Mesh Smoothing by Adaptive and Anisotropic Gaussian Filter Applied to Mesh Normals

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1 Mesh Smoothing by Adaptive and Anisotropic Gaussian Filter Applied to Mesh Normals Yutaka Ohtake Alexander Belyaev Hans-Peter Seidel Computer Graphics Group, Max-Planck-Institut für Informatik Stuhlsatzenhausweg 85, Saarbrücken, Germany s: Figure 1: Left: original noisy mesh (Caltech angel model). Middle: Taubin s smoothing is applied; the number of iterations is chosen manually in order to achieve the best-looking result. Right: smoothing by the method developed in this paper; fine shape features are preserved. Abstract In this paper, we develop a fully automatic mesh filtering method that adaptively smoothes a noisy mesh and preserves sharp features and features consisting of only few triangle strips. In addition, it outperforms other conventional smoothing methods in terms of accuracy. 1 Introduction Mesh smoothing, one of the most important mesh processing operations, remains to be an active research area, see, for instance, [1, 2, 6, 7, 9, 13, 15, 22, 24, 31, 32, 33, 34, 38]. The vast majority of mesh smoothing methods depend on one or more user-specified parameters which have to be selected properly in order to achieve a desired smoothing effect. For example, for the iterative mesh filtering schemes one of such parameters is the number of smoothing iterations. Several recent works on image smoothing explore statistical approaches for an automatic selection of parameters used in image-smoothing methods. In [16] it was proposed to select an optimal stopping time for iterative image filtering according a decorrelation criterion applied to noisy and smoothed images. An adaptive Gaussian filtering scheme combining a scale-space approach [36] and the minimal description length principle (MDL) [28, 29] was developed in [8]. In [8] it was demonstrated that the scheme worked somewhat better than a standard nonlinear diffusion method [25, 26]. This paper is an attempt to adapt and extend the image smoothing strategy proposed in [8] to noisy dense triangle meshes obtained by scanning real-world objects. We develop an adaptive and anisotropic Gaussian filter acting on the mesh normals. Given a noisy mesh, we compute its best smoothing scale for each mesh normal, then we smooth the mesh normals, and finally we update the On a leave from University of Aizu, Aizu-Wakamatsu Japan VMV Erlangen, Germany, November 20 22, 2002

