The joint bilateral filter, which enables feature-preserving signal smoothing according to the structural information from a guidance, has been applied for various tasks in geometry processing. Existing methods either rely on a static guidance that may be inconsistent with the input and lead to unsatisfactory results, or a dynamic guidance that is automatically updated but sensitive to noises and outliers. Inspired by recent advances in image filtering, we propose a new geometry filtering technique called static dynamic filter, which utilizes both static and dynamic guidances to achieve state-of-the-art results. The proposed filter is based on a nonlinear optimization that enforces smoothness of the signal within designated neighborhoods, while preserving signal variations that correspond to features of certain scales. We develop an efficient iterative solver for the problem, which unifies existing filters that are based on static or dynamic guidances. The filter can be applied to mesh face normals followed by vertex position update, to achieve scale-aware and feature-preserving filtering of mesh geometry. It also works well for other types of signals defined on mesh surfaces, such as texture colors. Extensive experimental results demonstrate the effectiveness of the proposed filter for various geometry processing applications such as mesh denoising, geometry feature enhancement, and texture color filtering.
@cite_22: 
@cite_27: We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.
@cite_31: Spectral mesh processing is an idea that was proposed at the beginning of the 1990s to port the "signal processing toolbox" to the setting of 3D mesh models. Recent advances in both computing power and numerical software make it possible to fully implement this vision. In the classical context of sound and image processing, Fourier analysis was a cornerstone in development of a wide spectrum of techniques, such as filtering and recognition, to name but a few. In this course, attendees learn how to transfer the underlying concepts to setting a mesh model, how to implement the "spectral mesh processing" toolbox, and how to use it for real applications, including filtering, shape matching, remeshing, segmentation, and parameterization, among others.
@cite_16: We propose a novel point signature based on the properties of the heat diffusion process on a shape. Our signature, called the Heat Kernel Signature (or HKS), is obtained by restricting the well-known heat kernel to the temporal domain. Remarkably we show that under certain mild assumptions, HKS captures all of the information contained in the heat kernel, and characterizes the shape up to isometry. This means that the restriction to the temporal domain, on the one hand, makes HKS much more concise and easily commensurable, while on the other hand, it preserves all of the information about the intrinsic geometry of the shape. In addition, HKS inherits many useful properties from the heat kernel, which means, in particular, that it is stable under perturbations of the shape. Our signature also provides a natural and efficiently computable multi-scale way to capture information about neighborhoods of a given point, which can be extremely useful in many applications. To demonstrate the practical relevance of our signature, we present several methods for non-rigid multi-scale matching based on the HKS and use it to detect repeated structure within the same shape and across a collection of shapes.
@cite_10: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
From a signal processing point of view, meshes can be seen as a combination of signals with multiple frequency bands, which also relates with the scale space analysis @cite_10 . Previous work separate geometry signals of different frequencies using eigenfunctions of the heat kernel @cite_16 or the Laplace operator @cite_27 @cite_31 . Although developed with sound theoretical foundations, such approaches are computationally expensive. Moreover, as specific geometric features can span across a wide range of frequencies, it is not easy to preserve or manipulate them with such approaches. The recent work from @cite_22 provides an efficient way to separate and edit geometric features of different scales, harnessing the scale-aware property of the rolling guidance filter. Our SD filter also supports scale-aware processing of geometry signals, with more robustness than RGNF thanks to the incorporation of both static and dynamic guidances.