In this paper, we introduce a Mumford-Shah framework to restore the face normal field on the triangulated surface. To effectively discretize $\Gamma$-convergence approximation of the Mumford-Shah model, we first define an edge function space and its associated differential operators. They are helpful for directly diffusing the discontinuity function over mesh edges instead of computing the approximated discontinuity function via pointwise diffusion in existing discretizations. Then, by using the operators in the proposed function space, two Mumford-Shah-based denoising methods are presented, which can produce denoised results with neat geometric features and locate geometric discontinuities exactly. Our Mumford-Shah framework overcomes the limitations of existing techniques that blur the discontinuity function, be less able to preserve geometric features, be sensitive to surface sampling, and require a postprocessing to form feature curves from located discontinuity vertices. Intensive experimental results on a variety of surfaces show the superiority of our denoising methods qualitatively and quantitatively.

在本文中，我们引入了Mumford-Shah框架来还原三角表面上的面法线场。有效离散化$\Gamma$-Mumford-Shah模型的收敛逼近，我们首先定义边缘函数空间及其关联的微分算子。它们有助于直接在网格边缘上扩散不连续函数，而不是通过现有离散化中的逐点扩散来计算近似的不连续函数。然后，通过在提出的函数空间中使用算子，提出了两种基于Mumford-Shah的去噪方法，它们可以产生具有整齐的几何特征的去噪结果，并精确地定位几何不连续点。我们的Mumford-Shah框架克服了现有技术的局限性，这些技术模糊了不连续性函数，难以保留几何特征，对表面采样敏感，并且需要后处理才能从定位的不连续性顶点形成特征曲线。