SSegmentation Based Mesh Denoising
Chaofan Dai , Wei Pan and Xuequan Lu OPT Machine Vision Corp. School of Mechanical and Automotive Engineering, South China University of Technology School of Information Technology, Deakin University
Abstract
Feature-preserving mesh denoising has received noticeable attention recently. Many methods often design great weighting foranisotropic surfaces and small weighting for isotropic surfaces, to preserve sharp features. However, they often disregard the factthat small weights still pose negative impacts to the denoising outcomes. Furthermore, it may increase the di ffi culty in parameter tun-ing, especially for users without any background knowledge. In this paper, we propose a novel clustering method for mesh denoising,which can avoid the disturbance of anisotropic information and be easily embedded into commonly-used mesh denoising frameworks.Extensive experiments have been conducted to validate our method, and demonstrate that it can enhance the denoising results of someexisting methods remarkably both visually and quantitatively. It also largely relaxes the parameter tuning procedure for users, in termsof increasing stability for existing mesh denoising methods. Keywords:
Mesh Segmentation, Mesh Denoising, Surface Smoothing, Surface Clustering.
1. Introduction
Mesh denoising is a fundamental research problem in geom-etry processing. The denoised mesh models can be applied tofurther geometry processing, computer animation, industrial de-sign and so on. The main challenges lie in the removal of noisewhile preserving sharp features.Most existing mesh denoising methods focused on the useof local information (e.g., [1, 2, 3, 4, 5, 6, 7, 8]). However, itstill remains a challenge in recovering sharp features from noisymesh models to date. In particular, the face normal based meth-ods [1, 3, 4, 6] often consider the design of small weights foranisotropic faces and large weights for isotropic weights, whilesmall weights still a ff ect the denoising outcomes. Furthermore,they sometimes require remarkable e ff orts and even tremendouse ff orts in parameter tuning. That is, it is sometimes not easy toachieve desired outcomes, especially for those who have insu ffi -cient familiarity with the algorithms. A few other methods utilizemore information for mesh denoising [9, 10]. Nevertheless, theyinvolve significant computation and are usually slow.To address the above issues, we introduce a novel segmenta-tion approach to greatly facilitate existing mesh denoising meth-ods, with reducing the di ffi culty in parameter tuning for users.The key idea is to cluster the surface of a mesh model beforereal denoising. Specifically, we realize the segmentation by sim-ply contrasting an edge indicator. The cluster information canessentially eliminate the disturbance of anisotropic surfaces. Asa result, some commonly-used mesh denoising methods can befurther employed for mesh denoising. We conduct extensive ex-periments to validate our approach, and found it is able to boostmesh denoising outcomes. It is simple and easy to be embeddedinto plenty of existing mesh denoising frameworks.Our main contributions are summarized as follows. • We design a region-growing segmentation approach to di-vide a noisy input mesh model into pieces. • The achieved segmentation is applied to facilitate and boostsome commonly-used mesh denoising methods.
2. Related Work
Mesh segmentation, a.k.a. mesh clustering, is to decomposean input mesh into smaller and meaningful subsets. It is usuallydivided into two categories: semantic segmentation and geomet-ric segmentation. The former one is to segment the mesh intomeaningful clusters based on semantic information, while thelatter clusters based on geometric criteria such as curvatures andnormals. It has been a very active area of research in computergraphics [11]. Readers may refer to [12, 13, 14] for comprehen-sive surveys.In this work, we mainly review geometric segmentation whichincludes region-based and boundary-based segmentation meth-ods. Region-based segmentation is a clustering method that gath-ers similar regions together based on geometric information. Thecommon methods are K-means and its variant [15]. VariationalShape Approximation (VSA) is an error-driven method whichfits planar proxies to geometry [16]. Distortion error is iterativelyreduced through repeated clustering of faces into best-fitting re-gions. Each iteration consists of a region growing phase basedon the L or L metric. In the work of [17, 18], SLIC super-pixel technique is used to compute super facets e ffi ciently withthe K-means approach. Similar to superpixel in image process-ing, triangles with similar geometric metrics are then groupedinto super facets. Other region-based clustering methods containmean-shift [19], medoid shift [20], quick shift [21], hierarchicaldecomposition [22], primitive fitting [23], random walks [24],and so on. Boundary-based methods extract or detect the geo-metric feature