Watermarking Applications of Krawtchouk–Sobolev Type Orthogonal Moments
Abstract
:1. Introduction
2. Krawtchouk and Krawtchouk–Sobolev Type Orthogonal Polynomials
2.1. Basic Definitions
2.2. Krawtchouk Polynomials
 1.
 The following recurrence relation holds for all $n\ge 0$:$$x{K}_{n}^{p,N}\left(x\right)={K}_{n+1}^{p,N}\left(x\right)+{\alpha}_{n}^{p,N}{K}_{n}^{p,N}\left(x\right)+{\beta}_{n}^{p,N}{K}_{n1}^{p,N}\left(x\right),$$$${\alpha}_{n}^{p,N}=p(Nn)+n(1p),\phantom{\rule{2.em}{0ex}}{\beta}_{n}^{p,N}=np(1p)(Nn+1).$$Here, we have considered ${K}_{1}^{p,N}\left(x\right)=0$, and ${K}_{0}^{p,N}\left(x\right)=1$.
 2.
 Squared norm. For every $n\in \mathbb{N}$, it holds that$${\u2225{K}_{n}^{p,N}\u2225}^{2}=n!{(N)}_{n}{p}^{n}{(p1)}^{n}.$$
 3.
 Let $0\le k\le N$. The forward shift operator is defined by$${\Delta}^{k}{K}_{n}^{p,N}\left(x\right)={\left[n\right]}_{k}{K}_{nk}^{p,Nk}\left(x\right),$$$${\left[z\right]}_{n}:={\left(1\right)}^{n}{\left(z\right)}_{n},\phantom{\rule{1.em}{0ex}}n\ge 1.$$
 4.
 The sequence of classical Krawtchouk monic orthogonal polynomials satisfies the following secondorder difference equation (hypergeometric type equation)$$(1p)x\Delta \nabla {K}_{n}^{p,N}\left(x\right)+(Npx)\Delta {K}_{n}^{p,N}\left(x\right)+n{K}_{n}^{p,N}\left(x\right)=0,$$$${\Delta}^{n}f\left(x\right)=\Delta \left[{\Delta}^{n1}f\left(x\right)\right]\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\nabla}^{n}f\left(x\right)=\nabla \left[{\nabla}^{n1}f\left(x\right)\right],\phantom{\rule{1.em}{0ex}}n\in \mathbb{N}.$$
2.3. Krawtchouk–Sobolev Type Orthogonal Polynomials
 1.
 Hypergeometric representation. Given positive integers $n\le N$ and j, one has$${\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)={p}^{n1}{\left(N\right)}_{n1}{h}_{n}^{\left(j\right)}\left(x\right){\phantom{\rule{0.166667em}{0ex}}}_{3}{F}_{2}\left(\begin{array}{cc}n,x,{f}_{n}^{\left(j\right)}\left(x\right)& \\ N,{f}_{n}^{\left(j\right)}\left(x\right)1\end{array}{p}^{1}\right),$$$${f}_{n}^{\left(j\right)}\left(x\right)=\frac{np\left(Nn+1\right){\phantom{\rule{4pt}{0ex}}\mathcal{C}}_{1,n}^{\left(j\right)}\left(x\right)}{{\mathcal{D}}_{1,n}^{\left(j\right)}\left(x\right)}n+1,$$$${h}_{n}^{\left(j\right)}\left(x\right)=\left(p\left(Nn+1\right){\phantom{\rule{4pt}{0ex}}\mathcal{C}}_{n}^{\left(j\right)}\left(x\right){\mathcal{D}}_{n}^{\left(j\right)}\left(x\right)\right).$$
 2.
