A Technique to Evaluate Upper Bounds on Performance of Pixel–prediction Based Reversible Watermarking Algorithms

Abstract

Reversible watermarking algorithms allow distortion–free recovery of the cover image after watermark extraction. Current state–of–the–art does not allow the prediction of the upper bounds of the embedding capacity and distortion characteristics of reversible watermarking algorithms for a given image. In this work, we develop a statistical modelling technique to derive closed form expressions for upper bounds on these performance metrics of pixel–prediction based reversible watermarking algorithms, independent of the actual algorithm used. Comparison of the derived metrics and performance trends with those obtained from two recently reported reversible watermarking algorithms show that the developed model is accurate and consistent.

Appendix: Detailed Derivations

The general form of a linear combination of k Gaussian functions (Gaussian–k), generated by the MATLAB Curve Fitting Toolbox, is:

$$ f^{k}(x) = \displaystyle \sum\limits_{i=1}^{k} a_{i} e^{{- \left( \frac{x - b_{i}}{c_{i}}\right)}^{2}} $$

(A.1)

In this paper, functions f _p and f _Δ are empirically estimated as a linear combination of multiple Gaussian functions, i.e. f _p =f ^k(p) and f _Δ=f ^l(Δ) respectively, where 1≤k,l≤8.

In this section, we derive the general forms of the results presented in Eqs. 30 and 37, for E[C a p a c i t y] and E[M S E] respectively. In our derivations the following two standard integrals [2] have been used:

$$\begin{array}{@{}rcl@{}} \int e^{-(ax^{2}\! +\! 2bx + c)} \,dx &=& \frac{1}{2} \sqrt{\frac{\pi}{2}} \cdot e^{\frac{b^{2} - ac}{a}} \cdot erf\left( \sqrt{a}x \,+\, \frac{b}{\sqrt{a}}\right)\\ &&+ \text{constant}, \quad (a \neq 0) \end{array} $$

(A.2)

$$ \int erf(x) \,dx = x ~erf(x) + \frac{1}{\sqrt{\pi}}e^{-x^{2}} + \text{constant} $$

(A.3)

where e r f(x) is a special mathematical function called the Error Function [2], which is defined as $erf(x)= \frac {2}{\sqrt {\pi }} {\displaystyle {\int }_{0}^{x}} e^{-t^{2}} \,dt$.

Next we compute six integrals I ₁ through I ₆, using Eqs. A.2 and A.3 above, which will be useful in the subsequent steps to derive the closed form expressions for E[C a p a c i t y] and E[M S E].

$$\begin{array}{@{}rcl@{}} I_{1} &=& \int e^{-x^{2}} \,dx \\ &=& \frac{1}{2} \sqrt{\pi} ~erf(x) + \text{constant} \end{array} $$

(A.4)

$$\begin{array}{@{}rcl@{}} I_{2} &=& \int x e^{-x^{2}} \,dx \\ &=& \int \frac{1}{2} e^{-t} \,dt \left[ \text{substituting \(t=x^{2}\)} \right] \\ &=& -\frac{1}{2} e^{-t} + \text{constant} \\ &=& -\frac{1}{2} e^{-x^{2}} + \text{constant} \end{array} $$

(A.5)

I ₃ is evaluated by repeated application of the rule of integration by parts:

$$\begin{array}{@{}rcl@{}} I_{3} &=& \int x^{2} e^{-x^{2}} \,dx = x^{2} \int e^{-x^{2}} \,dx - \int 2x \left[ \int e^{-x^{2}} \,dx \right] \,dx \\ &=& \frac{\sqrt{\pi}}{2} x^{2} ~erf(x) - \sqrt{\pi} \int x ~erf(x) \,dx \\ &=& \frac{\sqrt{\pi}}{2} x^{2} ~erf(x) - \sqrt{\pi} \left[ \frac{x^{2}}{2} ~erf(x) \,+\, \frac{x}{2 \sqrt{\pi}}e^{-x^{2}} \,-\, \frac{1}{4} erf(x) \right]\\ &&+ \text{constant} \\ &=& \frac{\sqrt{\pi}}{4} erf(x) - \frac{x}{2} e^{-x^{2}} + \text{constant} \end{array} $$

