# Design of Robust Sensing Matrix for UAV Images Encryption and Compression
## 4 Experiment  
The UAV images for compression and recovery using the proposed sensing matrix are experimented on in this section. As there is no ground-truth dictionary for image data, the dictionary \(\mathbf{\Psi}\in\Re^{N\times L}\) is learned by applying KSVD algorithm [34] from a set of 6000 training samples, which extract 15 patches (size: \(8\times 8\)) from arbitrary 400 images randomly in the \"UAV123\" training datasets [40]. Each patch of \(8\times 8\) is re-arranged as a vector of \(64\times 1\) to achieve the training samples \(\mathbf{X}\in\Re^{N\times Q}\) with \(N=64,Q=6000\). The parameters in \(\mathbf{\Psi}_{opt}\) are \(N=64\), \(L=100\). Then, the sparse representation error \(\mathbf{E}\) is achieved from the sparse representation \(\{\mathbf{\Psi}_{opt},\mathbf{A}^{[1]}\}\) in the above training samples. With \(C-1\) iterations, groups of sparse representation are extracted from the SRE, and hence the LSRE \(\Delta\mathbf{E}\) is obtained. In our experiments, five sensing matrices are compared: Gaussian random sensing matrix \(CS_{RAN}\), three designed sensing matrices \(CS_{LZY}\), \(CS_{LLL}\), \(CS_{HZ}\) corresponding to the reference [24; 28; 30], and the proposed sensing matrix \(CS_{LESM}\) is optimized with (18) using Algorithm 1. The test images \(\mathbf{X}_{t}\) from the \"UAV123\" test datasets [40] are extracted in \(8\times 8\) patches and re-arranged to \(N\times q\) with \(N=64\). With above-mentioned sensing matrices, the test images from \"UAV123\" are compressed and encrypted to measurements of a low dimension. Then, the images are recovered by adopting the Orthogonal Matching Pursuit (OMP) algorithm. Three evaluation indexes are adopted in this experiment. They are _Peak Signal-to-Noise Ratio_ (PSNR), _sparse representation error_ (SRE), and _Structure Similarity Index Measure_ (SSIM) [41]. PSNR represents the ratio of maximum power and noise power. The higher the PSNR is, the smaller the noise effect is. SRE measures the difference between the sparse representation of the recovered image and the original image. SSIM measures the similarity between the original image and the recovery image, which is more in line with the intuitive feeling of human eyes. The definition of PSNR can be expressed as follows:  
\[PSNR=10\times\log_{10}(\frac{(2^{r}-1)^{2}}{MSE}), \tag{31}\]  
with \(r=8\) bits per pixel. The _Mean Square Error_ (MSE) [26] is defined as:  
\[MSE=\frac{\|\mathbf{X}_{t}-\widehat{\mathbf{X}}_{t}\|_{F}^{2}}{N\times q}, \tag{32}\]  
where \(\widehat{\mathbf{X}}_{t}\) is the recovery image and \(\mathbf{X}_{t}\) is the original image, \(N\times q\) is the size of the image. The definition of SRE can be expressed as:  
\[SRE=\frac{\|\mathbf{X}_{t}-\mathbf{\Psi}_{opt}\widehat{\mathbf{A}}_{t}\|_{F}^{2}}{N\times q }, \tag{33}\]  
where \(\mathbf{\Psi}_{opt}\widehat{\mathbf{A}}_{t}\) is the sparse representation of \(\widehat{\mathbf{X}}_{t}\). \(\mathbf{X}_{t}\) is the original image, and \(N\times q\) is the size of the image. With the parameter setting and evaluation indexes, some experiments are carried out in the following to verify the performance of the proposed sensing matrix and its applications in the UAV images to resist the large SRE. _Case 1: Performance evaluation of the sensing matrix._ The behavior of these five sensing matrices is listed in Table 1 in terms of cost function, measurement error, averaged mutual coherence, and mutual coherence. **Remark 1**. :
* The cost function, measurement error and averaged mutual coherence are in the low range through the sensing matrix design. In particular, the measurement error of the proposed \(CS_{LESM}\) algorithm is the smallest, which means the projected error is rare in the measurement. * The mutual coherence for the real images is usually large, which has the same theoretical explanation of why we use the averaged mutual coherence as the criterion to design the sensing matrix.