Modern geometric generation methods rely heavily on deep learning methods that, while powerful, often lack interpretability and require extensive training data. This work introduces MeshCone, a convex optimization framework for mesh enhancement from partially deformed meshes that requires no training data. We formulate the problem as a second-order cone program where vertex positions are optimized to align with target geometry while enforcing smoothness through convex edge-length regularization. Our convex relaxation enables deterministic, interpretable solutions with proven convergence properties via the Splitting Conic Solver (SCS). We demonstrate robust performance across 56 diverse object categories from ShapeNet and ThreeDScans, achieving superior refinement quality compared to classical baselines while maintaining sub-second inference times. This work establishes a principled baseline demonstrating what convex optimization alone can achieve, providing mathematical guarantees and interpretability that complement data-driven approaches.

翻译：现代几何生成方法严重依赖深度学习方法，这些方法虽然强大，但通常缺乏可解释性且需要大量训练数据。本文提出MeshCone——一种无需训练数据的凸优化框架，用于从部分变形网格中进行网格增强。我们将该问题构建为二阶锥规划问题，通过优化顶点位置使其与目标几何对齐，同时通过凸边缘长度正则化强制平滑性。我们的凸松弛方法通过分裂锥规划求解器（SCS）实现了具有可证明收敛性质的确定性、可解释解。我们在ShapeNet和ThreeDScans的56个不同物体类别上展示了稳健的性能，相比经典基线方法获得了更优的细化质量，同时保持亚秒级推理时间。这项工作建立了一个原则性基线，展示了仅靠凸优化所能实现的成果，提供了数学保证和可解释性，从而与数据驱动方法形成互补。