## 1 Introduction  
Commercial hybrid PET-MRI imaging systems, first introduced in 2010 [1, 2], have rapidly advanced the field of clinical imaging by synergistically combining the high soft-tissue contrast of MRI with the metabolic and functional imaging capabilities of PET. This multimodal integration leverages the complementary strengths of each modality, facilitating improved diagnostic accuracy through correlated rather than redundant information. Historically, PET has been combined with CT to enhance oncological imaging, particularly using \({}^{18}\)_F-FDG_ to quantify metabolic activity of malignant cells [3]. While CT provides exquisite anatomical detail and can detect even subtle lesions missed by PET due to its limited resolution or motion-related artifacts (e.g., respiration, patient movement), MRI offers superior soft tissue contrast without ionizing radiation, making it preferable for neuro-oncology, craniofacial abnormalities, and abdominal imaging. Despite these advances, fundamental challenges remain in the seamless integration of MRI and PET data acquisition [1]. PET and MRI fundamentally differ in their signal generation and detection mechanisms; for example, strong magnetic fields essential for MRI can disrupt PET photomultiplier tube operation. Consequently, current architectures often rely on sequential scanning or physically separate scanners connected by a transport table, resulting in longer examination times and potential patient discomfort. True simultaneous MR-PET imaging could overcome these limitations but necessitates novel image reconstruction methodologies capable of dealing with sparsely sampled, noisy, and motion-corrupted data streams. Sparse sampling addresses the limitations of prolonged acquisition times and patient discomfort by strategically reducing the number of measurements while preserving essential information through intelligent priors and reconstruction algorithms. This enables accelerated imaging without a significant loss in diagnostic fidelity. In recent years, super-resolution reconstruction (SRR), Refs. [4; 5; 6; 7] and compressed sensing (CS) methods [8; 9] have emerged as powerful strategies to accelerate data acquisition while enhancing image quality. Deep learning, particularly convolutional neural networks (CNNs) [10], has revolutionized single-image super-resolution (SISR), modeling complex mappings from low-resolution to high-resolution images, as demonstrated in natural image processing [11; 12]. However, translating these methods to medical imaging presents unique challenges: volumetric 3D data, high memory and computational demands, and the need to preserve clinically relevant anatomical details. Conventional pixel-wise loss functions such as mean squared error (MSE) often fail to capture perceptual quality, leading to overly smooth reconstructions [13]. Several recent developments in multimodal image fusion have contributed foundational insights that guided the present work. Notably, [PERSON] et al. proposed an extensive benchmark for evaluating the fusion of overexposed infrared and visible images [14]. Their dataset and baseline serve as an important reference for evaluating perceptual fidelity in multi-source fusion tasks. Furthermore, [PERSON] et al. introduced STFNet, a self-supervised transformer-based model tailored for infrared and visible image fusion [15]. Their approach leverages attention mechanisms to enhance feature correlation across modalities, a concept which parallels the perceptual alignment pursued in the GAN architecture proposed herein. Both studies emphasize the critical role of cross-modality consistency, a theme directly addressed through the joint sparsity priors and perceptual loss design in the proposed reconstruction framework. By building upon the methodological foundations established in these works, this study contributes a novel integration of structural sparsity and generative modeling for high-fidelity MR-PET image synthesis.