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Decoupling local geometric features from the spatial location of a mesh is crucial for feature-preserving mesh denoising. This paper focuses on first order features, i.e., facet normals, and presents a simple yet effective anisotropic mesh denoising framework via normal field denoising. Unlike previous denoising methods based on normal filtering, which process normals defined on the Gauss sphere, our method considers normals as a surface signal defined over the original mesh. This allows the design of a novel bilateral normal filter that depends on both spatial distance and signal distance. Our bilateral filter is a more natural extension of the elegant bilateral filter for image denoising than those used in previous bilateral mesh denoising methods. Besides applying this bilateral normal filter in a local, iterative scheme, as common in most of previous works, we present for the first time a global, noniterative scheme for an isotropic denoising. We show that the former scheme is faster and more effective for denoising extremely noisy meshes while the latter scheme is more robust to irregular surface sampling. We demonstrate that both our feature-preserving schemes generally produce visually and numerically better denoising results than previous methods, especially at challenging regions with sharp features or irregular sampling.