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.
Feature-preserving signal smoothing can also be achieved via optimization. Notable examples include image smoothing algorithms that induce sparsity of image gradients via @math -norm @cite_4 or @math -norm @cite_25 regularization. These approaches were later adapted for mesh smoothing and denoising @cite_28 @cite_1 @cite_2 . Although effective in many cases, their optimization formulation only regularizes the signal difference between immediately neighboring faces. In comparison, our optimization compares signals within a neighborhood with user-specified size, which provides more flexibility and achieves better preservation of large-scale features.
@cite_4 @cite_28 @cite_1 @cite_2 @cite_25
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In this paper, we introduce a feature-preserving denoising algorithm. It is built on the premise that the underlying surface of a noisy mesh is piecewise smooth, and a sharp feature lies on the intersection of multiple smooth surface regions. A vertex close to a sharp feature is likely to have a neighborhood that includes distinct smooth segments. By defining the consistent subneighborhood as the segment whose geometry and normal orientation most consistent with those of the vertex, we can completely remove the influence from neighbors lying on other segments during denoising. Our method identifies piecewise smooth subneighborhoods using a robust density-based clustering algorithm based on shared nearest neighbors. In our method, we obtain an initial estimate of vertex normals and curvature tensors by robustly fitting a local quadric model. An anisotropic filter based on optimal estimation theory is further applied to smooth the normal field and the curvature tensor field. This is followed by second-order bilateral filtering, which better preserves curvature details and alleviates volume shrinkage during denoising. The support of these filters is defined by the consistent subneighborhood of a vertex. We have applied this algorithm to both generic and CAD models, and sharp features, such as edges and corners, are very well preserved. We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edge-preserving regularizers-one convex, the other nonconvex-are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as the choice of error norm, angular sampling rate and amount of noise, affect the reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results. Image denoising is a central problem in image processing and it is often a necessary step prior to higher level analysis such as segmentation, reconstruction, or super-resolution. The nonlocal means (NL-means) perform denoising by exploiting the natural redundancy of patterns inside an image; they perform a weighted average of pixels whose neighborhoods (patches) are close to each other. This reduces significantly the noise while preserving most of the image content. While it performs well on flat areas and textures, it suffers from two opposite drawbacks: it might over-smooth low-contrasted areas or leave a residual noise around edges and singular structures. Denoising can also be performed by total variation minimization-the Rudin, Osher and Fatemi model-which leads to restore regular images, but it is prone to over-smooth textures, staircasing effects, and contrast losses. We introduce in this paper a variational approach that corrects the over-smoothing and reduces the residual noise of the NL-means by adaptively regularizing nonlocal methods with the total variation. The proposed regularized NL-means algorithm combines these methods and reduces both of their respective defaults by minimizing an adaptive total variation with a nonlocal data fidelity term. Besides, this model adapts to different noise statistics and a fast solution can be obtained in the general case of the exponential family. We develop this model for image denoising and we adapt it to video denoising with 3D patches. In many signal processing applications, the aim is to reconstruct a signal that has a simple representation with respect to a certain basis or frame. Fundamental elements of the basis known as “atoms” allow us to define “atomic norms” that can be used to formulate convex regularizations for the reconstruction problem. Efficient algorithms are available to solve these formulations in certain special cases, but an approach that works well for general atomic norms, both in terms of speed and reconstruction accuracy, remains to be found. This paper describes an optimization algorithm called CoGEnT that produces solutions with succinct atomic representations for reconstruction problems, generally formulated with atomic-norm constraints. CoGEnT combines a greedy selection scheme based on the conditional gradient approach with a backward (or “truncation”) step that exploits the quadratic nature of the objective to reduce the basis size. We establish convergence properties and validate the algorithm via extensive numerical experiments on a suite of signal processing applications. Our algorithm and analysis also allow for inexact forward steps and for occasional enhancements of the current representation to be performed. CoGEnT can outperform the basic conditional gradient method, and indeed many methods that are tailored to specific applications, when the enhancement and truncation steps are defined appropriately. We also introduce several novel applications that are enabled by the atomic-norm framework, including tensor completion, moment problems in signal processing, and graph deconvolution.