2 mesh vertex positions by fitting the mesh to the field of smoothed normals. The proposed method is fully automatic and outperforms conventional mesh filtering methods in terms of accuracy. 2 Adaptive Gaussian filtering. Gaussian filtering, the convolution of a 2D image Á Ü Ýµ Áµ Ü Ýµ Ü Ù Ý ÚµÁ Ù Úµ ÙÚ with the two-dimensional Gaussian function Ü Ýµ ܵ ݵ ܵ Ô ÜÔ ¾ Ü ¾ ¾ ¾ is a basic image processing technique used for image denoising. The main drawback of Gaussian filtering consists of uniform blurring of image regions. A natural extension of Gaussian smoothing is a adaption of the Gaussian kernel shape to the image structure [18] (see also [35] for a review of similar ideas). Recently in [8] it was proposed to use the convolution with Gaussian Üݵ Ü Ýµ where deviation Ü Ýµ is automatically determined via combining a scale-space approach [36] with a variant of the minimal description length principle (MDL) [28, 29]. The main advantage of the adaptive Gaussian smoothing method developed in [8] is that it is fully automatic: no manually selected parameters are required. 3 Mesh Smoothing via Adaptive Diffusion of Mesh Normals. A natural way to adapt image filtering techniques to surface smoothing consists of considering the surface normals as a vector-valued image, smoothing the field of normals, and then smoothing the surface by fitting it to the field of smoothed normals. Smoothing normals for a robust estimation of the needle diagram seems to be a standard technique in computer vision [10, 27, 37]. Various combinations of a diffusion of the surface normals with surface fitting to the smoothed normals were considered in [3, 4, 11, 12, 20, 22, 30, 33, 38]. Consider a triangle mesh Å. First we use a scale-space approach and for each mesh triangle Ì and its unit normal n Ì µ determine the best scale nµ, and smooth n Ì µ by taking a weighted average of n and neighboring normals with Gaussianlike weights,see Section 3.2. In Section 3.3 we show how to compute an anisotropic neighborhood of a triangle centroid (a vertex of the dual mesh) such that the normals in the neighborhood do not differ much from n Ì µ. We describe a procedure to fit the mesh to a set of smoothed normals in Section Fitting Mesh to Modified Normals In this section, we describe how to modify a mesh in order to fit it to a field of smoothed normals [19]. Assume that a certain smoothing procedure (we will explain it later) was applied to the set of mesh normals n Ì µ and a new set of unit vectors m Ì µ (smoothed normals) associated with mesh triangles was generated. Let us introduce an error function measuring how good the mesh fits the field of smoothed normals m Ì µ. For each mesh vertex È, we define Ø È µ Ì µ n Ì µ m Ì µ ¾ (1) where Ì µ is the area of Ì and the sum is taken over all triangles adjacent to È. A mesh error function is now defined by Ø Åµ Ø È µ A new position for È is found by minimizing Ø È µ: È ÒÛ ÖÑÒ Ø È µ (2) È and new mesh Å ÒÛ is obtained as Å ÒÛ ÖÑÒ Å Ø Åµ We use a conjugate gradient descent method to approach a minimum of Ø È µ. Since Ì µ n Ì µ m Ì µ ¾ ¾ Ì µ n mµ deriving the gradient Ö È Ø È µ is reduced to computing the gradient of the area Ì µ and its projection onto the plane orthogonal to m. A formula for the area gradient can be found in [7] (see also [21] for a simple derivation of the formula). 2

3 3.2 Adaptive Gaussian Smoothing Mesh Normals Consider a mesh triangle Ì and several rings of neighboring triangles Ì. Denote by and the centroids of Ì and Ì, respectively. The centroids form the dual mesh, as shown in Fig Figure 3: Graph of finite-support Gaussian-like kernel à ܵ,. Let us define a local variance corresponding to (4) by Primal mesh Dual mesh Figure 2: Primal and dual meshes. Let µ be a geodesic distance between and. The distance È Éµ between two neighboring centroids È and É is approximated by È Éµ È Å ÅÉ (3) where Å is the midpoint of the edge separated È and É. Now µ is found via Dijkstra s algorithm on the dual mesh. The averaged (smoothed) normal m Ì µ is computed as m È Û n È Û n (4) where n are the normals of triangles Ì and weights Û are defined by Û Ì µ à µµ Here Ì µ is the area of Ì and à ܵ is the following smooth finite-support Gaussian-like function [17] à ܵ Ô ¾ ÜÔ Ü¾ µ if Ü ¾ ¾ ¾ Ü ¾ µ if ¾ Ü ¼ if Ü where ¾¾ is the base of natural logarithms. Function à ܵ is obtained from Gaussian function ܵ by smoothly splining its tails to zero, as seen in Fig. 3 The amount of smoothing in (4) is controlled by the parameter, the standard deviation of the Gaussian-like function à ܵ. ¾ È Û m n µ ¾ Û ¾ È Û Û n Now we select optimal Ø according to the following rule Ø ÖÑÒ ¾ ¾ (5) where ¼ ¼ Ð ¾, Ð is the arithmetic mean of the edge lengths of the mesh. To find Ø in (5) ten scale-spaces with from ÑÒ ¼¾ Ð to ÑÜ ¾¼ Ð with step-size ¼¾ Ð are used. Fig. 4 demonstrates how the proposed smoothing procedure works. Notice that for a mesh triangle Ì its best scale Ø Ì µ is closer to its maximum value ÑÜ if Ì is located at a relatively flat mesh region corrupted by a slight noise and Ø Ì µ is closer to its minimal value ÑÒ if Ì is located at a noisy and/or curved mesh region. Thus the local scale map Ø Ì µ provides valuable information on the smoothing procedure, as demonstrated by the topright image of Fig. 4. Fig. 5 compares our smoothing method with the implicit mean curvature flow proposed in [7]. Our method produces better smoothing and does not require any user s intervention (for deciding an optimal number of smoothing iterations, for example). Smoothing a real-world model, an eye part of the scan of the St. Matthew statue [14], is exposed in Fig. 6. Notice that Michelangelo s chisel marks, important model features, are well preserved by our smoothing procedure. 3

