Accurate digital heart mesh models are crucial for cardiac electromechanical simulations, commonly derived from Cardiac Magnetic Resonance (CMR) or Computed Tomography (CT) imaging. CMR offers high tissue contrast but suffers from large inter-slice distances and breath-hold misalignments, while CT is isotropic but uses harmful radiation and has limited temporal resolution. To leverage the merits of both, our study introduces MorphiNet. This novel network enhances heart model reconstruction by learning heart anatomy from high-resolution CT images, enabling more accurate CMR reconstructions. MorphiNet decodes the anatomy as a modality-independent distance field, transforming a template mesh into patient-specific geometries. A multi-layer graph subdivision network refines these geometries while maintaining dense point correspondence. MorphiNet generates high-fidelity models with reduced training and inference times, rivalling top methods like nnU-Net and NISF and enabling detailed, controllable surface refinement for accurate simulations. The code is available at MorphiNet .