This paper presents a method for implicit fairing of irregularly triangulated meshes to remove noise and uneven edges while preserving desirable geometric features. The approach uses implicit integration of the diffusion equation, which allows for efficient, stable, and large time-step computations. A scale-dependent Laplacian operator improves the diffusion process, and a robust curvature flow operator achieves smoothing of the shape itself. The method also includes automatic exact volume preservation and constraints on mesh vertex positions. The method is compared to previous operators and algorithms, and it is shown that the curvature and Laplacian operators have desirable mathematical properties that improve the appearance of the resulting surface. The paper also introduces a scale-dependent diffusion operator for smoothing while preserving the triangulation's structure and a curvature flow operator for smoothing without affecting the sampling rate. These methods are demonstrated on various examples, showing improved results over previous techniques. The paper discusses the use of implicit integration for diffusion, which avoids time-step limitations and allows for larger time steps. It also introduces a scale-dependent Laplacian operator that provides better filtering performance. The method is applied to meshes with irregular connectivity, and it is shown that the implicit integration approach is more efficient and stable than explicit methods. The paper also addresses the issue of mesh shrinkage by introducing an automatic anti-shrinking fairing technique that preserves the volume of the mesh. The paper further discusses the inadequacy of the umbrella operator for approximating the Laplacian on irregular meshes and proposes a more accurate scale-dependent approximation. It also introduces a curvature flow method for noise removal, which uses the mean curvature of the surface to smooth the mesh. The curvature flow method is shown to produce better results than diffusion-based methods, especially for preserving the shape of the mesh. The paper concludes that the proposed methods provide a comprehensive set of tools for mesh fairing, with the implicit fairing method offering efficiency, quality, and stability. The scale-dependent umbrella operator and curvature flow method are shown to be effective in preserving the shape and volume of the mesh while removing noise and uneven edges. The methods are applicable to a wide range of meshes and can be used for various types of fairing tasks.This paper presents a method for implicit fairing of irregularly triangulated meshes to remove noise and uneven edges while preserving desirable geometric features. The approach uses implicit integration of the diffusion equation, which allows for efficient, stable, and large time-step computations. A scale-dependent Laplacian operator improves the diffusion process, and a robust curvature flow operator achieves smoothing of the shape itself. The method also includes automatic exact volume preservation and constraints on mesh vertex positions. The method is compared to previous operators and algorithms, and it is shown that the curvature and Laplacian operators have desirable mathematical properties that improve the appearance of the resulting surface. The paper also introduces a scale-dependent diffusion operator for smoothing while preserving the triangulation's structure and a curvature flow operator for smoothing without affecting the sampling rate. These methods are demonstrated on various examples, showing improved results over previous techniques. The paper discusses the use of implicit integration for diffusion, which avoids time-step limitations and allows for larger time steps. It also introduces a scale-dependent Laplacian operator that provides better filtering performance. The method is applied to meshes with irregular connectivity, and it is shown that the implicit integration approach is more efficient and stable than explicit methods. The paper also addresses the issue of mesh shrinkage by introducing an automatic anti-shrinking fairing technique that preserves the volume of the mesh. The paper further discusses the inadequacy of the umbrella operator for approximating the Laplacian on irregular meshes and proposes a more accurate scale-dependent approximation. It also introduces a curvature flow method for noise removal, which uses the mean curvature of the surface to smooth the mesh. The curvature flow method is shown to produce better results than diffusion-based methods, especially for preserving the shape of the mesh. The paper concludes that the proposed methods provide a comprehensive set of tools for mesh fairing, with the implicit fairing method offering efficiency, quality, and stability. The scale-dependent umbrella operator and curvature flow method are shown to be effective in preserving the shape and volume of the mesh while removing noise and uneven edges. The methods are applicable to a wide range of meshes and can be used for various types of fairing tasks.