boundaries of the input mesh, such that the ob-ject can be divided into di ff erent parts by the boundaries. Rel-evant methods include randomized cuts [25], fuzzy-clustering-and-cuts (FCC) [22], shape diameter function [26], 3D meshscissoring [27, 28], etc. These methods highly rely on the lo-cal geometric information of the input mesh, and easily fail oncethe structure of the mesh becomes complicated or noises levelbecomes high. Preprint submitted to Elsevier August 25, 2020 a r X i v : . [ c s . G R ] A ug ecent advances in deep learning lead to new data-drivenmethods for mesh segmentation [29]. The majority of them dealwith the semantic segmentation problem; see [30] for a compre-hensive review. There exists a large body of literature for mesh denoising;readers are referred to [31, 32] for comprehensive reviews.The Laplacian smoothing methods [33, 34] are early researchfor mesh denoising. However, its isotropic nature results infeature-wiping and shrinking artifacts. Taubin [35] proposed atwo-step smoothing method for non-shrinking mesh denoising.Later, a fairing method based on di ff usion and curvature flow[36] was proposed to handle irregular meshes. Various isotropicsmoothing methods [37, 38, 39, 40, 41] have been further intro-duced based on volume preservation, pass frequency controlling,di ff erential properties etc.The above isotropic methods often wipe out features. Vari-ous anisotropic methods have been proposed to mitigate this is-sue. Some common approaches consist of di ff usion / di ff erential-based methods [42, 43, 44, 45, 46, 47, 48, 49, 50], bilateral meth-ods [51, 52, 3, 53, 54], two-step methods involving normal filter-ing and vertex update [3, 53, 55, 56, 57, 2, 1, 58, 51, 59, 60,4, 5, 6, 7]. The two-step methods including normal smoothingand vertex update have been proved to be promising in preserv-ing sharp features [3, 53, 55, 56, 57, 2, 1, 58, 51, 59, 60, 4, 5,6, 7, 61, 62]. In recent years, some researchers attempted toconduct vertex and face classification for better mesh denois-ing [63, 64, 65, 66, 67, 68, 69]. These classification strategiesare mainly focused on local neighborhood and usually sensitiveto noise. The ideas of pre-filtering [70, 6, 7] are introduced forbetter denoising outcomes. Arvanitis et al. [71] introduced anovel coarse-to-fine graph spectral processing approach for meshdenoising.Sparsity was introduced into mesh smoothing in some recentworks [50, 72, 70, 73]. He et al. [50] developed an L mini-mization framework. It is non-convex and slow. An improvedalternating optimization strategy [73] was designed to solve the L minimization, which involves vertex positions and face nor-mals. Wang et al. [72] proposed a method to decouple noise andfeatures by weighted L -analysis compressed sensing. Lu et al.[70] introduced a novel L minimization to detect features. Alow-rank matrix approximation approach was proposed for ge-ometry filtering [9]. The low-rank idea was further extended tomesh denoising [10, 74]. Li et al. [75] described a method to ex-tract feature lines with Laplacian preprocessing on noisy models,and the feature lines are used to identify neighbours for guidednormal estimation. By constructing half window of local neigh-borhood for each vertex, Pan et al. [8] proposed a half-kernelLaplacian operator to reduce the damages on features while re-moving noise. This method is fast and e ff ective, but has limitedcapability for sharp edge preservation in CAD-like models.There are also some data-driven methods, such as Wang etal. [5] and [76]. They defined the filtered facet normal descriptor(FDN) according to the neighborhood of a noisy mesh facet tothe noise-free mesh facet normal, and then modeled non-linearregression functions by mapping the FDN. The modeled func-tion is used to compute new facet normals. Multiple iterations ofmesh denoising are required which forms the cascaded normalregression.
3. Method
Fig. 1 provides an overview of our method. The first stepis pre-filtering which provides good initialization. This step isonly required for input with relatively large noise (e.g., 0 . Figure 1: Overview of our approach.
We simply adopt the pre-filtering algorithm introduced by [70]for the pre-processing purpose, when the noise level of the inputmesh is relatively high. The objective function ismin (cid:88) i || ˜ p i − p i || + α (cid:88) e w ( e ) || D ( e ) || + β (cid:88) e w ( e ) || R ( e ) || , (1)where ˜ p i is the new position which is unknown and i denotes the i -th vertex. α and β are the weights of two di ff erent terms. D ( e )and R ( e ) are the area-based edge operator and the regularizer in[50]. w ( e ) is the weighting function for the edge e , which utilizesnormal information.The above objective function can be easily solved with lin-ear equation systems. It should be noted that this step is onlyrequired for input meshes corrupted with relatively large noise.Fig. 2 shows two examples for pre-filtering. Figure 2: Two pre-filtering examples.
