Coarse‐to‐Fine Normal Filtering for Feature‐Preserving Mesh Denoising Based on Isotropic Subneighborhoods

State‐of‐theart normal filters usually denoise each face normal using its entire anisotropic neighborhood. However, enforcing these filters indiscriminately on the anisotropic neighborhood will lead to feature blurring, especially in challenging regions with shallow features. We develop a novel mesh denoising framework which can effectively preserve features with various sizes. Our idea is inspired by the observation that the underlying surface of a noisy mesh is piecewise smooth. In this regard, it is more desirable that we denoise each face normal within its piecewise smooth region (we call such a region as an isotropic subneighborhood) instead of using the anisotropic neighborhood. To achieve this, we first classify mesh faces into several types using a face normal tensor voting and then perform a normal filter to obtain a denoised coarse normal field. Based on the results of normal classification and the denoised coarse normal field, we segment the anisotropic neighborhood of every feature face into a number of isotropic subneighborhoods via local spectral clustering. Thus face normal filtering can be performed again on the isotropic subneighborhoods and produce a more accurate normal field. Extensive tests on various models demonstrate that our method can achieve better performance than state‐of‐theart normal filters, especially in challenging regions with features.     

Mean shift denoising of point-sampled surfaces

This paper presents an anisotropic denoising/smoothing algorithm for point-sampled surfaces. Motivated by the impressive results of mean shift filtering on image denoising, we extend the concept to 3D surface smoothing by taking the vertex normal and the curvature as the range component and the vertex position as the spatial component. Then the local mode of each vertex on point-based surfaces is computed by a 3D mean shift procedure dependent on local neighborhoods that are adaptively obtained by a kdtree data structure. Clustering pieces of point-based surfaces of similar local mode provides a meaningful model segmentation. Based on the adaptively clustered neighbors, we finally apply a trilateral point filtering scheme that adjusts the position of sample points along their normal directions to successfully reduce noise from point-sampled surfaces while preserving geometric features.     

具有特殊效果的混合细分方法

提出了基于三角形和四边形的混合控制网格的细分曲面尖锐特征、半尖锐特征生成和控制方法,避免了已有方法仅局限于初始控制网格为单一的三角形或单一的四边形网格的缺陷.通过局部修改混合细分规则,在光滑混合曲面上产生了刺、尖、折痕、角的尖锐特征效果,并对尖锐特征处局部细分矩阵进行了详细的特征分析,讨论了极限曲面的收敛性及光滑性.同时,用特征处的离散曲率来控制特征处的尖锐程度,实现了半尖锐的特征效果,并通过自适应细分方法,把尖锐特征、半尖锐特征的生成统一起来.该方法具有多分辨率表示能力强、局部性好、简单易操作的特点.实验结果表明,该算法效果好,成功地解决了混合曲面特殊效果生成问题.     

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Active B-Spline Curve developed by Pottmann et al. We have also developed a theoretically demonstrated approach, analysing curvature properties of B-Splines, which computes a near optimal evaluation of the initial number and positions of control points. Moreover, our original Active Footpoint Parameterization method prevents wrong matching problems occurring particularly for self-intersecting curves. Thus, the stability of the algorithm is highly increased. Our method was tested on different sets of curves and gives satisfying results regarding to approximation error, convergence speed and compression rate. This method is in line with a larger 3D CAD object compression scheme by piecewise subdivision surface approximation. The objective is to fit a subdivision surface on a target patch by first fitting its boundary with a subdivision curve whose control polygon will represent the boundary of the surface control polyhedron.     

Surface mesh denoising with normal tensor framework

In this paper, we propose a novel method for feature-preserving mesh denoising based on the normal tensor framework. We utilize the normal tensor voting directly for the mesh denoising whose eigenvalues and eigenvectors are used for detecting saliency, and introduce an algorithm that updates a vertex by the Laplacian of curvature which minimizes a difference of the curvature in one neighborhood. By connecting the feature saliency with a distance metric in the normal tensor space, our algorithm preserves sharp features more robustly and clearly for noisy mesh data. Comparing our method with the existing ones, we demonstrate the effectiveness of our algorithm against some synthetic noisy data and real-world scanned data.     

保特征的联合滤波网格去噪算法

目的 在去噪的过程中保持网格模型的特征结构是网格去噪领域研究的热点问题。为了能够在去噪中保持模型特征,本文提出一种基于变分形状近似（VSA）分割算法的保特征网格去噪算法。方法 引入变分形状近似分割算法分析并提取噪声网格模型的几何特征,分3步进行去噪。第1步使用变分形状近似算法对网格进行分割,对模型进行分块降噪预处理。第2步通过分析变分形状近似算法提取分割边界中的特征信息,将网格划分为特征区域与非特征区域。对两个区域用不同的滤波器联合滤波面法向量。第3步根据滤波后的面法向量,使用非迭代的网格顶点更新方法更新顶点位置。结果 相较于现有全局去噪方法,本文方法可以很好地保持网格模型的特征,引入的降噪预处理对于非均匀网格的拓扑结构保持有着很好的效果。通过对含有不同程度高斯噪声的网格模型进行实验表明,本文算法无论在直观上还是定量分析的结果都相较于对比的方法有着更好的去噪效果,实验中与对比算法相比去噪效果提升15%。结论 与现有的网格去噪算法对比,实验结果表明本文算法在中等高斯噪声下更加鲁棒,对常见模型有着比较好的去噪效果,能更好地处理不均匀采样的网格模型,恢复模型原有的特征信息和拓扑结构。     

