Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Field Published Mathematics. Laplacian smoothing is attractive for its simplicity but it does require an existing triangulation.

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A method of triangular surface mesh smoothing is presented to improve angle quality by extending the original optimal Delaunay triangulation ODT to surface meshes. The mesh quality is improved by solving a quadratic optimization problem that minimizes the approximated interpolation error between a parabolic function and its piecewise linear interpolation defined on the mesh.

A suboptimal problem is derived to guarantee a unique, analytic solution that is significantly faster with little loss in accuracy as compared to the optimal one. In addition to the quality-improving capability, the proposed method has been adapted to remove noise while faithfully preserving sharp features such as edges and corners of a mesh.

Numerous experiments are included to demonstrate the performance of the method. Triangular surface meshes are widely used in computer graphics, industrial design and scientific computing. In computer graphics and design, people are typically interested in the smoothness low variation in curvature and sharp features edges, corners, etc. In many applications of scientific computing, however, the quality of a mesh is a key factor that significantly affects the numerical result of finite or boundary element analysis.

One of the most common criteria for mesh quality is the uniformity of angles, although this may not be the best in some cases where anisotropic meshes are desired [ 1 ]. For its popularity, however, we shall adopt the angle-based criterion in the present work. The main interest and contribution of the present work is to improve the quality of triangular surface meshes. Additionally our method will be extended to be able to remove noise and preserve sharp features on surface meshes.

For simplicity, we refer to both mesh quality improvement and mesh denoising as mesh smoothing unless otherwise specified. Mesh denoising has a long history in computer graphics and the related methods include three main categories: 1 geometric flows [ 2 - 6 ], 2 spectral analysis [ 7 , 8 ], and 3 optimization methods [ 9 , 10 ].

Due to its simplicity and low computational cost, Laplacian smoothing has established itself as one of the most common methods among all the geometric flow-based methods. In this method, every node is updated towards the barycenter of the neighborhood of the node. However, volume shrinkage often occurs during this process. The shrinkage problem may be tackled by methods utilizing spectral analysis of the mesh signal, which is the main idea of the second category.

Optimization-based methods guarantee the smoothness of the mesh by minimizing different types of energy functions. But the iterative process searching for optimal solutions can be time-consuming.

A variety of techniques on mesh quality improvement have been developed [ 11 , 12 ]. Two or more of the above techniques are sometimes combined to achieve better performance. For instance, Dyer et al. Among these methods, Laplacian smoothing in its simplest form that moves a vertex to the center or barycenter of the surrounding vertices [ 19 ] is one of the fastest methods but it may fail in improving mesh quality and is often equipped with other techniques such as optimizations [ 24 , 25 ].

Ohtake et al. Nealen et al. Both methods, as shown in [ 28 ], cannot warrant mesh quality or feature-preservation. Wang et al. Among all the repositioning-based methods for mesh quality improvement, the optimal Delaunay triangulation ODT [ 29 , 1 , 30 ] has been proved to be effective on 2D triangular meshes.

However, the extension from 2D meshes to 3D surface meshes is non-trivial in both mathematical analysis and algorithm design. For 3D surface meshes we need to consider not only angle quality but also mesh noise that causes bumpiness on surfaces, which was not taken into account in the original ODT method or its variants in tetrahedral mesh smoothing [ 31 , 32 ]. In addition, sharp surface features must be well preserved during the processes of mesh denoising and quality improvement.

There have been extensive studies on feature-preserving surface mesh processing [ 33 - 38 ]. However, most of the previous work was focused on the mesh denoising problem but only a few dealt with both mesh denoising and quality improvement with feature preservation [ 28 ]. The main goal of the present paper is to generalize the 2D ODT idea to 2-manifold surface meshes by formulating the mesh quality improvement as an optimization problem that minimizes the interpolation error between a parabolic function and its piecewise linear interpolation at each vertex of the surface mesh.

Unfortunately there is no analytical solution to this optimization problem. To solve the minimization problem faster, we consider a suboptimal problem by simplifying the objective function into a quadratic formula such that an analytical solution can be derived. The proposed suboptimal Delaunay triangulation or S-ODT is then extended to include two other capabilities: removing mesh noise as well as preserving sharp features on the original meshes.

The remainder of this paper is organized as follows. In Section 2, we extend the original ODT method [ 29 , 1 ] to improve the angle quality of a surface mesh. Several variants of the new algorithm are also introduced to warrant additional desirable properties such as noise removal and feature preservation. Numerous mesh examples are included and comparisons are given in Section 3 to demonstrate the performance of the proposed algorithms, followed by our conclusions in Section 4.

Some mathematical details of the algorithms are provided in the Appendices. Like many other mesh smoothing approaches, our method is iterative and vertex-based, meaning that all mesh vertices are repositioned in each iteration and the process is repeated until the mesh quality meets some predefined criteria or a maximum number of iterations is reached.

In this section we shall describe three algorithms with the basic one addressing the mesh quality improvement using the proposed sub-optimization formulation and two extended algorithms dealing additionally with the issues of feature preservation and noise removal.

For completeness, we shall begin with a brief introduction to Delaunay triangulation and the original ODT method [ 29 ]. In computational geometry, Delaunay triangulation DT is a well known scheme to triangulate a finite set of fixed points P , satisfying the so-called empty sphere condition.

That is, no point in P can be inside the circumsphere of any simplex e. Consider, for example, the four points p 0, p 1, p 2 and p 3 in Fig. There are obviously two ways to triangulate this point set, but only the one in Fig.

If we lift the point set onto a parabolic function x 2 , any triangulation on the lifting points q 0, q 1, q 2 and q 3 will result in a unique piecewise linear interpolation of the parabolic function. The one that minimizes the interpolation error can be projected back to the original point set and makes the Delaunay triangulation.