(A.6)

$$\begin{array}{@{}rcl@{}} \left[ \begin{array}{l} \displaystyle \because \int x ~erf(x) \,dx = x \int erf(x) \,dx - \int \left[ \int erf(x) \,dx \right] \,dx \\ \displaystyle \quad\quad= x \left[ x ~erf(x) + \frac{1}{\sqrt{\pi}}e^{-x^{2}} \right] - \int \left[ x ~erf(x) + \frac{1}{\sqrt{\pi}}e^{-x^{2}} \right] \,dx \\ \displaystyle \quad\quad= \frac{x^{2}}{2} ~erf(x) + \frac{x}{2 \sqrt{\pi}}e^{-x^{2}} - \frac{1}{4} erf(x) + \text{constant} \end{array} \right] \end{array} $$

We find the next three integrals I ₄, I ₅, I ₆ as definite integrals, with their upper and lower limits of integration set to U and L respectively.

$$\begin{array}{@{}rcl@{}} \left. {I_{4}^{k}}\right|^{U}_{L} &=& {{\int}^{U}_{L}} f^{k}(x) \,dx \\ &=& \sum\limits_{i=1}^{k} {{\int}_{L}^{U}} a_{i} e^{- \left( \frac{x - b_{i}}{c_{i}} \right)^{2}} \,dx \\ &=& \sum\limits_{i=1}^{k} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} a_{i} c_{i} e^{-t^{2}} \,dt \left[ \text{substituting}{\kern5pt} t\,=\,\frac{x - b_{i}}{c_{i}}\right] \\ &=& \sum\limits_{i=1}^{k} a_{i} c_{i} \left[ \frac{\sqrt{\pi}}{2} erf(t) \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} \\ &=& \frac{\sqrt{\pi}}{2} \sum\limits_{i=1}^{k} a_{i} c_{i} \left[ erf\left( \frac{U - b_{i}}{c_{i}} \right) \,-\, erf\left( \frac{L - b_{i}}{c_{i}} \right) \right]\\ \end{array} $$

(A.7)

$$\begin{array}{@{}rcl@{}} \left. {I_{5}^{k}}\right|^{U}_{L} &=& {{\int}^{U}_{L}} x f^{k}(x) \,dx = \displaystyle\sum\limits_{i=1}^{k} {{\int}_{L}^{U}} x ~a_{i} e^{- \left( \frac{x - b_{i}}{c_{i}} \right)^{2}} \,dx \\ &=& \displaystyle\sum\limits_{i=1}^{k} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} (c_{i} t + b_{i}) a_{i} e^{-t^{2}} c_{i} \,dt\\ &&\left[ \text{substituting}{\kern5pt} t=\frac{x - b_{i}}{c_{i}}\right] \\ &=& \displaystyle\sum\limits_{i=1}^{k}\left[ a_{i} {c_{i}^{2}} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} te^{-t^{2}} \,dt + a_{i} b_{i} c_{i} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} e^{-t^{2}} \,dt \right] \\ &=& \displaystyle\sum\limits_{i=1}^{k} a_{i} {c_{i}^{2}} \left[ - \frac{1}{2} e^{-t^{2}} \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} + \displaystyle\sum\limits_{i=1}^{k} a_{i} b_{i} c_{i} \left[ \frac{\sqrt{\pi}}{2} erf(t) \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} \\ &=& - \frac{1}{2} \displaystyle\sum\limits_{i=1}^{k} a_{i} {c_{i}^{2}} \left[ e^{-\left( \frac{U - b_{i}}{c_{i}}\right)^{2}} - e^{-\left( \frac{L - b_{i}}{c_{i}}\right)^{2}} \right] \\ &&+ \frac{\sqrt{\pi}}{2} \displaystyle\sum\limits_{i=1}^{k} a_{i} b_{i} c_{i} \left[ erf\left( \frac{U - b_{i}}{c_{i}}\right) - erf\left( \frac{L - b_{i}}{c_{i}}\right) \right]\\ \end{array} $$

