# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 1 Introduction  
Unmanned aerial vehicles (UAV) [1; 2; 3] mainly orient to low-altitude remote sensing, which has advantages in complex scenarios exploration. Compared to manual detection, UAV can reach the place where a ground search is difficult to approach by controlling the flight attitude [4; 5]. Hence, adopting a UAV to capture images can reduce costs and increase flexibility. Improving the transmission security and efficiency of captured images from UAV has become our concern. The compressed sensing (CS) technique [6; 7; 8] can compress and encrypt UAV images simultaneously to achieve a low-dimensional matrix. The low-dimensional matrix can be transmitted safely and efficiently due to its incomprehensible meaning. At the decoding stage, the low-dimensional matrix can be recovered with high accuracy by employing the CS technique. In our paper, how to design a robust compressed sensing framework to deal with the large SRE in the UAV images is the main focus of our research. CS technique [9; 10; 11; 12; 13] projects compressible data \(\mathbf{x}\in\Re^{N\times 1}\) into low dimensional measurements \(\mathbf{y}\in\Re^{M\times 1}\) by adopting the sensing matrix \(\mathbf{\Phi}\in\Re^{M\times N}\), which can be expressed as:  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{x}. \tag{1}\]  
Equation (1) describes the process for data compression and encryption by using the sensing matrix \(\mathbf{\Phi}\in\Re^{M\times N}\). As \(M\ll N\), it is an underdetermined problem that has infinite solutions. The constraint of sparseness can be added to the data \(\mathbf{x}\), and its form of sparse representation can be written as:  
\[\mathbf{x}=\sum_{l=1}^{L}\mathbf{a}(l)\mathbf{\
abla}(;l)=\mathbf{\
abla}\mathbf{\alpha}, \tag{2}\]  
in which the matrix \(\mathbf{\
abla}\in\Re^{N\times L}\) is named dictionary with its column atoms \(\{\mathbf{\
abla}(;l)\}_{l=1}^{L}\), and the \(\mathbf{\alpha}(l)\) is the \(l\)th element of the sparse coefficient \(\mathbf{\alpha}\in\Re^{L\times 1}\). Then, the vector \(\mathbf{x}\) is said\(K\)-sparse in \(\mathbf{\Psi}\) if the sparse coefficient \(\mathbf{\alpha}\) satisfying \(\|\mathbf{a}\|_{0}\leq K\). \(\|\cdot\|_{0}\) denotes the number of non-zero elements. Following the data \(\mathbf{x}\) in (1) being substituted with (2), the mathematical framework of CS is concluded as:  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{\Psi}_{\mathbf{\alpha}}\triangleq\mathbf{D}\mathbf{\alpha}, \tag{3}\]  
in which the matrix \(\mathbf{D}\in\Re^{M\times L}\) is the so-called equivalent dictionary. Under the sparsity condition on \(\mathbf{x}\), a unique mapping between the measurement \(\mathbf{y}\) and the original data \(\mathbf{x}\)[14; 15] is established. Hence, with the given sensing matrix \(\mathbf{\Phi}\) and dictionary \(\mathbf{\Psi}\), the process of decryption and recovery can be executed as:  
\[\widehat{\mathbf{\alpha}}=\arg\min_{\mathbf{\alpha}}\|\mathbf{\alpha}\|_{0}\;\;s.t.\;\bm {y}=\mathbf{D}\mathbf{\alpha}. \tag{4}\]  
the estimate \(\widehat{\mathbf{\alpha}}\) is achieved by solving the above problem. The greedy algorithms deal with the above optimization problem of \(\ell_{0}\)-norm constraint, such as Orthogonal Matching Pursuit (OMP) [16]. When the constraint in (4) becomes \(\ell_{1}\)-norm, it is solved by employing Basis Pursuit (BP) [17] and a Least Absolute Shrinkage and Selection Operator (LASSO) [18]. Then, the estimated data \(\widehat{\mathbf{x}}\) is recovered via calculating \(\widehat{\mathbf{x}}=\mathbf{\Psi}\widehat{\mathbf{\alpha}}\). In order to improve the recovery performance, some optimal regulations are proposed to obtain a better property for the equivalent dictionary, such as mutual coherence [19; 20; 21] and the averaged mutual coherence denoted as \(\mu_{\varpi}(\mathbf{D})\)[22]. While mutual coherence demonstrates the worst case for successful recovery, it does not emphasize the actual performance for the recovery precise. Hence, the author in [22] proposed the averaged mutual coherence to play better behavior.