## 7 Registration of MR Scans
### MR-PET Images Denoising Procedure  
The presence of noise in MR and PET images poses significant challenges for accurate joint super-resolution reconstruction [51; 52; 53]. In particular, MR magnitude images are corrupted by Rician noise [54], characterized by a non-Gaussian distribution which introduces signal-dependent bias, especially in low signal-to-noise ratio (SNR) regions. The noisy observation \(I_{obs}\) can be modeled as follows:  
\[I_{obs}=\sqrt{(I_{clean}+n_{r})^{2}+n_{l}^{2}}, \tag{9}\]  
where \(I_{clean}\) is the underlying noise-free image, and \(n_{r},n_{i}\sim\mathcal{N}(0,\sigma^{2})\) represent independent Gaussian noise components in the real and imaginary channels, respectively. In this formulation,  
* \(I_{obs}\) denotes the observed MR image magnitude after noise contamination. * \(I_{clean}\) is the underlying noise-free image that we aim to recover through denoising. * \(n_{r}\) and \(n_{i}\) are independent Gaussian-distributed noise components with zero mean and variance \(\sigma^{2}\). * The real and imaginary parts of the complex MR signal are corrupted separately by \(n_{r}\) and \(n_{i}\). * The final observed image is computed as the magnitude of the complex signal, \(I_{obs}=|I_{clean}+n_{r}+jn_{i}|\), which mathematically leads to the above square root form. * Due to this formulation, the noise in \(I_{obs}\) follows a Rician distribution rather than a simple Gaussian, especially at low signal-to-noise ratios (SNR). * The Rician nature of the noise introduces signal-dependent bias and makes traditional Gaussian denoising methods sub-optimal, motivating the use of data-driven or specialized techniques. Traditional denoising approaches [51] often fail to capture the complex, signal-dependent nature of Rician noise, motivating the use of data-driven methods. A convolutional neural network \(G_{\theta}\) is employed as the denoiser, mapping noisy images \(I_{obs}\) to denoised estimates \(\hat{I}\)[52; 53]:  
\[\hat{I}=G_{\theta}(I_{obs}). \tag{10}\]  
In this expression,  
* \(I_{obs}\) is the noisy observed image, following the Rician noise model as previously defined. * \(G_{\theta}\) denotes the generator network (typically a convolutional neural network) parameterized by weights \(\theta\). * The generator is trained to map the noisy input \(I_{obs}\) to a denoised estimate \(\hat{I}\) that closely approximates the clean image \(I_{clean}\). * The goal of the network is to implicitly learn the nonlinear transformation required to remove both signal-dependent and signal-independent noise while preserving anatomical structures. * The denoised image \(\hat{I}\) serves as a high-quality input for subsequent processing stages, including joint sparsity modeling and super-resolution reconstruction. To encourage the generator to produce realistic, artifact-free images that closely match the true data distribution, a discriminator network \(D_{\phi}\) is adversarially trained alongside \(G_{\theta}\)[17; 18; 19]. The discriminator aims to differentiate between denoised outputs and clean images, forming the basis of a Wasserstein GAN (WGAN) framework with gradient penalty for stable training [19]. The composite loss function guiding the denoising training is formulated as follows:  
\[L=\delta_{1}L_{MSE}+\delta_{2}L_{perc}+\delta_{3}L_{adv}, \tag{11}\]  
where \(\delta_{1},\delta_{2},\delta_{3}\) are empirically determined weights balancing the different objectives. In practice, the weights \(\delta_{1}\), \(\delta_{2}\), and \(\delta_{3}\) are selected to balance pixel fidelity, perceptual quality, and adversarial realism. A commonly effective choice is as follows:  
* \(\delta_{1}=1\)--giving strong importance to the Mean Squared Error (MSE), ensuring that low-level pixel-wise accuracy is preserved, which is critical for quantitative medical imaging tasks. * \(\delta_{2}=0.01\)--giving a small but meaningful contribution from the perceptual loss. This encourages the network to capture important structural and anatomical patterns without introducing hallucinated details.