# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 3 Proposed LESM Sensing Matrix Design
### The LESM Algorithm for Sensing Matrix
Hence, \(\epsilon(\mathbf{H},\mathbf{W})\) satisfies  
\[\epsilon(\mathbf{H},\mathbf{W})\geq-\sum_{j=1}^{M}\tilde{\mathbf{W}}^{2}(j,j)\geq-\sum_{j =1}^{M}\sigma_{j}^{2}.\]  
When the diagonal elements of \(\tilde{\mathbf{W}}(j,j)\) are equal to its eigenvalues \(\sigma_{j}\), and the unknown quantity \(\lambda_{j}\) is going to be the diagonal elements of \(\tilde{\mathbf{W}}(j,j)\), the low bound of \(\epsilon(\mathbf{H},\mathbf{W})\) is achieved. The eigenvalues \(\sigma_{j}\) can be calculated by decomposing \(\mathbf{W}\) as \(\mathbf{W}=\mathbf{U}_{w}\mathbf{\Lambda}_{w}\mathbf{U}_{w}^{T}\) and assuming that  
\[\mathbf{V}=\mathbf{U}_{w} \begin{bmatrix}\mathbf{V}_{11}&\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{22}\end{bmatrix} , \tag{26}\]  
then \(\tilde{\mathbf{W}}\) can be written as  
\[\tilde{\mathbf{W}}= \begin{bmatrix}\mathbf{V}_{11}^{T}&\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{22}^{T}\end{bmatrix} \mathbf{\Lambda}_{w} \begin{bmatrix}\mathbf{V}_{11} &\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{22}\end{bmatrix} , \tag{27}\]  
so \(\tilde{\mathbf{W}}\) and the \(\mathbf{W}\) possess the same eigenvalues of matrix \(\mathbf{\Lambda}_{w}\). As a result, the optimal model of the sensing matrix is minimized in such a case when \(\mathbf{\Lambda}_{h}=\mathbf{\Lambda}_{w}(1:M,1:M)\), \(\mathbf{V}_{11}\) is the identity matrix of dimension \(M\), and \(\mathbf{V}_{22}\) is an arbitrary orthonormal matrix of dimension \(N-M\). (2) If the eigenvalues \(\sigma_{1},\cdots,\sigma_{M}>0\) and \(\sigma_{M+1},\cdots,\sigma_{M}<0\) in \(\tilde{\mathbf{W}}\) with \(\bar{M}<M\), the \(\tilde{\mathbf{W}}\) can be expressed as:  
\[ \begin{split}\tilde{\mathbf{W}}&=\mathbf{V}^{T}\mathbf{U}_{w}\mathbf{ \Lambda}_{w}\mathbf{U}_{w}^{T}\mathbf{V}\\ &=\mathbf{V}^{T}\mathbf{U}_{w}(\mathbf{\Lambda}_{w}{}^{(1)}-\mathbf{\Lambda}_{w}{} ^{(2)})\mathbf{U}_{w}^{T}\mathbf{V}\\ &=\tilde{\mathbf{W}}_{1}-\tilde{\mathbf{W}}_{2},\end{split}  \tag{28}\]  
where  
\[ \begin{split}\mathbf{\Lambda}_{w}{}^{(1)}&=diag(\sigma_{1}, \cdots,\sigma_{M},0,\cdots,0),\\ \mathbf{\Lambda}_{w}{}^{(2)}&=diag(0,\cdots,0,-\sigma_{M+1}, \cdots,-\sigma_{N}).\end{split} \]  
So the problem in (23) becomes:  
\[ \begin{split}\epsilon(\mathbf{H},\mathbf{W})&=\sum_{j=1}^{M}[ \lambda_{j}^{4}-2\lambda_{j}^{2}\tilde{\mathbf{W}}_{1}(j,j)+2\lambda_{j}^{2}\tilde {\mathbf{W}}_{2}(j,j)]\\ &\geq\sum_{j=1}^{M}[\lambda_{j}^{4}-2\lambda_{j}^{2}\tilde{\mathbf{W} }_{1}(j,j)]\\ &\geq-\sum_{j=1}^{M}\tilde{\mathbf{W}}_{1}^{2}(j,j)\geq-\sum_{j=1}^{ \bar{M}}\sigma_{j}^{2}.\end{split}  \tag{29}\]  
in which the Lemma mentioned above is also used to guarantee that the inequality in (29) is true. In order to reach the low bound in this case, \(\lambda_{j}^{2}=0,\forall j>\bar{M}\) and \(\lambda_{j}^{2}=\tilde{\mathbf{W}}_{1}^{2}(j,j)\), \(\forall j\leq\bar{M}\)_s.t._\(\{\tilde{\mathbf{W}}(j,j)\}=\{\sigma_{j}\}\). Therefore, \(\mathbf{\Lambda}_{h}(1:\bar{M},1:\bar{M})=\mathbf{\Lambda}_{w}(1:\bar{M},1:\bar{M})\) and \(\mathbf{\Lambda}_{h}(\bar{M}:M,\bar{M}:M)=\mathbf{0}\), \(\mathbf{V}_{11}\) can be the identity matrix of dimension \(\bar{M}\), and \(\mathbf{V}_{22}\) is an arbitrary orthonormal matrix of dimension \(N-\bar{M}\). (3) If the eigenvalues \(\sigma_{1},\cdots,\sigma_{M}<0\) in \(\bar{\mathbf{W}}\), the symmetric \(\mathbf{W}\) is semi-negative-definite, so \(\mathbf{W}(j,j)\leq\sigma_{j}\leq 0\). In such a case, (19) can be minimized when \(\lambda_{j}^{2}=0\). That is, the sensing matrix \(\mathbf{\Phi}_{LESM}=\mathbf{0}\). In summary, corresponding to each of the above cases, the solution of (19) is:  
\[\mathbf{\Phi}_{opt}=\mathbf{U}_{n}\big{[}\mathbf{\Lambda}_{h}\quad\mathbf{0}\big{]} \begin{bmatrix} \mathbf{I}_{M}&\mathbf{0}\\ \mathbf{0}&\mathbf{V}_{22}^{T}\end{bmatrix} \mathbf{U}_{w}^{T}\mathbf{G}_{d}^{-\frac{1}{2}}, \tag{30}\]  
where \(\mathbf{U}_{n}\in\Re^{M\times M}\) and \(\mathbf{V}_{22}\in\Re^{(N-M)\times(N-M)}\) are arbitrary orthonormal matrices with proper dimensions. \(\mathbf{I}_{M}\) is the identity matrix of dimension \(M\).