## 7 Registration of MR Scans
### MR-PET Blur Removal Network
* \(L_{adv}\) is the adversarial loss term, derived from a GAN framework (typically a WGAN-GP), which encourages the generator to produce visually realistic deblurred images that cannot be distinguished from real sharp images by the discriminator. * \(L_{perc}\) is the perceptual loss, computed as the \(\ell_{2}\) distance between feature activations of the generated and ground-truth sharp images, extracted from a pre-trained image classification network (commonly VGG-19). This promotes perceptual similarity beyond pixel-level agreement. * \(L_{pix}\) is the pixel-wise reconstruction loss, typically using the \(\ell_{1}\) or \(\ell_{2}\) norm between the generated and ground-truth images. It enforces direct fidelity in image intensity values. Together, these terms ensure that the generator learns to recover sharp, perceptually convincing images with minimal residual blur and noise, balancing adversarial realism with fidelity to the original ground-truth content. The adversarial loss \(L_{adv}\) follows a Wasserstein GAN (WGAN) formulation with gradient penalty [19], promoting stability:  
\[L_{adv}=\mathbb{E}_{I_{S}}[D_{\phi}(I_{S})]-\mathbb{E}_{I_{B}}[D_{\phi}(G_{\theta }(I_{B}))]+\lambda_{gp}\mathbb{E}_{I}[(\|\
abla_{I}D_{\phi}(\hat{I})\|_{2}-1)^{2 }], \tag{6}\]  
where \(\hat{I}\) denotes random interpolations between real and generated samples. In this adversarial loss formulation,  
* \(L_{adv}\) is the adversarial component of the total loss optimized during training of the generator \(G_{\theta}\) and discriminator \(D_{\phi}\). * \(D_{\phi}(I_{S})\) represents the discriminator's output when presented with a real sharp image \(I_{S}\). The goal is for the discriminator to assign high scores to real images. * \(D_{\phi}(G_{\theta}(I_{B}))\) is the discriminator's output on a generated (deblurred) image \(I_{S}=G_{\theta}(I_{B})\). The discriminator is trained to assign low scores to generated images, while the generator is trained to maximize these scores. * \(\lambda_{gp}\) is the weighting factor for the gradient penalty term. A typical value is \(\lambda_{gp}=10\). * The term \(\mathbb{E}_{I}\big{[}\big{(}\|\
abla_{I}D_{\phi}(\hat{I})\|_{2}-1\big{)}^{2} \big{]}\) is the gradient penalty introduced in the Wasserstein GAN with Gradient Penalty (WGAN-GP) framework [19]. It enforces the Lipschitz constraint required for the WGAN formulation, which stabilizes training and improves convergence. * \(\hat{I}\) denotes images interpolated linearly between pairs of real images \(I_{S}\) and generated images \(G_{\theta}(I_{B})\); these interpolations are used to compute the gradient penalty. This loss encourages the generator to produce deblurred images that are indistinguishable from real sharp images in the discriminator's feature space, promoting perceptual realism and high-fidelity restoration. The perceptual loss \(L_{perc}\) is defined as the feature-space distance between \(I_{S}\) and \(\hat{I}_{S}\), computed over pre-trained VGG-19 network features [13]:  
\[L_{perc}=\sum_{I}\|\phi_{I}(I_{S})-\phi_{I}(\hat{I}_{S})\|_{2}^{2}, \tag{7}\]  
where \(\phi_{I}(\cdot)\) represents feature activations at layer \(I\). In this formulation of the perceptual loss,  
* \(L_{perc}\) is the perceptual loss that measures the difference between the real sharp image \(I_{S}\) and the deblurred image \(\hat{I}_{S}\) in a deep feature space. * \(\phi_{I}(\cdot)\) denotes the feature activations extracted from the \(l\)-th layer of a pre-trained VGG-19 network [13], commonly used as a perceptual reference model. * The summation over \(l\) typically spans a set of selected VGG layers (such as conv1_2, conv2_, conv3_4, etc. ), which capture both low-level (edges, textures) and high-level (semantic content) image representations. * The \(\ell_{2}\) norm is applied to the difference between the corresponding feature maps of \(I_{S}\) and \(\hat{I}_{S}\), encouraging the generator to match the structure and appearance of the reference image beyond pixel-wise accuracy.