 Structure relations.$${\Theta}_{n}^{\left(j\right)}\left(x\right)\nabla {\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;2,1\right){\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)={\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right){\mathbb{K}}_{n1}^{\left(j\right)}\left(x\right)$$$${\Theta}_{n}^{\left(j\right)}\left(x\right)\nabla {\mathbb{K}}_{n1}^{\left(j\right)}\left(x\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;1,2\right){\mathbb{K}}_{n1}^{\left(j\right)}\left(x\right)={\Lambda}_{n}^{\left(j\right)}\left(x;2,2\right){\mathbb{K}}_{n}^{\left(j\right)}\left(x\right),$$$${\Theta}_{n}^{\left(j\right)}\left(x\right)=x\left\begin{array}{cc}{\mathcal{C}}_{1,n}^{\left(j\right)}\left(x\right)& {\mathcal{C}}_{2,n}^{\left(j\right)}\left(x\right)\\ {\mathcal{D}}_{1,n}^{\left(j\right)}\left(x\right)& {\mathcal{D}}_{2,n}^{\left(j\right)}\left(x\right)\end{array}\right,$$$${\Lambda}_{n}^{\left(j\right)}\left(x;i,k\right)={\left(1\right)}^{k}\left\begin{array}{cc}{\mathcal{E}}_{k,n}^{\left(j\right)}\left(x\right)& {\mathcal{C}}_{i,n}^{\left(j\right)}\left(x\right)\\ {\mathcal{F}}_{k,n}^{\left(j\right)}\left(x\right)& {\mathcal{D}}_{i,n}^{\left(j\right)}\left(x\right)\end{array}\right,\phantom{\rule{1.em}{0ex}}i=1,2,\phantom{\rule{1.em}{0ex}}k=1,2.$$
 3.
 Secondorder difference equations.$${\mathcal{F}}_{n}^{\left(j\right)}\left(x\right){\nabla}^{2}{\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\mathcal{G}}_{n}^{\left(j\right)}\left(x\right)\nabla {\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\mathcal{H}}_{n}^{\left(j\right)}\left(x\right){\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)=0,$$$${\tilde{\mathcal{F}}}_{n}^{\left(j\right)}\left(x\right)\Delta \nabla {\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\tilde{\mathcal{G}}}_{n}^{\left(j\right)}\left(x\right)\Delta {\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\tilde{\mathcal{H}}}_{n}^{\left(j\right)}\left(x\right){\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)=0,$$$${\mathcal{F}}_{n}^{\left(j\right)}\left(x\right)={\Theta}_{n}^{\left(j\right)}\left(x\right){\Theta}_{n}^{\left(j\right)}\left(x1\right),$$$$\begin{array}{ccc}\hfill {\mathcal{G}}_{n}^{\left(j\right)}\left(x\right)& =& {\Theta}_{n}^{\left(j\right)}\left(x\right)(\nabla {\Theta}_{n}^{\left(j\right)}\left(x\right)+{\Lambda}_{n}^{\left(j\right)}\left(x1;2,1\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;1,2\right))\hfill \\ & & \phantom{\rule{2.em}{0ex}}\frac{\nabla {\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right)({\Theta}_{n}^{\left(j\right)}\left(x\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;1,2\right)){\Theta}_{n}^{\left(j\right)}\left(x\right)}{{\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right)},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{H}}_{n}^{\left(j\right)}\left(x\right)& =& {\Theta}_{n}^{\left(j\right)}\left(x\right)\nabla {\Lambda}_{n}^{\left(j\right)}\left(x;2,1\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;1,2\right){\Lambda}_{n}^{\left(j\right)}\left(x;2,1\right)\hfill \\ & & \phantom{\rule{2.em}{0ex}}\frac{\nabla {\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right){\Lambda}_{n}^{\left(j\right)}\left(x;2,1\right)({\Theta}_{n}^{\left(j\right)}\left(x\right)+{\Lambda}_{n}^{\left(j\right)}\left(x;1,2\right))}{{\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right)}\hfill \\ & & \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{\Lambda}_{n}^{\left(j\right)}\left(x1;1,1\right){\Lambda}_{n}^{\left(j\right)}\left(x;2,2\right),\hfill \end{array}$$$${\tilde{\mathcal{F}}}_{n}^{\left(j\right)}\left(x\right)={\mathcal{F}}_{n}^{\left(j\right)}\left(x+1\right),\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\tilde{\mathcal{H}}}_{n}^{\left(j\right)}\left(x\right)={\mathcal{H}}_{n}^{\left(j\right)}\left(x+1\right),$$$${\tilde{\mathcal{G}}}_{n}^{\left(j\right)}\left(x\right)={\mathcal{G}}_{n}^{\left(j\right)}\left(x+1\right)+{\tilde{\mathcal{H}}}_{n}^{\left(j\right)}\left(x\right).$$
 4.