(A.8)

$$\begin{array}{@{}rcl@{}} \left. {I_{6}^{k}}\right|^{U}_{L} &=& {{\int}^{U}_{L}} x^{2} f^{k}(x) \,dx = \displaystyle\sum\limits_{i=1}^{k} {{\int}_{L}^{U}} x^{2} ~a_{i} e^{- \left( \frac{x - b_{i}}{c_{i}} \right)^{2}} \,dx \\ &=&\! \displaystyle\sum\limits_{i=1}^{k} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} (c_{i} t \,+\, b_{i})^{2} a_{i} e^{-t^{2}} c_{i} \,dt \!\left[ \text{substituting}{\kern2pt} t\,=\,\frac{x - b_{i}}{c_{i}}\!\right] \\ &=&\! \displaystyle\sum\limits_{i=1}^{k} a_{i} {c_{i}^{3}} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} t^{2} e^{-t^{2}} \,dt + \displaystyle\sum\limits_{i=1}^{k} 2 a_{i} b_{i} {c_{i}^{2}} {\int}_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}} t e^{-t^{2}} \,dt\\ &&+ \displaystyle\sum\limits_{i=1}^{k} a_{i} {b_{i}^{2}} c_{i} {\int}_{\frac{L - b_{i}} {c_{i}}}^{\frac{U - b_{i}}{c_{i}}} e^{-t^{2}} \,dt \\ &=&\! \displaystyle\sum\limits_{i=1}^{k} a_{i} {c_{i}^{3}} \left[ \frac{\sqrt{\pi}}{4} erf(t) - \frac{t}{2} e^{-t^{2}} \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}}\\ &&+ \displaystyle\sum\limits_{i=1}^{k} 2 a_{i} b_{i} {c_{i}^{2}} \left[ -\frac{1}{2} e^{-t^{2}} \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}}\\ &&+ \displaystyle\sum\limits_{i=1}^{k} a_{i} {b_{i}^{2}} c_{i} \left[ \frac{\sqrt{\pi}}{2} erf(t) \right]_{\frac{L - b_{i}}{c_{i}}}^{\frac{U - b_{i}}{c_{i}}}\\ &=&\! \sum\limits_{i=1}^{k} a_{i} {c_{i}^{3}} \left[ \frac{\sqrt{\pi}}{4} erf\left( \frac{U - b_{i}}{c_{i}}\right) - \left( \frac{U - b_{i}}{2c_{i}}\right) e^{-\left( \frac{U - b_{i}}{c_{i}}\right)^{2}}\right.\\ &&- \left.\frac{\sqrt{\pi}}{4} erf\left( \frac{L - b_{i}}{c_{i}}\right) + \left( \frac{L - b_{i}}{2c_{i}}\right) e^{-\left( \frac{L - b_{i}}{c_{i}}\right)^{2}} \right]\\&& - \displaystyle\sum\limits_{i=1}^{k} a_{i} b_{i} {c_{i}^{2}} \left[ e^{-\left( \frac{U - b_{i}}{c_{i}}\right)^{2}} - e^{-\left( \frac{L - b_{i}}{c_{i}}\right)^{2}} \right]\\ &&+ \frac{\sqrt{\pi}}{2} \displaystyle\sum\limits_{i=1}^{k} a_{i} {b_{i}^{2}} c_{i} \left[ erf\left( \frac{U - b_{i}}{c_{i}}\right) - erf\left( \frac{L - b_{i}}{c_{i}}\right) \right]\\ \end{array} $$

(A.9)

Finally, we derive the general, closed form expressions for E[C a p a c i t y] and E[M S E] in terms of I ₁, I ₂ ⋯ I ₆, from the results of Eqs. 30 and 37 respectively. In our derivations, f _p and f _Δ are considered as linear combinations of k and l Gaussian functions respectively (1≤k,l≤8). By Eq. 30,

$$\begin{array}{@{}rcl@{}} E[Capacity] &=& \frac{\mathcal{N}}{mn}{\int}_{0}^{255}f_{p}\,dp {\int}_{-\mathcal{T}}^{\mathcal{T}} f_{\Delta} \,d{\Delta} \\ &=& \frac{\mathcal{N}}{mn} {\int}_{0}^{255} f^{k}(p) \,dp {\int}_{-\mathcal{T}}^{\mathcal{T}} f^{l}({\Delta}) \,d{\Delta} \\ &=& \frac{\mathcal{N}}{mn} \times \left. ~{I_{4}^{k}} \right|^{255}_{0} \times \left. ~{I_{4}^{l}} \right|_{-\mathcal{T}}^{\mathcal{T}} \end{array} $$

