Mesh Denoising via a Novel Mumford–Shah Framework

In this paper, we introduce a Mumford–Shah framework to restore the face normal field on the triangulated surface. To effectively discretize Γ-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. [Display omitted] •Two coupled function spaces and associated operators are given out over meshes, which can describe the edge function space and its operators for directly diffusing the function over edges.•Two Mumford–Shah-based models are formulated in the proposed function spaces, which are more able to produce high quality denoised results with neat features and at the same time locate discontinuities accurately.•Two efficient algorithms based on alternating minimization are presented to solve the proposed Mumford–Shah regularizations.