We propose to use an edge metric for mesh segmentation. Anintuitive way is to take account the normal information on themesh surface. Nevertheless, normals are quite sensitive to noiselevel. We resort to the di ff erential edge operator defined in [50]as the edge metric in segmentation, which is one term appearedin Eq.(1). The edge operator is2 ( e ) = (cid:52) ( p − p ) · ( p − p ) + (cid:52) ( p − p ) · ( p − p )( (cid:12)(cid:12)(cid:12)(cid:12) p − p (cid:12)(cid:12)(cid:12)(cid:12) ) ( (cid:52) + (cid:52) ) (cid:52) (cid:52) + (cid:52) (cid:52) ( p − p ) · ( p − p ) + (cid:52) ( p − p ) · ( p − p )( (cid:12)(cid:12)(cid:12)(cid:12) p − p (cid:12)(cid:12)(cid:12)(cid:12) ) ( (cid:52) , , + (cid:52) , , ) (cid:52) (cid:52) + (cid:52) T p p p p (2)where (cid:52) denotes the area of the triangle defined by p , p and p , and (cid:52) is the area of the triangle formed by p , p and p . D ( e ) is a vector (1 × / non-feature property of the edge e . Its L -norm shouldbe 0 when the corresponding two triangles are on a plane (twotriangles sharing the edge). Thus, the ideal threshold D thr is 0 fordetermining feature regions or non-feature regions. While meet-ing noisy input, we can relax this constraint by setting a greaterthreshold. Figure 3: The edge operator metric can distinguish sharp edges from non-sharpedges. Triangular faces with sharp edges are rendered in red and those withoutsharp edges are rendered in green.
After having the edge metric, we propose to do region-growsegmentation based on the values of the di ff erential edge oper-ator for each edge, which to some extent determines the edgeis a feature or non-feature edge. Experiment shows that ourregion-growing scheme is more robust in segmentation (Fig. 5).To achieve the region-growing segmentation, we first set a seed-ing triangle face in the input mesh and assign a label to it. Wethen compute each edge operator of this face and determine ifthe edge-connected faces belong to the same cluster as the seed-ing face, by comparing the L -norm of edge operators with thethreshold D thr . If the L -norm is smaller than D thr , we assignthe cluster label of the seeding face to that connected triangleface. The new faces with the same label are all viewed as seedingfaces. We further calculate the edge operators of new faces, andcontinue this procedure until the norm of a new edge operator isgreater than or equal to D thr . If it is stopped, we randomly selecta seeding face from the rest area of the input mesh and continuethe above procedure again. We stop it until all faces on the meshare clustered. Fig. 4 shows a demonstration of our segmentationprocess.Fig. 5 shows the segmentation results of several methods.Since some other segmentation methods such as mean-shift [19]is using the same metrics as K-means, we only display the re-sult of K-means here. The region-growing segmentation basedon distance and normal metric is highly sensitive to noise, whileour segmentation method generates better results. Fig. 6 showsthat our segmentation results are influenced by D thr . Refinement.
