## 4 Experimental Results and Analysis
### Two-Dimensional Image  
In this part, we evaluate the performance of various measurement matrices for image reconstruction under compressed sensing. As shown in Figure 5, the tested images included six commonly used test images, two medical images, and two aerial images. The size of all images was 223*223. We utilized block-based compressive sensing to recover the tested images. As compressed sensing relies on sparsity, we achieved image sparsity by retaining a certain proportion of coefficients under the discrete Fourier transform. In this paper, this proportion was set to 30%. In the block-based compressive sensing, the size of image sub-blocks was set as \(223\times 1\). This means that each column of the original image was treated as a sub-block. Each sub-block was compressed individually in the measurement process. In the recovery process, each image block was separately reconstructed using OMP and then assembled to reconstruct the complete image. Figure 5: Test images: (**a**) [PERSON], (**b**) [PERSON], (**c**) [PERSON], (**d**) [PERSON], (**e**) [PERSON], (**f**) [PERSON], (**g**) Bone1, (**h**) Bone2, (**i**) Aerial image1, and (**j**) Aerial image2. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & **Measurement** & **This Paper** & **Gaussian** & **Random** & **Random** & **Bipartite** \\  & **Number** & **This Paper** & **Random** & **Symmetric** & **Sparse** & **Graph** \\ \hline \multirow{3}{*}{Gaussian random sparse signals} & 220 & 7.77 & 7.62 & 7.66 & 7.69 & 7.76 \\  & 280 & 17.80 & 17.59 & 17.75 & 17.77 & 17.77 \\  & 340 & 20.03 & 19.79 & 19.99 & 19.98 & 20.00 \\  & 400 & 21.40 & 21.16 & 21.37 & 21.38 & 21.38 \\ \hline \multirow{3}{*}{Uniform random sparse signals} & 220 & \(-0.90\) & \(-0.91\) & \(-0.92\) & \(-0.92\) & \(-0.91\) \\  & 280 & 1.62 & 1.47 & 1.47 & 1.47 & 1.49 \\ \cline{1-1}  & 340 & 5.16 & 4.91 & 4.97 & 5.01 & 5.01 \\ \cline{1-1}  & 400 & 10.28 & 10.01 & 10.13 & 10.12 & 10.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Recovery SNRs of various measurement matrices in noisy case with \(k=100\) and \(N=431\). The peak signal-to-noise ratio (PSNR) was used to assess the quality of the recovered image. For two images of \(m\times n\), \(\mathbf{G}\), and \(\mathbf{\hat{G}}\), the PSNR was calculated using the following expression:  
\[\text{PSNR}=10\text{log}_{10}\bigg{(}\frac{223^{2}}{\text{MSE}}\bigg{)} \tag{14}\]  
where the mean square error (MSE) is defined as  
\[\text{MSE}=\big{\|}\mathbf{G}-\mathbf{\hat{G}}\big{\|}_{2}^{2}/(m\times n) \tag{15}\]  
Table 4 presents the PSNR value for various images under different measurement matrices with the measurement number of 120. Our proposed measurement matrix achieved the highest peak signal-to-noise ratio (PSNR) among the compared measurement matrices. In addition to the PSNR, we employed the structural similarity measure (SSIM) as a complementary evaluation metric. As shown in Table 5, the results indicate that with a measurement number of 120, our matrix achieved the highest SSIM value compared to the other measurement matrices. This indicates that our proposed measurement matrix outperformed other measurement matrices in preserving the structural details. Figures 6-8 individually show the visually reconstructed results of the images \"Lena\", \"Bone2\", and \"Aerial Image2.\" The details of the areas within the red boxes are shown in Figures 9-11, respectively. For the Lena and Aerial Image2 images, the block effects in the reconstructed images using our proposed measurement matrix were significantly reduced, resulting in smoother images. In the Bone2 image, our matrix effectively minimized artifacts that were present with other matrices, demonstrating its superior performance in preserving intricate textures and details.