4 Figure 6: Left: an eye part of the scan of the St. Matthew statue [14]. Middle: the mesh is smoothed by the method described in Sections 3.2 and 3.1. Right: the original (noisy) mesh is shaded according to the best-scale map Ø : light grays correspond to large values of and dark grays correspond to small values of. (a) (b) Figure 4: Top-left: a noisy mesh. Top-right: mesh is shaded according to the best-scale map Ø : light grays correspond to large and dark grays correspond to small. Bottom-left: the mesh is shaded according to the smoothed normals m Ø. Bottom-right: the smoothed mesh obtained by fitting m Ø. (c) (d) Figure 5: (a) The Stanford bunny model with noise added. (b) The model is smoothed adaptively according to (5) chosen automatically for each mesh triangle. (c) The implicit mean curvature flow [7] is used for smoothing. A number of iterations is decided manually according to the best visual result. (d) Typical over-smoothing caused by a wrong number of smoothing iterations. 4

5 3.3 Anisotropic Neighborhoods The smoothing procedure described in the previous section does not remove noise near sharp edges and corners: if a mesh triangle Ì is close to a sharp feature, Ø Ì µ is small and smoothing is insufficient to eliminate the noise. Fig. 7 demonstrates this drawback. Figure 8: meshes. Anisotropic neighborhoods on noisy neighborhoods. According to our experiments, the choice ¼ ¼ ¾ Ð ¾ with ten scale-spaces from ÑÒ ¼ Ð to ÑÜ ¼ Ð works well. Fig. 9 compares our isotropic and anisotropic approaches for smoothing a mesh with sharp edges. Figure 7: Top-left: a noisy mesh with sharp features. Top-right: the best-scale map Ø Ì µ. Bottom-left: the smoothed mesh. Bottom-right: a magnified part of the smoothed mesh. Isotropic Anisotropic One possible way to overcome this limitation of the proposed mesh smoothing technique consists of generating anisotropic neighborhoods avoiding regions with abrupt changes of the mesh normals. In neighborhood of a vertex Ç of the dual mesh let us define a distance between two neighboring vertices È and É of the dual mesh by È Éµ n È µ n ɵ ¾ n ǵ n È µ¾ n ǵ n ɵ¾ ¾ ¾ where È Éµ is given by (3) and is a positive constant. Now an anisotropic neighborhood of Ç is found using Dijkstra s algorithm. According to our numerical experiments, ¾¼ leads to generating desired anisotropic neighborhoods, as demonstrated in Fig. 8. We change the thresholds in (5) and ÑÒ and ÑÜ when using our technique of anisotropic Figure 9: The isotropic (left) and anisotropic (right) smoothing methods are applied to a noisy model shown in the right image of Fig. 8. Notice how well anisotropic smoothing eliminates noise near the sharp features. Smoothing the angel model (the right image of Fig. 1) performed by our anisotropic neighborhood technique demonstrates a high fidelity of the proposed method. 4 Comparison with Other Smoothing Techniques To compare quantitatively our smoothing technique with conventional mesh filtering methods we con- 5