A new feature-preserving mesh-smoothing algorithm

We present a novel mesh denoising and smoothing method in this paper. Our approach starts by estimating the principal curvatures and mesh saliency value for each vertex. Then, we calculate the uniform principal curvature of each vertex based on the weighted average of local principal curvatures. After that, we use the weighted bi-quadratic Bézier surface to fit the neighborhood of each vertex using the least-square method and obtain the new vertex position by adjusting the parameters of the fitting surface. Experiments show that our smoothing method preserves the geometric feature of the original mesh model efficiently. Our approach also prevents the volume shrinkage of the input mesh and obtains smooth boundaries for non-closed mesh models.     

Creating MPU implicit surfaces from unoriented point sets with orientation inference

In this paper, we extend the MPU implicits algorithm to deal with unoriented point sets while preserving its desirable properties, such as sharp feature preservation. An orientation inference algorithm is introduced to orientate the local implicit patches by solving a graph labeling problem through energy minimization. Sign consistency between local functions is exploited to infer the globally consistent orientation. To precisely model the features, we employ the affinity propagation clustering algorithm to identify the local surface patches composing the features by considering orientation consistency between data points. Sharp features can then be accurately reconstructed by performing piecewise smooth surface fitting. Experimental results are shown to demonstrate the performance of the proposed algorithm.     

Sharp feature preserving MLS surface reconstruction based on local feature line approximations

Sharp features in manufactured and designed objects require particular attention when reconstructing surfaces from unorganized scan point sets using moving least squares (MLS) fitting. It is an inherent property of MLS fitting that sharp features are smoothed out. Instead of searching for appropriate new fitting functions our approach computes a modified local point neighborhood so that a standard MLS fitting can be applied enhanced by sharp features reconstruction.We present a two-stage algorithm. In a pre-processing step sharp feature points are marked first. This algorithm is robust to noise since it is based on Gauss map clustering. In the main phase, the selected feature points are used to locally approximate the feature curve and to segment and enhance the local point neighborhood. The MLS projection thus leads to a piecewise smooth surface preserving all sharp features. The method is simple to implement and able to preserve line-type features as well as corner-type features during reconstruction.     

Robust mesh smoothing

This paper proposes a vertex-estimation-based, feature-preserving smoothing technique for meshes. A robust mesh smoothing operator called mean value coordinates flow is introduced to modify mean curvature flow and make it more stable. Also the paper proposes a three-pass vertex estimation based on bilateral filtering of local neighbors which is transferred from image processing settings and a Quasi-Laplacian operation, derived from the standard Laplacian operator, is performed to increase the smoothness order of the mesh rapidly whilst denoising meshes efficiently, preventing volume shrinkage as well as preserving sharp features of the mesh. Compared with previous algorithms, the result shows it is simple, efficient and robust.     

基于差异性累积与子空间传播的法向估计算法

以分割为基础的法向估计算法主要是通过法向的差异来构造点之间的相似性.针对由于距离属性的缺失,使这类算法对于紧邻面及一些光滑曲面的估计结果并不理想的问题,提出基于差异性累积与子空间传播的法向估计算法,利用最短路将法向的差异性和点的位置信息相融合.首先,对于部分点的邻域,找到邻域点间的最短路,通过叠加最短路中点的法向差异,计算点之间的相似性;然后,利用谱分割对邻域进行分割,选择一子邻域估计此点的法向;最后,为了提高效率,提出法向约束的子空间结构传播算法,其余邻域的分割结果由已有的分割结果进行推断.在Fandisk等仿真数据和Armadillon等真实扫描数据上的实验结果表明,文中算法能准确地恢复模型的尖锐特征,有效地克服噪声及非均匀采样.     

Fast and effective feature-preserving mesh denoising

We present a simple and fast mesh denoising method, which can remove noise effectively, while preserving mesh features such as sharp edges and corners. The method consists of two stages. Firstly, noisy face normals are filtered iteratively by weighted averaging of neighboring face normals. Secondly, vertex positions are iteratively updated to agree with the denoised face normals. The weight function used during normal filtering is much simpler than that used in previous similar approaches, being simply a trimmed quadratic. This makes the algorithm both fast and simple to implement. Vertex position updating is based on the integration of surface normals using a least-squares error criterion. Like previous algorithms, we solve the least-squares problem by gradient descent, but whereas previous methods needed user input to determine the iteration step size, we determine it automatically. In addition, we prove the convergence of the vertex position updating approach. Analysis and experiments show the advantages of our proposed method over various earlier surface denoising methods.     

