Reversible fragile watermarking for locating tampered blocks in 2D vector maps

Abstract

For 2D vector maps, obtaining good tamper localization performance and original content recovery with existing reversible fragile watermarking schemes is a technically challenging problem. Using an improved reversible watermarking method and a fragile watermarking algorithm based on vertex insertion, we propose a reversible fragile watermarking scheme that detects and locates tampered blocks with high accuracy while ensuring recovery of the original content. In particular, we propose dividing the features of the vector map into different blocks, calculating the block authentication watermarks and embedding the watermarks with different watermarking schemes. While the block division ensures superior accuracy of tamper localization, the reversible watermarking method and the fragile watermarking algorithm based on vertex insertion provide recovery of the original content. Experimental results show that the proposed scheme could detect and locate malicious attacks such as vertex/feature modification, vertex/feature addition, and vertex/feature deletion.

Appendix. The proof of the algorithm in Section 2.2

Here, we prove that no watermarked coordinate exceeds the original coordinate range as long as the maximum and the minimum coordinates do not carry watermark bits.

We first explain that all watermarked coordinates of X are greater than x _min (the minimum coordinate of X) as long as x _min does not carry watermark bits. Suppose $x_{{\min }}^{r}\left( {x_{{\min }}^{r} = x_{{l,k}}^{s} + x_{{m,k}}^{s},\;1 \le k \le n} \right)$ is the regulated coordinate transformed from x _min , $x_{{l,\min}}^r\left( {x_{{l,\min}}^r=x_{l,k}^s} \right)$ and $x_{{m,\min}}^r\left( {x_{{m,\min}}^r=x_{m,k}^s} \right)$ are the decimal part and the integer part of $x_{\min}^r$, respectively. As shown in Fig. 16a, two cases may arise when embedding the watermark, namely embedding the watermark into $x_{l,i}^s\left( {1\leq i<k\leq n} \right)$ and embedding the watermark into $x_{l,t}^s\left( {1\leq k<t\leq n} \right)$.

Fig. 16

a The position of $x_{{l,\min}}^r$ in $X_l^s$ and the position of $x_{{m,\min}}^r$ in $X_m^s$ and b the position of $x_{{l,\max}}^r$ in $X_l^s$ and the position of $x_{{m,\max}}^r$ in $X_m^s$

If the watermark is embedded into $x_{l,i}^s$, the following will hold,

$$ \begin{array}{*{20}c} {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,i}^s<x_{l,k}^s} \\ {x_{l,i}^s+x_{m,i}^s>x_{l,k}^s+x_{m,k}^s=x_{\min}^r} \\ \end{array}} \right.} \hfill \\ \Rightarrow \hfill & {\left\{ {\begin{array}{*{20}c} {x_{m,i}^s>x_{m,k}^s} \\ {x_{l,i}^s+x_{m,i}^s>x_{l,k}^s+x_{m,k}^s=x_{\min}^r} \\ \end{array}} \right.} \hfill \\ \end{array}. $$

(26)

Thus, after embedding the watermark into $x_{l,i}^s$, x _min is still the minimum coordinate of X.

If the watermark is embedded into $x_{l,t}^s$, we have the following,

$$ \begin{array}{*{20}c} {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,t}^s\geq x_{l,k}^s} \hfill \\ {x_{l,t}^s+x_{m,t}^s>x_{l,k}^s+x_{m,k}^s=x_{\min}^r} \hfill \\ \end{array}} \right.} \hfill \\ \Rightarrow \hfill & {\left\{ {\begin{array}{*{20}c} {} \hfill & {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,t}^s>x_{l,k}^s} \hfill \\ {x_{m,t}^s\geq x_{m,k}^s} \hfill \\ \end{array}\Rightarrow \left\{ {\begin{array}{*{20}c} {x_{l,t}^s{{\prime}} >x_{l,k}^s} \\ {x_{l,t}^s{\prime} +x_{m,t}^s>x_{l,k}^s+x_{m,k}^s=x_{\min}^r} \\ \end{array}} \right.} \right.} \hfill \\ {or} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,t}^s=x_{l,k}^s} \hfill \\ {x_{m,t}^s>x_{m,k}^s} \hfill \\ \end{array}\Rightarrow x_{l,t}^s{\prime} +x_{m,t}^s>x_{l,k}^s+x_{m,k}^s=x_{\min}^r} \right.} \hfill \\ \end{array}} \right.} \hfill \\ \end{array}. $$

(27)

Thus, after embedding the watermark into $x_{l,t}^s,{x_{min }}$ is still the minimum coordinate of X.

Next, we will show that all watermarked coordinates are less than x _max (the maximum coordinate of X), as long as x _max does not carry watermark bits. Suppose $x_{{\max }}^{r}\left( {x_{{\max }}^{r} = x_{{l,h}}^{s} + x_{{m,h}}^{s}\;1 \le h \le n} \right)$ is the regulated coordinate transformed from ${x_{max }},x_{{l,\max}}^s\left( {x_{{l,\max}}^s=x_{l,h}^s} \right)$ and $x_{{m,\max}}^s\left( {x_{{m,\max}}^s=x_{m,h}^s} \right)$ are the decimal part and the integer part of $x_{\max}^r$, respectively. As shown in Fig. 16b, there may also exist two cases when embedding the watermark, namely embedding the watermark into $x_{l,i}^s\left( {1\leq i < h\leq n} \right)$ and embedding the watermark into $x_{l,t}^s\left( {1\leq h<t\leq n} \right)$.

If the watermark is embedded into $x_{l,i}^s$, the following will hold,

$$ \begin{array}{*{20}c} {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,i}^s<x_{l,h}^s} \\ {x_{l,i}^s+x_{m,i}^s<x_{l,h}^s+x_{m,h}^s=x_{\max}^r} \\ \end{array}} \right.} \hfill \\ \Rightarrow \hfill & {\left\{ {\begin{array}{*{20}c} {x_{m,i}^s\leq x_{m,h}^s} \\ {x_{l,i}^s{\prime} <x_{l,h}^s} \\ {x_{l,i}^s{\prime} +x_{m,i}^s<x_{l,h}^s+x_{m,h}^s=x_{\max}^r} \\ \end{array}} \right.} \hfill \\ \end{array} $$

(28)

Thus, after embedding the watermark into $x_{l,i}^s,\;{x_{max }}$ is still the maximum coordinate of X.

If the watermark is embedded into $x_{l,t}^s$, we have the following,

$$ \begin{array}{*{20}c} {} \hfill & {\left\{ {\begin{array}{*{20}c} {x_{l,t}^s\geq x_{l,h}^s} \\ {x_{l,t}^s+x_{m,t}^s<x_{l,h}^s+x_{m,h}^s=x_{\max}^r} \\ \end{array}} \right.} \hfill \\ \Rightarrow \hfill & {\left\{ {\begin{array}{*{20}c} {x_{m,t}^s<x_{m,h}^s} \\ {x_{l,t}^s{\prime} +x_{m,t}^s<x_{l,h}^s+x_{m,h}^s=x_{\max}^r} \\ \end{array}} \right.} \hfill \\ \end{array} $$

(29)

Thus, after embedding the watermark into $x_{l,t}^s,\;{x_{max }}$ is still the maximum coordinate of X.

Therefore, as long as x _min and x _max do not carry watermark bits, our method can guarantee that the watermarked coordinates are still within the original coordinate range (x _min , x _max ). With the original sorted order and the original coordinate range unchanged, extracting the embedded watermarks from a watermarked block of a 2D vector map correctly can be guaranteed.