Small clusters are sometimes observed afterregion-growing segmentation, which is inevitable due to noise.We further found that such small clusters usually pose negativeimpacts to mesh denoising (Fig. 7). We simply identify smallclusters with less than 50 triangles, and merge them into otherclusters. Specifically, for each triangle face in the small cluster,we calculate the cosine of the current face normal and each ofits 2-ring neighborhood face normal and sum the cosine within
Figure 4: Demonstration of our region-growing segmentation method based onthe edge operator metric. A seed triangle will be di ff used to its three co-sidedneighbors, and the di ff usion process is based on the value of the L -norm of thedi ff erential edge operator. An edge with a greater width indicates a larger L -norm of the edge operator. (a) K-means (b) Region grow (c) Ours Figure 5: Segmentation results of several di ff erent methods. (a) K-means ( K = (a) D thr = . D thr = . D thr = . D thr = . Figure 6: Segmentation results by our method with di ff erent D thr . LGORITHM 1:
Region-growing Segmentation
Input:
Mesh with low noises or preprocessed mesh, Threshold D thr Output:
Mesh with clustersCompute di ff erential edge operator D ( e ) for each edge; repeat Randomly select an unprocessed face F i as a seed for a new cluster C ;Get F i ’s edge-connected neighbors { F j } and corresponding edges { e j } ; while D e j < D thr do cluster F j into C ;mark F j as a new seed; enduntil all faces are clustered ; the same cluster. For simplicity, the cluster label which inducesthe greatest sum is assigned to this triangle face. The function isdefined as arg max k (cid:88) j ∈ S ( i ) cos( n i , n j ) , (3)where k indicates the cluster label k . n i is the current face nor-mal, and { j } = S ( i ) represents the face set with a certain clusterlabel in the 2-ring neighborhood. n j denotes a face normal cor-responding to index j in this set. Algorithm 1 summarizes ourregion-growing segmentation algorithm.(a) (b) (c) (d) Figure 7: Denoising results with / without cluster refinement. (a) Cluster resultwithout refinement; (b) denoising result based on (a); (c) cluster result with re-finement; (d) denoising result based on (c). Regarding mesh denoising applications, our segmentation canbe easily embedded into some commonly-used mesh denoisingframeworks, and is able to help boost the performance. In par-ticular, the output clusters provide useful information in exclud-ing anisotropic neighbors and avoid the negative influence fromanisotropic neighbors. In this respect, many local based meshdenoising techniques can benefit from our segmentation, such as[1, 3, 4, 6]. We constrain and update the neighbors within thesame cluster for mesh denoising, which also reduces the di ffi -culty in parameter tuning and increases stability to mesh denois-ing algorithms.
4. Experimental Results
The selected state-of-the-art techniques are: the unilateral nor-mal filter (UNF) [1], the bilateral normal filter (BNF) [3], L minimization (L0) [50], the guided normal filter (GNF) [4], andthe L -Median filter (L1) [6]. We select them since the involvedsource codes available online or provided by the original authors.We embed our clustering approch into each of these methods. The parameters of these methods used in our experiments aresummarized in Table 1. Note that we have one extra parameter(segmentation threshold ( D thr )) which is easy to tune. Desiredmesh denoising results can be easily obtained, because the in-fluences of the original parameters are significantly reduced byour clustering method. For reasonable and fair comparisons, weset the same parameters for the original methods and the adaptedcluster-driven methods. Table 1: Parameters of some commonly used mesh denoising methods.
Methods Parameters ParametersUNF 3 T : threshold for controlling the averaging weights n iter : number of iterations for normal update v iter : number of iterations for vertex updateBNF(Local) 3 v iter : number of iterations for vertex update σ s : variance parameter for the spatial kernel n iter : number of iterations for normal updateGNF 5 v iter : number of iterations for vertex update σ r : variance parameter for the range kernel σ s : variance parameter for the spatial kernel r : radius parameter r for finding a geometrical neighborhood n iter : number of iterations for normal updateL0 6 µ β : update ratios for ββ : initial values for ββ max : maximum value of βµ α : update ratios for αα : initial values for α . λ : weight for the L0 term in the target functionL1 Median 3 v iter : number of iterations for vertex update σ s : variance parameter for the spatial kernel n iter : number of iterations for normal updateOur + Others 1 + X D thr : Segmentation threshold X : the parameters of other methods We compare our method with theselected state-of-the-art methods on various models corruptedwith synthetic noise. According to state-of-the-art mesh smooth-ing techniques, noisy synthetic models are generated by addingzero-mean Gaussian noise with standard deviation σ n to the cor-responding ground truth. σ n is proportional to the mean edgelength l e of the input mesh.From Fig. 8 to Fig. 13, we display the results by originalmethods and by the embedded versions (i.e., ours + original meth-ods). It is obvious that the cluster-driven approaches have a bet-ter preservation of shapr features than the original methods. Todeal with heavy noise, current denoising methods such as BNFand UNF usually require a large number of iterations for normalsmoothing and vertex update (e.g., 100 or 200), which sometimesleads to over-smooth results (see Fig 9). By contrast, the embed-ded versions (ours + original methods) enable better preservationof sharp features. Synthetic non-CAD models.
The improvement of our methodon dealing with non-CAD models is less significant than that onCAD-like models. This is because that growing variations onshapes lead to an increasing di ffi culty on segmentation. Fig. 14shows comparisons on the Atenean model corrupted by syntheticGaussian noise ( σ n = . l e ). Compared with the original meth-ods, the embedded versions can preserve more sharp creasesand edges, though the improvement is not remarkable. Fig. 15shows comparisons on the Octaflower model corrupted by syn-thetic Gaussian noise ( σ n = . l e ). The edges are sharper. Raw scanned models.