6 sider two error metrics measuring deviations of mesh vertices and normals [23]. Let Å be a reference mesh (we assume that Å is dense enough) and Å ¼ be a mesh obtained from Å by adding noise and applying then a smoothing process. Consider a vertex È ¼ of the smoothed mesh Å ¼. Let us set Ø È ¼ ŵ equal to the distance between È ¼ and a triangle of the reference mesh Å which is closest to È ¼. Our Ä ¾ vertex-based meshto-mesh error metric is then given by Ú Å ¼ µ È ¼ ¾Å ¼ È ¼ µ Ø È ¼ ŵ ¾ where È ¼ µ is the sum of areas of all triangles of Å ¼ incident with È ¼ and Å ¼ µ is the total area of Å ¼. Our Ä ¾ normal-based mesh-to-mesh error metric measuring deviations between the corresponding normals of two meshes Å and Å ¼ is defined in a similar way. Consider a triangle Ì ¼ of the mesh Å ¼ and let us find a triangle Ì of Å closest to the centroid of Ì ¼. Let n Ì µ and n Ì ¼ µ be the orientation unit normals of Ì and Ì ¼, respectively. The metric is defined by Ò Å ¼ µ Ì ¼ ¾Å ¼ Ì ¼ µn Ì µ n Ì ¼ µ ¾ where Ì ¼ µ denotes the area of Ì ¼. Note that Ò is scale-independent. Fig. 10 displays two error graphs obtained for a noisy Stanford bunny (the top-left image of Fig. 5) smoothed by the implicit mean curvature flow [7], three variations of the Taubin method ( ¼ ¼ ) [31, 33] and isotropic version of the method developed in this paper (Sections 3.2 and 3.1). The horizontal axis shows the number of smoothing iterations required the and mean curvature flow methods. According to the graphs, our method demonstrates the best performance. In addition, it does not require specifying the number of iterations (only one iteration is needed) and other smoothing parameters. 5 Discussion In this paper, we have developed a fully automatic mesh filtering method that adaptively smoothes a Figure 10: Graphs of Ú (top) and Ò (bottom) for smoothing the model shown in the top-left image of Fig. 5. (Æ) method with equal weights, ( ) method with weights equal to inverse edge lengths, (¾) method with cotangent weights, () implicit mean curvature flow [7], ( ) our method. noisy mesh and preserves sharp features and features consisting of only few triangle strips. In addition, it outperforms other conventional smoothing methods in terms of accuracy. We have proposed two versions of the method: isotropic and anisotropic. Both the versions demonstrate equal performance on models without sharp edges and corners. If a model has sharp features, the anisotropic version is preferable, as it was demonstrated in Fig. 9. Fig. 1 and Fig. 11 provide a reader with a visual comparison of our method with Taubin s method. For these models, both the isotropic and anisotropic versions of our method produce similar results (results produced by anisotropic version are displayed in both the figures). Our method can be also used iteratively. Although it produces a smoother mesh, oversmoothing can be observed, as seen in Fig. 12. Our method has the following drawbacks. It is not capable to remove large noise. It is time-consuming to compare with such 6

7 (a) (b) (c) (d) Figure 11: (a) A noisy monk mesh obtained using Minolta Vivid 700 laser scanner. (b) The best visual result achieved with Taubin s smoothing method. (c) Anisotropic smoothing developed in this paper was applied. (d) The best-scale map Ø Ì µ of (a). one iteration two iterations Figure 12: Second iteration produces a smoother and, perhaps, better-looking mesh. Notice however that high curvature features are oversmoothed. conventional mesh filtering techniques as Laplacian and bilaplacian [13] smoothing flows, mean curvature flow [7], Taubin s method [31, 33]. For example, for the Stanford bunny our method is approximately ¼ times slower than the method with equal weights. Its mathematical foundation is not well established. We were unable to adapt an MDL approach [8, 28, 29] to the mesh normals and our rule for deciding the best scale (5) may be incorrect from a statistical point of view. Fixing these shortcomings is a promising direction for future research. We have tested our method on dense meshes with a small variance of edge length distribution. Such meshes are usually produced by modern dataacquisition hardware. Adapting the method for nonuniform meshes also constitutes a future research direction. Acknowledgments We are grateful to Marc Levoy and Stanford University for providing us with the St. Matthew model scanned within the Digital Michelangelo Project [14]. The angel model is from Caltech 3D Gallery [5]. We would like to thank the anonymous reviewers of this paper for their valuable and constructive comments. 7

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