# The Prediction Performance Analysis of the Lasso Model with Convex Non-Convex Sparse Regularization
## 4 Numerical Experiment
### MRI Reconstruction  
The utilization of high-dimensional sparse linear regression serves as the foundation for numerous signal and image processing techniques, including MRI reconstruction [43; 44; 45]. The MRI is a powerful medical imaging technique, featuring high soft tissue contrast and the advantage of being radiation-free. It is widely used in clinical diagnosis and scientific research fields. In this experimental, we employ GMC sparse regularization (15) for MRI reconstruction and compare its performance against \(\ell_{1}\) and \(\ell_{1/2}\) regularization. To make the statement clear, we define the design matrix \(X=R\times F\), where \(R\in\mathbb{R}^{n\times p}\) is the sampling template, and \(F\in\mathbb{R}^{p\times p}\) is the sparse Fourier operator. We tested different reconstruction models on three MRI images with variable density sampling and Cartesian sampling templates, respectively. For comparison, all images are set to \(256\times 256\) pixels, with grayscale values ranging from \(0\) to \(255\). The parameters were \(\lambda=0.001\), \(\rho=150\), the iteration step size \(\alpha\) was chosen \(\alpha=1\). For the GMC regularization, it is necessary to explicitly specify matrix \(B\) and set it using (16) with \(\omega=0.9\). In this experiment, we selected relative error (RE) and peak signal-to-noise ratio (PSNR) as evaluation metrics to quantitatively assess the quality and accuracy of reconstructed images. The RE is defined as  
\[\text{RE}=\frac{\|\hat{\beta}-\beta^{*}\|_{2}}{\|\hat{\beta}^{*}\|_{2}}, \tag{34}\]  
where \(\beta^{*}\) and \(\hat{\beta}\) are the original and reconstructed images, respectively. The PSNR is defined as  
\[\text{PSNR}=10\cdot\lg\frac{\text{MAX}^{2}}{\frac{1}{N}\|\hat{\beta}-\beta^{* }\|_{2}^{2}}, \tag{35}\]  
where \(N=256\times 256\), \(\text{MAX}=255\), and \(\hat{\beta}\), \(\beta^{*}\) are the reconstructed, and the original image, respectively. For these two types of quantitative evaluation criteria, the lower the RE value, the more superior the reconstruction effect, while for PSNR, an increase in numerical value indicates a more outstanding reconstruction ability. To further enhance the visual contrast effect, we calculated the difference between the reconstructed image and the original image. Furthermore, we magnified a small portion of the local image to show more details. The reconstruction results of three regularizations are shown in Figures 2-4. It can be found that the reconstructed images based on \(\ell_{1}\) regularization have problems such as blurred edges and residual shadows, and the \(\ell_{1/2}\) regularization reconstruction model also shows similar phenomena. In contrast, the reconstructed images based on GMC sparse regularization are closer to the original images and exhibit higher reconstruction quality. Additionally, we conducted a quantitative evaluation of the reconstruction results in Table 1. It is obvious that compared with \(\ell_{1}\) and \(\ell_{1/2}\) regularization, GMC sparse regularization has the best performance in MRI reconstruction and can obtain the lowest RE and highest PSNR. Figure 3: (**a**), Original image; (**b**), undersampling template with 30% sampling rate; (**c**–**e**), reconstructed images using \(\ell_{1}\) regularization, \(\ell_{1/2}\) regularization and GMC regularization, respectively; (**f**), difference between (**a**) and (**c**); (**g**), difference between (**a**) and (**d**); (**h**), difference between (**a**) and (**e**).