Triangulated Surface Denoising using High Order Regularization with Dynamic Weights

Mesh denoising is a fundamental problem in geometry processing. The main challenge is to preserve sharp features (such as edges and corners) and smooth regions (such as smoothly curved regions and fine details) while removing the noise. State‐of‐the‐art denoising methods still struggle with this issue. In this paper, we first propose a new variational model combining total variation and anisotropic Laplacian regularization to filter the normal vector field of the mesh. This model can preserve sharp features and simultaneously handle smooth regions well. Then, a new vertex updating scheme is presented to reconstruct the mesh according to the filtered face normals. It prevents the orientation ambiguity problem introduced by existing schemes. Experiments show that our denoising method outperforms all compared methods visually and quantitatively, especially for meshes consisting of both sharp features and smooth regions.

This paper describes a graph-spectral method for shape-from-shading. The algorithm takes as its input a field of initial surface normal estimates computed using the Lambertian irradiance cone and the Canny edge gradient. We illustrate how to refine this field of surface normals using a graph-spectral method. From the initial field of surface normals we make curvature estimates. The curvature estimates are in turn used to compute a transition probability matrix. From the theory of random walks on graphs, the leading eigenvector of this matrix is a curvature minimising path through the field of surface normals. We impose curvature consistency on the initially noisy field of surface normals by rotating them about their irradiance cones so that they follow the path defined by the leading eigenvector. This method is applied to a variety of real-world images and is shown to lead to improved surface reconstruction.

Establishing a consistent normal orientation for point clouds is a notoriously difficult problem in geometry processing, requiring attention to both local and global shape characteristics. The normal direction of a point is a function of the local surface neighborhood; yet, point clouds do not disclose the full underlying surface structure. Even assuming known geodesic proximity, calculating a consistent normal orientation requires the global context. In this work, we introduce a novel approach for establishing a globally consistent normal orientation for point clouds. Our solution separates the local and global components into two different sub-problems. In the local phase, we train a neural network to learn a coherent normal direction per patch ( i.e. , consistently oriented normals within a single patch). In the global phase, we propagate the orientation across all coherent patches using a dipole propagation. Our dipole propagation decides to orient each patch using the electric field defined by all previously orientated patches. This gives rise to a global propagation that is stable, as well as being robust to nearby surfaces, holes, sharp features and noise.

<p style='text-indent:20px;'>Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we propose a semismooth Newton based augmented Lagrangian method for solving this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency heavily depends on solving the corresponding coupled and nonlinear system together and simultaneously. With efficient primal-dual semismooth Newton methods for the challenging and highly coupled nonlinear subproblems involving total generalized variation, we develop a highly efficient and competitive augmented Lagrangian method compared with some fast first-order method. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods.</p>

In this paper we explore the use of complex numbers as means of representing angular statistics for surface normal data. Our aim is to use the representation to construct a statistical model that can be used to describe the variations in fields of surface normals. We focus on the problem of representing facial shape. The fields of surface normals used to train the model are furnished by range images. We compare the complex representation with one based on angles, and demonstrate the advantages of the new method. Once trained, we illustrate how the model can be fitted to brightness images by searching for the set of parameters that both satisfy Lambert’s law and minimize the integrability error.KeywordsZenith AngleAzimuth AngleRange ImageIntegrability ErrorStatistical Shape ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

A feature-preserving framework for point cloud denoising

Most existing image coding and communication systems aim to minimize the mean square error (MSE) of the pixels reconstructed at receivers. However, the quality metric MSE has long been criticized for not being consistent with the perception of human vision systems. This paper considers a gradient-based image SoftCast (G-Cast) scheme, based on the recent advancements in image quality assessment which indicate that gradient similarity is highly correlated with perceptual image quality. To reconstruct the image from the received noisy gradient data, we exploit the statistical characteristics of image gradients. Instead of using the very simple Laplacian distribution for image gradient as in the total variation (TV) model, we further exploit the non-local similarity of image patches. A non-local gradient sparsity regularization (NLGSR) method is developed and solved using augmented Lagrangian method. Experimental results show that the proposed scheme provides promising perceptual image quality, and the NLGSR reconstruction scheme outperforms the existing schemes remarkably.

Point cloud denoising is a fundamental and challenging problem in geometry processing. Existing methods typically involve direct denoising of noisy input or filtering raw normals followed by point position updates. Recognizing the crucial relationship between point cloud denoising and normal filtering, we re-examine this problem from a multitask perspective and propose an end-to-end network called PCDNF for joint normal filtering-based point cloud denoising. We introduce an auxiliary normal filtering task to enhance the network's ability to remove noise while preserving geometric features more accurately. Our network incorporates two novel modules. First, we design a shape-aware selector to improve noise removal performance by constructing latent tangent space representations for specific points, taking into account learned point and normal features as well as geometric priors. Second, we develop a feature refinement module to fuse point and normal features, capitalizing on the strengths of point features in describing geometric details and normal features in representing geometric structures, such as sharp edges and corners. This combination overcomes the limitations of each feature type and better recovers geometric information. Extensive evaluations, comparisons, and ablation studies demonstrate that the proposed method outperforms state-of-the-art approaches in both point cloud denoising and normal filtering.

