A high capacity combined reversible watermarking scheme for 2-D CAD engineering graphics

Abstract

Compared with the wide research on reversible watermarking scheme for raster image, the research on its counterpart for 2-D CAD engineering graphics lags far behind. In this paper, based on improved quantization index modulation (IQIM) and improved difference expansion (IDE), a combined reversible watermarking scheme is proposed for 2-D CAD engineering graphics. The proposed scheme can solve the embedding-limitation problem existing in IQIM technique and increase the watermark embedding capacity greatly. Theoretical analysis and experimental results also show that the proposed scheme has good imperceptibility and robustness.

Appendix

In this part, we present the process to determine the values of t and k in detail as a supplement to Section 4.1.

Since we embed watermark in the relative distance r as described in step 2 of Section 3.1, we can calculate the range of the watermarked relative distance r ^w as follows.

According to the Eqs. (6) ~ (7), we get r = (I + F) × 2^s × 10^p, and according to the Eq. (10), we get r ^w = (I ^w + F) × 10^p, then

$$ \begin{array}{c}\hfill {r}^w-r=\left({I}^w+F\right)\times {10}^p-\left(I+F\right)\times {2}^s\times {10}^p\hfill \\ {}\hfill =\left(I\times {2}^s+w+F\right)\times {10}^p-\left(I+F\right)\times {2}^s\times {10}^p\hfill \\ {}\hfill =\left(w+\left(1-{2}^s\right)\times F\right)\times {10}^p.\hfill \end{array} $$

(31)

Given w ∈ {0, 1, …, 2^s − 1} and F ∈ [0,1), we obtain r ^w − r ∈ (1 − 2^s, 2^s − 1] × 10^p. With r ∈ [0, 2^b × Δ), we can get r ^w ∈ [(1 − 2^s) × 10^p, 2^b Δ + (2^s − 1) × 10^p].

By substituting the range of r ^w into Eq. (24), we can obtain

$$ \left\{\begin{array}{l}\left(1-{2}^s\right)\times {10}^p\ge 5\cdot t\cdot {10}^{-\left(q+1\right)}-k\\ {}{2}^b\varDelta +\left({2}^s-1\right)\times {10}^p<t\left(\varDelta -5\cdot {10}^{-\left(q+1\right)}\right)-k\end{array}\right. $$

(32)

Then, it follows that

$$ \frac{2^b\varDelta +\left({2}^s-1\right)\times {10}^p+k}{\varDelta -5\cdot {10}^{-\left(q+1\right)}}<t\le \frac{\left(1-{2}^s\right)\times {10}^p+k}{5\cdot {10}^{-\left(q+1\right)}}. $$

(33)

To solve the above inequalities, two linear equations are firstly denoted as

$$ t=g(k)=\frac{k}{\varDelta -5\cdot {10}^{-\left(q+1\right)}}+\frac{2^b\varDelta +\left({2}^s-1\right)\times {10}^p}{\varDelta -5\cdot {10}^{-\left(q+1\right)}}, $$

(34)

$$ t=f(k)=\frac{k}{5\cdot {10}^{-\left(q+1\right)}}+\frac{\left(1-{2}^s\right)\times {10}^p}{5\cdot {10}^{-\left(q+1\right)}}. $$

(35)

We use a geometric method shown in Fig. 4 which considers the equation geometric meaning. To guarantee Inequality (33) solvable, the slope of Line t = f(g) must be smaller than that of Line t = f(k), that is $\frac{1}{\varDelta -5\cdot {10}^{-\left(q+1\right)}}<\frac{1}{5\cdot {10}^{-\left(q+1\right)}}$, then we obtain the range which ∆ must satisfy is ∆ > 10^-q.

Fig. 4

A geometric method to determinate parameters k and t

Let f(k ₀) = g(k ₀), then we obtain X-coordinate of the intersection between the two lines

$$ {k}_0=\frac{5\varDelta {2}^b\cdot {10}^{-\left(q+1\right)}+\left({2}^s-1\right)\varDelta \cdot {10}^p}{\varDelta -{10}^{-q}}. $$

(36)

As seen in Fig. 4, when k is larger than k ₀, any value of t within (t1, t2] can satisfy Inequality (33).

So finally we manage to obtain the range of the two parameters

$$ k>{k}_0\kern0.5em ,\kern0.5em \frac{2^b\varDelta +\left({2}^s-1\right)\times {10}^p+k}{\varDelta -5\cdot {10}^{-\left(q+1\right)}}<t\le \frac{\left(1-{2}^s\right)\times {10}^p+k}{5\cdot {10}^{-\left(q+1\right)}}. $$

(37)

where ${k}_0=\frac{5\varDelta {2}^b\cdot {10}^{-\left(q+1\right)}+\left({2}^s-1\right)\varDelta \cdot {10}^p}{\varDelta -{10}^{-q}}$.

Therefore, as long as t and k meet the above condition, Inequality (33) can be satisfied. Then our watermark scheme will be reversible.