## 3 The Proposed \((k,n)\)-XVC Scheme with Adjustable Pixel Expansion
### Contrast of the Proposed Scheme  
To demonstrate that the proposed \((k,n)_{m^{\prime}}\)-XVC scheme is an effective \((k,n)\)-XVC scheme, we first prove that the contrast condition is satisfied. When any \(k\) rows of the basis matrices are superimposed, contrast \(\alpha^{\prime}\) of the proposed scheme must be large than 0. As the any \(m^{\prime}\) columns are randomly chosen, we use average contrast to describe the visual quality of the recovered secret image. Then, we prove that the average contrast of the scheme is equal in Theorem 2 no matter whatever pixel expansion is taken. **Theorem 2**: _Construction 1 above is a \((k,n)\)-XVC scheme. The pixel expansion is \(m^{\prime}\in[m-2^{k-1}m\alpha+1,m]\) and the average contrast is_  
\[\overline{\alpha}=2^{k-1}\alpha \tag{3}\]_for \(|U|=k\), if a original \((k,n)\)-OVC scheme has pixel expansion \(m\) and contrast \(\alpha\), where \(m\alpha\geq 1\)._  
Before proving the theorem, we will give some description and notations. For any basis matrices \(B_{0}\) and \(B_{1}\) of a \((k,n)\)-OVC scheme, it satisfies the relationship:  
\[H(\oplus(B_{1}|U))-H(\oplus(B_{0}|k)) = h-l=m\alpha\geq 1, \tag{4}\] \[H(\otimes(B_{1}|k))-H(\otimes(B_{0}|U)) = h^{\prime}-l^{\prime}=2^{k-1}m\alpha. \tag{5}\]  
When any \(k\) rows are superimposed, the secret image will be revealed with contrast \(\alpha\) and \(2^{k-1}\alpha\) when \"OR\" and \"XOR\" operation are used, separatively. The XORing results of any rows in \(U\) of \(B_{0}\) and \(B_{1}\) can be represented in the following form regardless of the arrangement order:  
\[\otimes(B_{0}|U) = (\underbrace{0,0,\ldots,0}_{\textit{$m^{\prime}$}-\textit{$l^{ \prime}$}},\underbrace{1,1,\ldots,1}_{\textit{$l^{\prime}$}},\underbrace{0,0, \ldots,0}_{\textit{$m$}-\textit{$l^{\prime}$}}), \tag{6}\] \[\otimes(B_{1}|U) = (\underbrace{1,1,\ldots,1}_{\textit{$l^{\prime}$}-\textit{$l^{ \prime}$}},\underbrace{1,1,\ldots,1}_{\textit{$l^{\prime}$}},\underbrace{0,0, \ldots,0}_{\textit{$m$}-\textit{$l^{\prime}$}}). \tag{7}\]  
for \(|U|=k\). When \(|U|<k\), the above relationship has the following form:  
\[\otimes(B_{0}|U) = (\underbrace{0,0,\ldots,0}_{\textit{$m$}-\textit{$p$}}, \underbrace{1,1,\ldots,1}_{\textit{$p$}}), \tag{8}\] \[\otimes(B_{1}|U) = (\underbrace{0,0,\ldots,0}_{\textit{$m$}-\textit{$q$}}, \underbrace{1,1,\ldots,1}_{\textit{$q$}}). \tag{9}\]  
Proof. : First, we show the scheme satisfies the contrast condition and compute the average contrast. The basis matrices, \(B_{0}^{\prime}\) and \(B_{1}^{\prime}\), of the proposed \((k,n)_{m^{\prime}}\)-XVC scheme are composed of \(m^{\prime}\) column vectors randomly selected from the basis matrices \(B_{0}\) and \(B_{1}\) are denoted as  
\[B_{0}^{\prime}=B_{0}[i_{1},\ldots,i_{m^{\prime}}],\text{ and }B_{1}^{\prime}=B_{1}[j_{1}, \ldots,j_{m^{\prime}}].\]  
The XORing results of any rows in \(U\) of \(B_{0}^{\prime}\) and \(B_{1}^{\prime}\) can be represented in the following form regardless of the arrangement order:  
\[\otimes(B_{0}^{\prime}|U) = (\underbrace{0,0,\ldots,0}_{\textit{$m^{\prime}$}-\textit{$x$}},\underbrace{1,1,\ldots,1}_{\textit{$x$}}), \tag{10}\] \[\otimes(B_{1}^{\prime}|U) = (\underbrace{0,0,\ldots,0}_{\textit{$m^{\prime}$}-\textit{$y$}}, \underbrace{1,1,\ldots,1}_{\textit{$y$}}). \tag{11}\]  
Considering the Hamming weight of \(\otimes(B_{1}^{\prime}|U)\) (resp. \(\otimes(B_{1}^{\prime}|U)\), we need only determine \(H(\otimes(B_{0}^{\prime}|U))\) (resp. \(H(\otimes(B_{1}^{\prime}|U))\)) with \(|U|=k\). Considering the extreme situation, \(\otimes(B_{1}^{\prime}|U)\) has the least \"\(1\)\"s of \(\otimes(B_{1}|U)\) and \(\otimes(B_{0}^{\prime}|U)\) has the most '\(1\)'s of \(\otimes(B_{0}|U)\). In such situation, if the contrast condition is satisfied, then all other possible basis matrices \(B_{1}^{\prime}\) and \(B_{0}^{\prime}\) must satisfy the contrast condition.