Decomposing a complex shape into geometrically simple primitives is a fundamental problem in geometry processing. We are interested in a shape decomposition problem where the simple primitives sought are generalized cylinders , which are ubiquitous in both organic forms and man-made artifacts. We introduce a quantitative measure of cylindricity for a shape part and develop a cylindricity-driven optimization algorithm, with a global objective function, for generalized cylinder decomposition. As a measure of geometric simplicity and following the minimum description length principle, cylindricity is defined as the cost of representing a cylinder through skeletal and cross-section profile curves. Our decomposition algorithm progressively builds local to non-local cylinders, which form over-complete covers of the input shape. The over-completeness of the cylinder covers ensures a conservative buildup of the cylindrical parts, leaving the final decision on decomposition to global optimization. We solve the global optimization by finding an exact cover, which optimizes the global objective function. We demonstrate results of our optimal decomposition algorithm on numerous examples and compare with other alternatives.

Shape completion is an intriguing problem in geometry processing with applications in CAD and graphics. This paper defines a new type of 3D curves, which can be utilized for curve completion. It can be considered as the extension to three dimensions of the 2D Euler spiral. We prove several properties of these curves - properties that have been shown to be important for the appeal of curves. We illustrate their utility in two applications. The first is fixing curves detected by algorithms for edge detection on surfaces. The second is shape illustration in archaeology, where the user would like to draw curves that are missing due to the incompleteness of the input model.

Biharmonic deformation transfer with automatic key point selection

Estimating dense point-to-point correspondences between two isometric shapes represented as 3D point clouds is a fundamental problem in geometry processing, with applications in texture and motion transfer. However, this task becomes particularly challenging when the shapes undergo non-rigid transformations, as is often the case with approximately isometric point clouds. Most existing algorithms address this challenge by establishing correspondences between functions defined on the shapes, rather than directly between points, because function mappings admit a linear representation in the spectral domain. State-of-the-art methods compute this linear representation using the eigenfunctions of the Laplace-Beltrami Operator (LBO) along with a small set of initial corresponding functions between the shapes. However, for approximately isometric point clouds, two key issues arise: (1) the eigenfunctions of the LBO may become misaligned, and (2) the initial corresponding functions may include outliers, both of which degrade the quality of the resulting correspondences. In this work, we propose an efficient approach to align the spectra of the LBOs of the two shapes, enabling the eigenfunctions to remain compatible even for approximately isometric 3D point clouds. Additionally, we introduce a technique to make function correspondence estimation robust to outliers. We validate our approach by comparing it with state-of-the-art 3D shape-matching algorithms on benchmark datasets, demonstrating its effectiveness.

We propose a novel method to regularize a normal vector field defined on a digital surface (boundary of a set of voxels). When the digital surface is a digitization of a piecewise smooth manifold, our method localizes sharp features (edges) while regularizing the input normal vector field at the same time. It relies on the optimisation of a variant of the Ambrosio‐Tortorelli functional, originally defined for denoising and contour extraction in image processing [AT90]. We reformulate this functional to digital surface processing thanks to discrete calculus operators. Experiments show that the output normal field is very robust to digitization artifacts or noise, and also fairly independent of the sampling resolution. The method allows the user to choose independently the amount of smoothing and the length of the set of discontinuities. Sharp and vanishing features are correctly delineated even on extremely damaged data. Finally, our method can be used to enhance considerably the output of state‐of‐the‐art normal field estimators like Voronoi Covariance Measure [MOG11] or Randomized Hough Transform [BM12].

—GeoFEM (Iizuka et al., 1999) is a parallel finite element analysis system intended for multi-physics, multi-scale problems of solid earth field phenomena. Very large linear elastic problems have already been resolved by parallel computation with GeoFEM. The next stage is to examine large-scale nonlinear problems using GeoFEM. The analysis of large-scale contact problems for fault zones is particularly important in the development of models that simulate the occurrence and cycle of earthquakes.¶This paper proposes a parallel FEM using an iterative solver with the augmented Lagrange method for solving large-scale contact problems. Direct solvers are presently applied to contact problem analysis because the matrix for such problems is ill-conditioned. However, direct solvers are not suitable for large-scale matrices. The augmented Lagrange method can improve the matrix conditions. The present study evaluates a parallel FEM using an iterative solver with the augmented Lagrange method. Analysis of a contact problem with the augmented Lagrange method revealed that an optimal penalty parameter exists and that large-scale parallel contact analysis using the iterative solver with localized preconditioning is promising.