Normal Vector Voting: Crease Detection and Curvature Estimation on Large,Noisy Meshes

This paper describes a robust method for crease detection and curvature estimation on large, noisy triangle meshes. We assume that these meshes are approximations of piecewise-smooth surfaces derived from range or medical imaging systems and thus may exhibit measurement or even registration noise. The proposed algorithm, which we call normal vector voting, uses an ensemble of triangles in the geodesic neighborhood of a vertex—instead of its simple umbrella neighborhood—to estimate the orientation and curvature of the original surface at that point. With the orientation information, we designate a vertex as either lying on a smooth surface, following a crease discontinuity, or having no preferred orientation. For vertices on a smooth surface, the curvature estimation yields both principal curvatures and principal directions while for vertices on a discontinuity we estimate only the curvature along the crease. The last case for no preferred orientation occurs when three or more surfaces meet to form a corner or when surface noise is too large and sampling density is insufficient to determine orientation accurately. To demonstrate the capabilities of the method, we present results for both synthetic and real data and compare these results to the G. Taubin (1995, in Proceedings of the Fifth International Conference on Computer Vision, pp. 902–907) algorithm. Additionally, we show practical results for several large mesh data sets that are the motivation for this algorithm.     

Feature-preserving mesh denoising based on vertices classification

In this paper, we present an effective surface denoising method for noisy surfaces. The two key steps in this method involve feature vertex classification and an iterative, two-step denoising method depending on two feature weighting functions. The classification for feature vertices is based on volume integral invariant. With the super nature of this integral invariant, the features of vertices can be fixed with less influence of noise, and different denoising degrees can be applied to different parts of the pending surface. Compared with other methods, our approach produces better results in feature-preserving.     

视觉特征保持的网格模型简化算法

针对二次误差测度算法存在几何特征消失等缺陷,提出了基于顶点视觉特 征度的新的网格模型简化算法。该算法采用半边折叠,通过引入顶点视觉特征度来优化了二 次误差测度,从而改变边折叠的顺序,使模型中的突出视觉特征更多的被保留下来。视觉特 征度通过顶点平均曲率熵来定义,它反映了顶点中心区域的视觉变化情况。实验表明,该算 法高效、可靠、能很好保持模型的视觉特征。     

Piecewise smooth reconstruction of normal vector field on digital data

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 ].     

Feature Preserving Mesh Generation from 3D Point Clouds

We address the problem of generating quality surface triangle meshes from 3D point clouds sampled on piecewise smooth surfaces. Using a feature detection process based on the covariance matrices of Voronoi cells, we first extract from the point cloud a set of sharp features. Our algorithm also runs on the input point cloud a reconstruction process, such as Poisson reconstruction, providing an implicit surface. A feature preserving variant of a Delaunay refinement process is then used to generate a mesh approximating the implicit surface and containing a faithful representation of the extracted sharp edges. Such a mesh provides an enhanced trade‐off between accuracy and mesh complexity. The whole process is robust to noise and made versatile through a small set of parameters which govern the mesh sizing, approximation error and shape of the elements. We demonstrate the effectiveness of our method on a variety of models including laser scanned datasets ranging from indoor to outdoor scenes.     

非平均化自适应Catmull-Clark细分算法

提出一种基于网格边的光滑度计算来进行Catmull-Clark自适应细分的新算法。该方法能够在满足显示需求的前提下较好地减小细分曲面过程中的网格生成数,同时解决了由于采用网格顶点曲率计算,来实现自适应细分方法中平均化生成顶点曲率带来的不足。通过对比试验,算法能更好地区别当前细分网格中光滑与非光滑区域,增加对非光滑区域网格加密密度,并且该算法能够普遍适用于较复杂的细分模式中,具有一定的推广价值。     

Surface interpolation of meshes by geometric subdivision

Subdivision surfaces are generated by repeated approximation or interpolation from initial control meshes. In this paper, two new non-linear subdivision schemes, face based subdivision scheme and normal based subdivision scheme, are introduced for surface interpolation of triangular meshes. With a given coarse mesh more and more details will be added to the surface when the triangles have been split and refined. Because every intermediate mesh is a piecewise linear approximation to the final surface, the first type of subdivision scheme computes each new vertex as the solution to a least square fitting problem of selected old vertices and their neighboring triangles. Consequently, sharp features as well as smooth regions are generated automatically. For the second type of subdivision, the displacement for every new vertex is computed as a combination of normals at old vertices. By computing the vertex normals adaptively, the limit surface is G¹ smooth. The fairness of the interpolating surface can be improved further by using the neighboring faces. Because the new vertices by either of these two schemes depend on the local geometry, but not the vertex valences, the interpolating surface inherits the shape of the initial control mesh more fairly and naturally. Several examples are also presented to show the efficiency of the new algorithms.