(A.10)

By Eq. 37,

$$\begin{array}{@{}rcl@{}} E[MSE] &=& \frac{\mathcal{N}}{mn} \left[ {\int}_{-\mathcal{T}}^{\mathcal{T}} \frac{{\Delta}^{2}}{2} f_{\Delta} \,d{\Delta} + {\int}_{-\mathcal{T}}^{0} \frac{({\Delta} - 1)^{2}}{2} f_{\Delta} \,d{\Delta}\right. \\ && + {\int}_{0}^{\mathcal{T}} \frac{({\Delta} + 1)^{2}}{2} f_{\Delta} \,d{\Delta} + \left.{\int}_{-255}^{-\mathcal{T}} (-\mathcal{T} - 1)^{2} f_{\Delta} \,d{\Delta}\right. \\ && + \left.{\int}_{\mathcal{T}}^{255} (\mathcal{T} + 1)^{2} f_{\Delta} \,d{\Delta} \right]\\ &=& \frac{\mathcal{N}}{mn} \left[ \frac{1}{2} {\int}_{-\mathcal{T}}^{\mathcal{T}} {\Delta}^{2} f^{l}({\Delta}) \,d{\Delta} + \frac{1}{2} {\int}_{-\mathcal{T}}^{0} {\Delta}^{2} f^{l}({\Delta}) \,d{\Delta} \right. \\ && -{\int}_{-\mathcal{T}}^{0} {\Delta} f^{l}({\Delta}) \,d{\Delta}+ \frac{1}{2} {\int}_{-\mathcal{T}}^{0} f^{l}({\Delta}) \,d{\Delta} \\ && + \frac{1}{2} {\int}^{\mathcal{T}}_{0} {\Delta}^{2} f^{l}({\Delta}) \,d{\Delta} + {\int}^{\mathcal{T}}_{0} {\Delta} f^{l}({\Delta}) \,d{\Delta} \\ && + \frac{1}{2} {\int}^{\mathcal{T}}_{0} f^{l}({\Delta}) \,d{\Delta} + (\mathcal{T}+1)^{2} {\int}_{-255}^{-\mathcal{T}} f^{l}({\Delta}) \,d{\Delta}\\ &&+ \left.(\mathcal{T}+1)^{2} {\int}^{255}_{\mathcal{T}} f^{l}({\Delta}) \,d{\Delta} \right] \\ &=& \frac{\mathcal{N}}{mn} \left[ \frac{1}{2} \left. {I_{6}^{k}} \right|_{-\mathcal{T}}^{\mathcal{T}} + \frac{1}{2} \left. {I_{6}^{k}} \right|_{-\mathcal{T}}^{0} - \left. {I_{5}^{k}} \right|_{-\mathcal{T}}^{0} + \frac{1}{2} \left. {I_{4}^{k}} \rule[14pt]{0pt}{0pt} \right|_{-\mathcal{T}}^{0} \right.\\ && + \frac{1}{2} \left. {I_{6}^{k}} \rule[14pt]{0pt}{0pt} \right|^{\mathcal{T}}_{0} + \left. {I_{5}^{k}} \rule[14pt]{0pt}{0pt} \right|^{\mathcal{T}}_{0} + \frac{1}{2} \left. {I_{4}^{k}} \right|^{\mathcal{T}}_{0} + (\mathcal{T}+1)^{2} \left. {I_{4}^{k}} \right|_{-255}^{-\mathcal{T}}\\ &&+ \left.(\mathcal{T}+1)^{2} \left. {I_{4}^{k}} \rule[14pt]{0pt}{0pt} \right|^{255}_{\mathcal{T}} \right] \end{array} $$

(A.11)