# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 3 Proposed LESM Sensing Matrix Design
### The NSSRE Structure on SRE  
Traditionally, the training samples \(\mathbf{X}\) can be represented sparsely with the sparse representation \(\mathbf{\
abla}\mathbf{A}\). However, the SRE in the image data can not be ignored. According to our limited knowledge, little research focuses on the modeling of SRE. For instance, the authors in [30] estimate the SRE as a particular matrix from a statistical point of view. Exploring SRE is promising and will improve the performance of the sensing matrix. In our view, for the image data, the SRE still contains some information that can be used for image representation, and this part of the information should not be treated as a term of projected error in the model of sensing matrix design. Hence, we should focus on the true error projected rather than the SRE. In addition, excessive errors are compressed into the measurements, which will bring a significant challenge to the recovery accuracy. In conclusion, it is better to eliminate the valid information from the sparse representation error \(\mathbf{E}\) to achieve the Lower SRE. Therefore, we construct a new structure to reduce SRE by eliminating groups of spares representation. The new structure can be expressed as follow:  
\[\mathbf{X}=\mathbf{\
abla}\mathbf{A}^{[1]}+\mathbf{\
abla}\Delta\mathbf{S}+\Delta\mathbf{E}. \tag{15}\]  
The image \(\mathbf{X}\) can be described in three parts. The first term \(\mathbf{\
abla}\mathbf{A}^{[1]}\) is the sparse representation learning from the training samples by dictionary learning methods [34; 35; 27; 36] and sparse decomposition algorithms [37; 16; 38]. As for the second term \(\mathbf{\
abla}\Delta\mathbf{S}\), it is defined as groups of sparse representation, which are extracted from sparse representation error \(\mathbf{E}\). We eliminate the valid information by calculating the groups of sparse representation, which means decomposing \(\mathbf{E}\) into sparse representation and residual \(\mathbf{E}\) by using the OMP algorithm iteratively. With several iterations, the residual \(\mathbf{E}\) is decomposed to achieve groups of sparse representation \(\mathbf{\
abla}\mathbf{A}^{[2]},\mathbf{\
abla}\mathbf{A}^{[3]},\cdots,\mathbf{\
abla}\mathbf{A}^{[C]}\). The groups of sparse representation can be calculated iteratively by the following mathematical model:  
\[ \begin{split}&\min_{\mathbf{A}^{[j+1]}}\lVert(\mathbf{X}-\sum_{j=1}^{J} \mathbf{\
abla}\mathbf{A}^{[j]})-\mathbf{\
abla}\mathbf{A}^{[J+1]})\rVert_{F}^{2}\\ & s.t.\;\;\lVert\mathbf{A}^{[J+1]}(;i)\rVert_{0}\leq K,J=1,\cdots,C-1,\;i=1,\cdots,Q.\end{split}  \tag{16}\]  
where \(\mathbf{X}-\sum_{j=1}^{J}\mathbf{\
abla}\mathbf{A}^{[j]}\) represents the residual \(\mathbf{E}\) after \(J\) iterations. With the given dictionary \(\mathbf{\
abla}\), the sparse matrices \(\mathbf{A}^{[J+1]}\) with \(J=1,\cdots,C-1\) are calculated iteratively along with reducing the SRE in every multi-stage. The iterative algorithm can avoid the operation on large matrices and reduce the computational complexity without affecting the perfor  
Figure 1: The CS framework of the proposed sensing matrix design (the red line) and the procedure for the UAV images compression and recovery (the blue line). mance of error removal from SRE. The groups of sparse matrices \(\Delta S\) can be written as: \(\Delta S=\sum_{l=1}^{C-1}\mathbf{A}^{[l+1]}\), and then the groups of sparse representation are \(\mathbf{\
abla}\Delta S\). After the groups of sparse representation are set, the original SRE is reduced to LSRE corresponding to the third term \(\Delta\mathbf{E}\). According to the above description, the SRE is split into the sparse part and dense part, and its expression is:  
\[\mathbf{E}=\mathbf{\
abla}\Delta\mathbf{S}+\Delta\mathbf{E}, \tag{17}\]  
in which \(\mathbf{\
abla}\Delta\mathbf{S}\) belongs to the sparse part with groups of sparse representation and \(\Delta\mathbf{E}\) is the dense part with true error.