## 7 Registration of MR Scans
### MR-PET Blur Removal Network  
The restoration of sharp MR-PET images from blurred observations constitutes a critical step in the proposed joint super-resolution framework. The presence of blur, caused by motion, system point spread functions (PSF), and limited resolution, disrupts the joint sparsity structure exploited in the structured Hankel representations [27; 29; 31]. Therefore, a dedicated blur removal network has been incorporated to recover high-fidelity images suitable for joint reconstruction, see Figure 4.  
The image degradation process is modeled as follows [49; 50]:  
\[I_{B}=I_{S}*k+n, \tag{3}\]  
where \(I_{B}\) denotes the observed blurred image, \(I_{S}\) represents the unknown sharp image, \(k\) is the unknown blur kernel, \(*\) denotes convolution, and \(n\) is additive noise. The task is to recover \(I_{S}\) from \(I_{B}\) without explicit knowledge of \(k\), i.e., a blind deblurring problem [49; 50]. In this equation, the image degradation process is modeled as a convolutional blur followed by additive noise:  
* \(I_{B}\) is the observed blurred image obtained from the imaging system. * \(I_{S}\) is the latent sharp image we aim to recover. * \(k\) is the unknown blur kernel (also referred to as the point spread function, PSF), which models the spread or distortion introduced by the imaging process, motion, or system imperfections. Figure 4: GAN-basedblur removal and denoising pipeline. * \(*\) represents the 2D or 3D convolution operator, applying the kernel \(k\) across the entire image domain. * \(n\) is additive noise, typically assumed to be Gaussian or signal-dependent, which further corrupts the observations. The goal of the deblurring task is to estimate \(I_{S}\) given only \(I_{B}\), without explicit prior knowledge of the blur kernel \(k\), making this a blind deblurring problem [49; 50], see Figure 4. Such problems are inherently ill-posed, as multiple combinations of \(I_{S}\) and \(k\) could produce the same \(I_{B}\). To address this, the proposed framework employs deep learning-based techniques to learn a direct mapping from \(I_{B}\) to an approximation of \(I_{S}\), implicitly learning to handle both the unknown blur and noise during training. A convolutional neural network \(G_{\theta}\), referred to as the generator, has been trained to perform this restoration:  
\[\hat{I}_{S}=G_{\theta}(I_{B}). \tag{4}\]  
In this formulation,  
* \(G_{\theta}\) represents the generator network, a convolutional neural network (CNN) parameterized by weights \(\theta\). * \(I_{B}\) is the input blurred image. * \(\hat{I}_{S}\) is the restored (deblurred) image produced by the generator. The network \(G_{\theta}\) is trained to approximate the unknown inverse mapping of the degradation process, learning to produce sharp images \(\hat{I}_{S}\) from their blurred and noisy counterparts \(I_{B}\). During training, \(G_{\theta}\) learns to implicitly handle both the effects of the unknown blur kernel \(k\) and the additive noise \(n\), enabling effective end-to-end blind deblurring. In order to promote perceptual realism and maintain consistency with the true data distribution, a discriminator network \(D_{\phi}\) has been employed in an adversarial training scheme [17; 18; 19]. The discriminator aims to distinguish between restored images \(\hat{I}_{S}\) and real sharp images. The overall training objective of the blur removal network is expressed as follows:  
\[L=\lambda_{adv}L_{adv}+\lambda_{perc}L_{perc}+\lambda_{pix}L_{pix}, \tag{5}\]  
where \(\lambda_{adv}\), \(\lambda_{perc}\), and \(\lambda_{pix}\) are weighting coefficients. In this objective function,  
* \(L\) is the total loss optimized during training of the blur removal network. * \(\lambda_{adv}\), \(\lambda_{perc}\), and \(\lambda_{pix}\) are positive hyperparameters (typically determined empirically) that balance the relative contributions of the corresponding loss terms.