{r}}\right) $$

(52)

Where:

$$ {\mathrm{C}}_{\mathrm{p}\mathrm{i}}={\left(-1\right)}^{\mathrm{p}-\mathrm{k}}\frac{\left(\mathrm{p}+\mathrm{k}\right)!}{\left(\mathrm{p}-\mathrm{k}\right)!{\left(\mathrm{k}!\right)}^2} $$

(53)

$$ {\mathrm{d}}_{\mathrm{pi}}=\frac{\left(2\mathrm{k}+1\right){\left(\mathrm{k}!\right)}^2}{\left(\mathrm{p}+\mathrm{k}+1\right)!\left(\mathrm{p}-\mathrm{k}\right)!} $$

(54)

Substitution equation (52) into equation (51) yields:

$$ {\displaystyle \begin{array}{l}{\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{S}}\right)\\ {}=\frac{2\mathrm{p}+1}{\uppi}\sum \limits_{\mathrm{k}=0}^{\mathrm{p}}\left(\sum \limits_{\mathrm{i}=\mathrm{k}}^{\mathrm{p}}{\mathrm{a}}^{2\mathrm{i}+2}{\mathrm{C}}_{\mathrm{p}\mathrm{i}}{\mathrm{d}}_{\mathrm{i}\mathrm{k}}{\int}_0^{2\uppi}{\int}_0^1\mathrm{f}\left(\widehat{\mathrm{r}},\uptheta \right){\overline{\mathrm{P}}}_{\mathrm{k}}\left(\widehat{\mathrm{r}}\right){\mathrm{e}}^{-\upmu \mathrm{q}\uptheta}\widehat{\mathrm{r}}\mathrm{d}\widehat{\mathrm{r}}\mathrm{d}\uptheta \right)\\ {}=\sum \limits_{\mathrm{k}=0}^{\mathrm{p}}\frac{2\mathrm{p}+1}{2\mathrm{k}+1}\left(\sum \limits_{\mathrm{i}=\mathrm{k}}^{\mathrm{p}}{\mathrm{a}}^{2\mathrm{i}+2}{\mathrm{C}}_{\mathrm{p}\mathrm{i}}{\mathrm{d}}_{\mathrm{i}\mathrm{k}}{\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left(\mathrm{f}\right)\right)\end{array}} $$

(55)

Where ${\mathrm{M}}_{\mathrm{pq}}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{s}}\right)$ and ${\mathrm{M}}_{\mathrm{pq}}^{\mathrm{R}}\left(\mathrm{f}\right)$ are the QLFM of f^s and f respectively. Use p = 0 and q = 0 in equation (55), yields:

$$ {\mathrm{M}}_{00}^{\mathrm{R}}\left({\mathrm{f}}^{\mathrm{s}}\right)={\mathrm{a}}^2\ {\mathrm{M}}_{00}^{\mathrm{R}}\left(\mathrm{f}\right) $$

(56)

The scale invariants can be constructed as follows:

$$ {\upvarphi}_{\mathrm{p}\mathrm{q}}=\sum \limits_{\mathrm{k}=0}^{\mathrm{p}}\frac{2\mathrm{p}+1}{2\mathrm{k}+1}\left(\sum \limits_{\mathrm{i}=\mathrm{k}}^{\mathrm{p}}\kern0.5em {\left({\mathrm{M}}_{00}^{\mathrm{R}}\left(\mathrm{f}\right)\right)}^{-\left(\mathrm{i}+1\right)}{\mathrm{C}}_{\mathrm{p}\mathrm{i}}{\mathrm{d}}_{\mathrm{i}\mathrm{k}}\right){\mathrm{M}}_{\mathrm{p}\mathrm{q}}^{\mathrm{R}}\left(\mathrm{f}\right) $$

(57)