Bridging Synthetic Priors and Variational Optimization for Robust 3D Medical Reconstruction

Deep learning models often struggle to generalize when trained on limited real-world datasets, particularly in high-stakes domains like medical imaging where "ground truth" is expensive or impossible to acquire. Conversely, classical variational optimization methods offer mathematical guarantees but suffer from slow convergence and sensitivity to parameter tuning. This paper proposes a hybrid framework that unifies these disparate approaches. We introduce a methodology for generating photorealistic synthetic anatomical datasets to pre-train a lightweight Convolutional Neural Network (CNN). The network's output is then treated not as a final prediction, but as a "warm start" initialization for a Preconditioned Alternating Direction Method of Multipliers (PADMM) solver. This approach enforces structural fidelity and physical consistency (e.g., tissue continuity) that pure deep learning often misses. Our comprehensive mathematical formulation and extensive experimental validation demonstrate that the proposed method reduces the required iterations of the ADMM solver by over 80% while improving reconstruction accuracy by 23.7% on standard benchmarks compared to end-to-end deep learning baselines. The framework establishes a new paradigm for combining data-driven learning with mathematically constrained optimization for ill-posed inverse problems.