In addition to the synthetic shapes, wealso experiment on scanned models corrupted with raw noise.Fig. 17 show the smoothing results of all methods on a rawscanned mesh. Compared with the original methods (first row),40 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 8: Results on the noisy Icosahedron model ( σ n = . l e ). L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 9: Results on the noisy Cube model ( σ n = . l e ). L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 10: Results on the noisy PartLP model ( σ n = . l e ). our methods (second row) preserve sharp features better. We ob-serve from Fig. 16 that the embedded versions can produce com-parable results to [52, 8]. Algorithms stability.
Our clustering increases the stability ofthe original algorithms, making them more robust against param- eter variations. Fig. 18 shows the results by UNF with di ff erentthresholds T . For the original UNF, the visual results are quitesensitive to di ff erent thresholds. By embedding our approach,the sharp edges are generally preserved well, even with di ff erent T (except T = . + BNF our + UNF our + GNF our + L1 Figure 11: Results on a noisy CAD model ( σ n = . l e ). L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 12: Results on the noisy Fandisk model ( σ n = . l e ). L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 13: Results on the noisy Dodecahedron ( σ n = . l e ). Boundaries are rendered in red for better visual e ff ect. and the embedded version (ours + GNF), with di ff erent σ r values.This demonstrates that our approach can significantly boost thestability of existing mesh denoising methods, thus relaxing users’ parameter tuning process.6 ime: 91ms L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 14: Results on the noisy Atenean model ( σ n = . l e ). L0 BNF UNF GNF L1Noisy our + BNF our + UNF our + GNF our + L1 Figure 15: Results on the noisy Octaflower model ( σ n = . l e ). Besides the above visual comparisons, we also show the quan-titative evaluations for the original methods and the embeddedversions. Specifically, we employ E v and MSAE (Mean SquareAngular Error) to respectively evaluate the errors on vertex posi-tions and face normals, as suggested by previous works [1, 3, 70]. E v is the L vertex-based mesh-to-mesh error metric, and MSAEmeasures the mean square angular error between the face nor- mals of the denoised mesh and those of the ground truth. Thesetwo metrics are calculated between the denoised results and theircorresponding ground-truth models. L0 has six parameters, weset µ β , β and µ α as suggested by its authors, and the remainingthree parameters are ( β max , α , λ ).Table 2 lists E v and MSAE over some models for all the com-pared methods. For most cases, the results produced by the em-bedded versions (ours + original method) have smaller MSAE and7oisy BMF [52] HLO [8] L0 [50]our + BNF our + UNF our + GNF our + L1 Figure 16: Results on the raw scanned Wilhelm model.
L0 BNF UNF GNF L1Scanned Raw our + BNF our + UNF our + GNF our + L1 Figure 17: Results on the raw scanned cube model. E v compared to the results denoised by the original methods, es-pecially for large noise. This shows the improvement of our clus-tering method upon the original methods, from the quantity per-spective. Analogous to previous research [1, 70], we also foundthat the visual comparisons might be inconsistent with E v . Itshould be noted that, for relatively small noise (e.g., σ = . l e ),the embedded methods usually have similar performance to theoriginal methods.
5. Conclusion
We have presented a cluster-driven mesh denoising frameworkwhich can generate clusters for mesh surfaces, especially forCAD-like models. It can be easily integrated with existing meshdenoising methods. Both visual and quantitative results confirmthat our method enables better mesh denoising outcomes, espe-cially for models corrupted with large noise, with a significantease for users.As with most existing clustering methods like K-means orMean-shift [19], it still has limited robustness to very large noise,8NF(T = = = = = + UNF(T = + UNF(T = + UNF(T = + UNF(T = + UNF(T = Figure 18: Results by setting di ff erent T for UNF and ours + UNF.
GNF( λ r = λ r = λ r = λ r = λ r = + GNF( λ r = + GNF( λ r = + GNF( λ r = + GNF( λ r = + GNF( λ r = Figure 19: Results by setting with di ff erent σ r for GNF and ours + GNF. and is not easy to output decent clusters without the aid of thepre-processing [70]. As the future work, we would like to designmore robust clustering techniques for noisy 3D shapes.
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