# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 3 Proposed LESM Sensing Matrix Design
### The LESM Algorithm for Sensing Matrix  
According to the above analysis, the model considering the LSRE for sensing matrix design is constructed. Designing a proper sensing matrix can prevent the error from being amplified. Hence, the model is constructed by simultaneously minimizing the averaged mutual coherence and the energy of projected LSRE. The function with variable \(\mathbf{\Phi}\) is described as:  
\[f(\mathbf{\Phi})=\|\mathbf{I}_{L}-\mathbf{\
abla}^{T}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{\Psi}\|_{F} ^{2}+\xi\|\mathbf{\Phi}\Delta\mathbf{E}\|_{F}^{2}\;\;s.t.\;\mathbf{E}=\mathbf{\
abla}\Delta\bm {S}+\Delta\mathbf{E} \tag{18}\]  
Our goal is to minimize the above cost function to obtain a more robust sensing matrix:  
\[\mathbf{\Phi}_{LESM}=\arg\min_{\mathbf{\Phi}\in\mathbb{R}^{M\times N}}f(\mathbf{\Phi}), \tag{19}\]  
Some manipulations are executed on (18):  
\[ \begin{split} f(\mathbf{\Phi})&=L-2 tr[\mathbf{\
abla}^{T} \mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{\Psi}]+\|\mathbf{\
abla}^{T}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{ \Psi}\|_{F}^{2}+\xi\|\mathbf{\Phi}\Delta\mathbf{E}\|_{F}^{2}\\ &=L+tr[\mathbf{H}\mathbf{H}]-2 tr[\mathbf{H}\mathbf{W}],\end{split}  \tag{20}\]  
supposed that  
\[\mathbf{G}_{d}\triangleq\mathbf{\
abla}\mathbf{\
abla}^{T}. \tag{21}\]  
Hence,  
\[\mathbf{H}=\mathbf{G}_{d}\mathbf{\overset{1}{2}}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{G}_{d}\mathbf{ \overset{1}{2}},\mathbf{W}=\mathbf{G}_{d}\mathbf{\overset{-1}{2}}[\mathbf{G}_{d}-\mathbf{\overset{ \varsigma}{2}}\Delta\mathbf{E}(\Delta\mathbf{E})^{T}]\mathbf{G}_{d}\mathbf{\overset{-1}{2}}. \tag{22}\]  
Observing the Equation (20), \(L\) is a fixed constant, and the minimized problem in (19) becomes:  
\[\mathbf{\Phi}_{LESM}=\arg\min_{\mathbf{\Phi}\in\mathbb{R}^{M\times N}}\{tr[\mathbf{H}\mathbf{H }]-2 tr[\mathbf{H}\mathbf{W}]=\epsilon(\mathbf{H},\mathbf{W})\}. \tag{23}\]  
Assume that the eigen decomposition of \(\mathbf{H}\in\Re^{N\times N}\) is \(\mathbf{H}=\mathbf{V} \begin{bmatrix}\Lambda_{h}^{2}&0\\ \mathbf{0}&\mathbf{0}\end{bmatrix} \mathbf{V}^{T}\) with \(\Lambda_{h}^{2}=diag(\lambda_{1}^{2},\cdots,\lambda_{M}^{2})\). The diagonal elements of \(\Lambda_{h}^{2}\) are arranged in decreasing order as \(\lambda_{1}^{2}\geq\cdots\geq\lambda_{M}^{2}\). The orthogonal matrix \(\mathbf{V}\), diagonal matrix \(\mathbf{\Lambda}_{h}\) can be solved via minimizing the problem of (23) as the following objective function:  
\[ \begin{split}\epsilon(\mathbf{H},\mathbf{W})&=tr[\mathbf{V}\begin{bmatrix} \Lambda_{h}^{4}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\mathbf{V}^{T}]-2 tr[\mathbf{V}\begin{bmatrix}\Lambda_{h}^{2}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{bmatrix}\mathbf{V}^{T}\mathbf{W}]\\ &=\sum_{j=1}^{M}[\lambda_{j}^{4}-2\lambda_{j}^{2}\mathbf{W}(j,j)],\end{split}  \tag{24}\]where \(\tilde{\mathbf{W}}=\mathbf{V}^{T}\mathbf{W}\mathbf{V}\) with \(\tilde{\mathbf{W}}\in\Re^{N\times N}\) and supposing its eigenvalues become \(\sigma_{1},\cdots,\sigma_{N}\) with \(M<N\). The solutions of \(\mathbf{\Phi}_{LESM}\) can be discussed in three cases. (1) If the eigenvalues \(\sigma_{1},\cdots,\sigma_{M}>0\) in \(\tilde{\mathbf{W}}\), the problem in (23) becomes:  
\[\epsilon(\mathbf{H},\mathbf{W})=\sum_{j=1}^{M}[\lambda_{j}^{2}-\tilde{\mathbf{W}}(j,j)]^{2 }-\sum_{j=1}^{M}\tilde{\mathbf{W}}^{2}(j,j). \tag{25}\]  
The Lemma in [28] proves that if \(\tilde{\mathbf{W}}=\{\tilde{\mathbf{W}}(k,l)\}\) is Hermitian with dimension \(N\) and its eigenvalues are supposed to be \(\{\sigma_{j}\}_{j=1}^{N}\) with decreasing order \(|\sigma_{j+1}|\geq|\sigma_{j}|\), then \(\Sigma_{j=1}^{N-m}|\tilde{\mathbf{W}}(j,j)|^{2}\leq\Sigma_{j=1}^{N-m}|\sigma_{j}|^ {2},\ \ \forall m\leq N-1\).