# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 1 Introduction
Based on this effective design criteria, some algorithms [23; 24; 25] are proposed to optimize the sensing matrix by minimizing the following cost function when the dictionary is given:  
\[\min_{\mathbf{\Phi},\mathbf{G}_{t}}\|\mathbf{G}_{t}-\mathbf{G}\|_{\mathrm{F}}^{2}\;\;s.t.\; \mathbf{G}=\mathbf{\Psi}^{T}\mathbf{\Phi}^{T}\mathbf{\Phi}\mathbf{\Psi}, \tag{5}\]  
where \(\|\cdot\|_{F}\) denotes the Frobenius-norm. \(\mathbf{G}\in\Re^{L\times L}\) is the Gram of the equivalent dictionary (denoted as \(\mathbf{G}=\mathbf{D}^{T}\mathbf{D}\)). \(\mathbf{G}_{t}\in\Re^{L\times L}\) is defined as a target Gram. The physical meaning is that the off-diagonal elements in \(\mathbf{G}\) represent the coherence between any atoms in equivalent dictionary, and approaching each value of the off-diagonal elements to the certain value of the target Gram, which results in the minimization of the averaged mutual coherence. Hence, minimizing \(\mu_{\varpi}(\mathbf{D})\) is the goal that we design the sensing matrix. The selection of target gram will influence the performance of data recovery. The tight frame (TF)-based matrix [26; 27] can provide better performance for the absolute sparse data. Besides, the references in [23; 24; 25] also elaborate that selecting the identity matrix as the target Gram can help recover image data better than the tight frame (TF)-based matrix ones. It is because the image data is not absolutely sparse that exists the sparse representation error. Therefore, the sparse representation for image data can be expressed as:  
\[\mathbf{X}=\mathbf{\Psi}\mathbf{A}+\mathbf{E}, \tag{6}\]  
in which \(\mathbf{X}\in\Re^{N\times Q}\) is the given image data, and \(\mathbf{A}\in\Re^{L\times Q}\) is its corresponding sparse matrix in dictionary \(\mathbf{\Psi}\in\Re^{N\times L}\). \(\mathbf{E}\in\Re^{N\times Q}\) is defined as the sparse representation error (SRE). Since the SRE can not be ignored in the image data, some research [28; 29; 30; 31] has focused on how the SRE impact the model of sensing matrix design. The general model can be expressed as:  
\[\min_{\mathbf{\Phi}}\|\mathbf{I}_{L}-\mathbf{G}\|_{F}^{2}+\lambda\|\mathbf{\Phi}\mathbf{E}\|_{F}^{ 2}, \tag{7}\]  
where \(\mathbf{I}_{L}\) is the target Gram \(\mathbf{G}_{t}\) that denotes the identity matrix with dimension \(L\). \(\mathbf{G}\in\Re^{L\times L}\) is the Gram of the equivalent dictionary (denoted as \(\mathbf{G}=\mathbf{D}^{T}\mathbf{D}\)). The idea in [28] indicates that the averaged mutual coherence should be minimized by the first term of the above cost function. As for the second term, the energy for SRE projected by the sensing matrix should also be minimized. For further analysis, an inappropriate sensing matrix will magnify the SRE itself, which leads to a significant error in the measurement, which will bring big challenges to the recovery accuracy of the image. Therefore, it is essential to consider the influence of SRE when the sensing matrix is designed. Observing Equation (6), the SRE is strongly associated with sparse representation. By considering the sparse representation error (SRE) with a new model, we are able to eliminate parts of the error, and hence achieve a lower sparse representation error (LSRE). A cost function is then designed for the proposed model, in which a robust sensing matrix can be obtained by simultaneously reducing the energy of projected LSRE and averaged mutual coherence. The above-designed sensing matrix can address the images with large SRE. For example, the UAV images contain large SRE due to the disturbance of the harsh environment or the instability of its flight [32; 33]. Hence, the new CS framework with the proposed sensing matrix to cope with the scenario of UAV images is dominant, which can achieve better recovery performance. The main contributions are summarized as follows:  
* A new structure where the **SRE** is reduced by eliminating groups of sparse representation is proposed, which is denoted as **NSSRE**.