# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 2 Preliminaries  
To ensure that the sparse coefficient is recovered successfully, the regulation of _mutual coherence_[19; 20; 21] is taken into account when the equivalent dictionary is given, that is:  
\[\mu(\mathbf{D})\triangleq\max_{1\leq i\
eq j\leq L}\frac{|(\mathbf{D}(:,i))^{T}\mathbf{D}( :,j)|}{\|\mathbf{D}(:,i)\|_{2}\|\mathbf{D}(:,j)\|_{2}}, \tag{8}\]  
where \(T\) denotes the transpose operator. Mutual coherence is defined to calculate the worst coherence between any two atoms of \(\mathbf{D}\). However, the relations between mutual coherence and the sparsity \(K\) will influence whether the decryption and recovery are successful or not. The author in [19] indicates that the sparse coefficient \(\mathbf{\alpha}\) can be recovered exactly via (4) as long as  
\[S<\frac{1}{2}\bigg{[}1+\frac{1}{\mu(\mathbf{D})}\bigg{]}. \tag{9}\]  
The Equation (9) inspires us to reduce the mutual coherence of \(\mathbf{D}\) by optimizing the sensing matrix when the dictionary is given, so that the sparsity \(K\) can achieve a larger value space. While mutual coherence provides the worst case for successful recovery, it does not emphasize the actual performance for precise recovery. Hence, the authors in [22] proposed the averaged mutual coherence to play better behavior. The definition of averaged mutual coherence can be written as:  
\[\mu_{av}(\mathbf{D})\triangleq\frac{\sum_{\forall(i,j)\in\mathcal{S}_{av}}|\tilde {\mathbf{G}}(i,j)|}{N_{gp}}, \tag{10}\]where \(\tilde{G}\) is the Gram matrix of the column-normalized equivalent dictionary \(\mathbf{D}\) and the off-diagonal elements \(\tilde{\mathbf{G}}(i,j)\) means the coherence of \(\mathbf{D}\). \(\mathcal{S}_{av}\) is a set that is defined as \(\mathcal{S}_{av}=\{(i,j):\bar{\mu}\leq|\tilde{G}(i,j)|<1,\forall i\
eq j\}\) with \(0\leq\bar{\mu}<1\). The low bound of \(\bar{\mu}\) is \(\sqrt{\frac{L-M}{M(L-1)}}\)[22]. \(N_{av}\) counts the number of elements in set \(\mathcal{S}_{av}\). Based on above regulations for sensing matrix design, the work in [24] proposed the cost function for optimizing sensing matrix, which can be expressed as:  
\[\mathbf{\Phi}_{LZY}=\arg\min_{\mathbf{\Phi}}\|\mathbf{I}_{L}-\mathbf{G}\|_{F}^{2}\;\;s.t.\; \mathbf{G}=\mathbf{\Psi}^{T}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{\Psi}. \tag{11}\]  
The above problem can be decomposed into two parts:  
\[\|\mathbf{I}_{L}-\mathbf{G}\|_{F}^{2}=\sum_{l=1}^{L}|1-\mathbf{G}(l,l)|^{2}+\sum_{i\
eq j }|\mathbf{G}(i,j)|^{2}. \tag{12}\]  
Observing the above equation, the physical meaning of the first term is to set the constraint that the columns of the dictionary approach to be normalized. The second term measures the average mutual coherence with high bound \(L(L-1)\). Hence, our optimization goal is to decrease the average mutual coherence by solving the problem of (11). Then, an analytical solution of the sensing matrix is obtained. Because the SRE cannot be ignored when the images are represented sparsely, the work in [28] proposed a more robust algorithm for the sensing matrix design. The cost function can be described as:  
\[\mathbf{\Phi}_{LLL}=\arg\min_{\mathbf{\Phi}}\|\mathbf{I}_{L}-\mathbf{G}\|_{F}^{2}+\gamma\|\mathbf{ \Phi}\mathbf{E}\|_{F}^{2}\;\;s.t.\;\mathbf{G}=\mathbf{\Psi}^{T}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{ \Psi}. \tag{13}\]  
The algorithm emphasizes that the SRE projected by the sensing matrix is prevented from being amplified. Hence, designing a proper sensing matrix can reduce the average mutual coherence and the energy of projected SRE simultaneously. As a result, the analytical solution of the sensing matrix is calculated in this work. Regarding to the SRE, the work in [30] indicates that the SRE calculated from a large number of samples will bring a huge computational burden to the optimization model. Hence, the authors propose that SRE can be identified as a certain matrix when the samples are large enough.