 The recurrence relation for the classical Krawtchouk monic orthogonal polynomials determines that, for the Krawtchouk–Sobolev type orthogonal polynomials, as follows:$${\tilde{\Theta}}_{n}^{\left(j\right)}\left(x\right){\mathbb{K}}_{n+1}^{\left(j\right)}\left(x\right)={\Xi}_{1,n}^{\left(j\right)}\left(x\right){\mathbb{K}}_{n}^{\left(j\right)}\left(x\right)+{\Xi}_{2,n}^{\left(j\right)}\left(x\right){\mathbb{K}}_{n1}^{\left(j\right)}\left(x\right),\phantom{\rule{1.em}{0ex}}n\ge 0,$$$${\tilde{\Theta}}_{n}^{\left(j\right)}\left(x\right)={\Theta}_{n}^{\left(j\right)}\left(x\right){\Lambda}_{n+1}^{\left(j\right)}\left(x;2,2\right),$$$${\Xi}_{1,n}^{\left(j\right)}\left(x\right)={\Theta}_{n}^{\left(j\right)}\left(x\right){\Lambda}_{n+1}^{\left(j\right)}\left(x;1,2\right){\Theta}_{n+1}^{\left(j\right)}\left(x\right){\Lambda}_{n}^{\left(j\right)}\left(x;2,1\right),$$$${\Xi}_{2,n}^{\left(j\right)}\left(x\right)={\Theta}_{n+1}^{\left(j\right)}\left(x\right){\Lambda}_{n}^{\left(j\right)}\left(x;1,1\right),$$
3. Weighted Krawtchouk–Sobolev Type Polynomials
4. Krawtchouk–Sobolev Type Orthogonal Moments
5. Application: Watermarking Scheme
5.1. Arnold Transform
5.2. Zigzag Scan
5.3. Dither Modulation
5.4. Embedding and Extraction Watermark Algorithm
Algorithm 1 Embedding Algorithm 

5.5. Experimental Analysis
5.6. Imperceptibility Test
5.7. Robustness Test
6. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Attacks  Parameters 

Cropping noise  5%, 10%, 12%, 15%, 17%, 20%, 23%, 25%, 27%, 30%, 33%, 35%, 37%, 40%. 
Gaussian noise  Percent noise: 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 
Salt and Pepper noise  Density: 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 
Median filter noise  Kernel size: 2 × 2, 4 × 4, 6 × 6, 8 × 7, 10 × 10, 12 × 12, 14 × 14, 16 × 16, 18 × 18, 20 × 20 
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Huertas, E.J.; Lastra, A.; SoriaLorente, A. Watermarking Applications of Krawtchouk–Sobolev Type Orthogonal Moments. Electronics 2022, 11, 500. https://doi.org/10.3390/electronics11030500
Huertas EJ, Lastra A, SoriaLorente A. Watermarking Applications of Krawtchouk–Sobolev Type Orthogonal Moments. Electronics. 2022; 11(3):500. https://doi.org/10.3390/electronics11030500
Chicago/Turabian StyleHuertas, Edmundo J., Alberto Lastra, and Anier SoriaLorente. 2022. "Watermarking Applications of Krawtchouk–Sobolev Type Orthogonal Moments" Electronics 11, no. 3: 500. https://doi.org/10.3390/electronics11030500