From this example, we can see that Delaunay triangulation of a fixed point set is equivalent to minimizing the following interpolation error, which can be achieved by swapping edges:. Illustration of minimizing interpolation error in two ways: edge swapping a and b and vertex-repositioning c and d. The mesh quality in b is improved by swapping the edges but keeping the vertices fixed. The interpolation error can also be reduced hence mesh quality is improved by moving the vertex p 0 in c to a new position in d , where the edge connections are kept unchanged.

Although Delaunay triangulation is optimal for a fixed set of points, it does not necessarily produce a high quality mesh if the given points are not nicely distributed. In addition to edge-swapping, there is actually another way, called vertex-repositioning, to minimize the error between a parabolic function and its piecewise linear interpolation.

Consider for example the point set in Fig. The triangulation is already optimal in terms of the DT criterion. However, the interpolation error can be further reduced by moving the vertex p 0 to a better position as shown in Fig. This strategy constitutes the core of the optimal Delaunay triangulation ODT method as detailed in [ 1 , 29 , 30 ]. It is worth noting that the vertex-repositioning alone does not produce a Delaunay-like triangulation.

For better mesh quality improvement, it is always wise to combine edge-swapping into vertex-repositioning, as in the original ODT method [ 29 ]. In the rest of the current paper, we shall extend the ODT method to surface meshes to improve the angle quality. However, we will not consider the edge-swapping technique in the descriptions of our algorithms as well as results, simply because our main focus in the current paper is how vertices are repositioned to achieve quality improvement and two other goals noise removal and feature preservation.

Note that f I x is always no less than f x so that we can remove the absolute-value operation in the first equation of 2. Thus 2 becomes the following equation see A for details :. The minimizer of 3 in general does not admit a closed-form expression. Although numerical methods may be used for solving 3 , it can be computationally inefficient, as will be demonstrated in Section 3.

Therefore, Laplacian smoothing is just a special case of 3. Please note that at this moment, we assume that the original mesh is smooth enough and noise-free, such that the tangent plane is well defined as above. For meshes with sharp features or noise, special care must be taken to calculate tangent planes or feature lines as will be discussed in the subsequent subsections. In these cases, the volume preservation is not guaranteed. We shall see in Section 3, especially Fig. Performance comparison between the analytical solution to the suboptimal problem the proposed S-ODT method: Algorithm 1 and the numerical solution to the optimal problem Eq.

While little difference is observed between the two smoothed meshes, the computational time is only about 36 s by using the analytical method for 20 iterations, in contrast to 1 min and 56 s by using the L-BFGS method for five iterations. This property makes the sum of all cubic terms in 5 a constant and thus minimizing 5 becomes an unconstrained quadratic optimization problem such that an analytic solution can be obtained.

The basic S-ODT algorithm for surface mesh quality improvement by minimizing 5 is summarized in Algorithm 1. Algorithm 1 performs well for surface meshes without sharp features such as creases or corners.

In reality, however, sharp features are commonly seen and crucial in precisely representing geometric features of a mesh. To this end, we classify the surface nodes into three categories: 1 smooth nodes with low curvature in the neighborhood, 2 crease nodes with low curvature in one direction and high curvature in another typically perpendicular to the first direction , and 3 corner nodes, where at least three creases intersect.

We define crease and corner nodes as feature nodes and impose some special restrictions on them during the mesh smoothing process. Specifically, a crease node moves only along the direction of the crease and a corner node remains unchanged. Motivated by [ 33 , 22 ], we distinguish between smooth and feature nodes by using the local structure tensor T at x 0 as defined below:.

Note that T is a semi-positive definite symmetric matrix and has three real eigenvalues. We decompose T using the eigen-analysis method and decide the type of x 0 based on the distribution of the eigenvalues of T.

In the mesh smoothing process, Algorithm 1 is still applicable when x 0 is a smooth node. When x 0 is a corner node, we just keep it unchanged. When x 0 is a crease node, however, we move x 0 to the optimal position by solving 5 along the direction of the crease. First, we compute the corresponding coefficients in the following way:. The process is summarized in Algorithm 2. Our method can be readily adapted to remove mesh noise while improving mesh quality and still retaining the feature-preserving property.

In the basic S-ODT algorithm Algorithm 1 , the optimal position is assumed to be on the tangent plane at x 0 of the surface mesh. When there is noise on the surface mesh, a common strategy is to fit a plane or higher order polynomials to the neighboring nodes of each vertex and project the vertex onto the plane [ 28 ]. Sharp features may be preserved by considering anisotropic local neighborhoods [ 35 ]. In our current work, we utilize a weighted least squares fitting strategy as detailed below [ 39 ].

As described in Section 2. We always keep the corner nodes unchanged.

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## Laplacian smoothing and Delaunay triangulations

Dar Mesh Smoothing Algorithms for Complex Who is online Users browsing this forum: Laplacian smoothing [8] [9], this method is applied only when the mesh quality is Laplacian Smoothing and Delaunay Triangulations ; In this delainay the effect of Laplacian smoothing on Delaunay triangulations is explored. Our mesh smoothing schemes also work well in the anisotropic case. A new technique for improving triangulations ; 1nternatlonal journal for numerical methods in engineering, vol. It will become clear that Why Should we Care: Delaunay triangulation of planar regions and, in particular, on how one selects the Feature-Preserving Reconstruction of Singular Surfaces ; that the sampled object is a collection of smooth surface patches with boundaries that can meet or intersect. Original triangulation of point cloud — very dense.

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## LAPLACIAN SMOOTHING AND DELAUNAY TRIANGULATIONS PDF

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