1 Introduction
With development of digital dentistry, optical scanning of dental models became essential for computer aided design (CAD) and computer aided manufacture (CAM) of dental prosthetics [1, 2, 3, 4] and for digital orthodontics [5], for example, digital indirect bonding technique [6], production of vacuum formed invisible aligners [7]. The optical scanners, based on principle of triangle measuring method, acquire model surface point cloud data and convert them to mesh (always triangular mesh). The mesh data contains vertex coordinates and their connections (edges and facets). Teeth in the mesh should be segmented before design of dental prosthetics or make virtual orthodontic setup, which may take enormous and strenuous manual labor.
There are several kinds of classical algorithm (without machine learning) for mesh segmentation
[8, 9, 10], such as region growing[11, 12, 13, 14], watershed segmentation[16, 17, 18, 19],contourbased mesh segmentation[15][20, 21, 22, 23, 24], iterative clustering[25, 26, 27, 28, 29], spectral analysis[30, 31] and implicit methods (define the boundaries between submeshes)[32, 33, 34, 35, 36], which are not reliable in complex cases and not easy to extend. With the development of artificial intelligence, machine learning methods
[37, 38, 39, 40, 41, 42, 43]were applied for mesh segmentation. Some researches collected hundreds of vertex or facet features to do classification on support vector machine or neural network
[37, 43], while the features are complex to define and calculate. Several recent researches aimed to take mesh as raw input and do segmentation work by converting to voxel for 3dimentional (3D) convolutional neural network (CNN)[44, 45, 46], or projection to 2dimentiional (2D) images for 2D CNN[39, 47], or direct point clouds deep learning[38, 41]. However, the innate vertex connections in the mesh were ignored in those methods, which is very important for the cognition of mesh shape and structure. Here we propose an artificial neural network named mesh CNN, where only a few features for each vertex need to be precalculated, and the mesh vertex feature matrix serve as raw input without voxelization or projection. In this model, the topological connections between mesh vertexes were utilized to produce an adjacent matrix, with which neighborhood features for each vertex were involved for calculate, expanding receptive filed iteratively, and finally, vertex features after convolution were sent to softmax layer. We challenged clinical dentition plaster casts segmentations and achieved good performances.In recent years, machine learning has been widely applicated in the field of mesh segmentations. Kalogerakis et al.[48] presented a datadriven approach to simultaneous segmentation and labeling of parts in 3D meshes based on Conditional Random Field model, where they achieved improvement on the Princeton Segmentation Benchmark. Guo et al.[43]
trained CNNs by a large pool of classical mesh facet geometric features, to initialize a label vector for each triangle indicating its probabilities of belonging to various object parts, but connections between mesh facets were unable to be concerned in these CNNs.Xu et al.
[49] constructed 2level hierarchical CNNs structure for tooth segmentation, consuming a set of geometry features (600dimension) as face feature representations to train the networks. George et al.[37]used multiscale features calculated as the mean of a set of neighboring facets to train multibranch 1D CNNs, thus to classify a facet by its multi scale local area features. However, hundreds of geometric and contextual features need to be defined and calculated aforehand for their networks.
Wu et al.[44]
proposed 3D shapeNets, which represent a geometric 3D shape as a probability distribution of binary variables on a 3D voxel grid based on Convolutional Deep Belief Network. Except transferring to voxel, multiview images were also used to represent 3D mesh geometric information. Le et al.
[47]introduced a multiview recurrent neural network approach for 3D mesh segmentation, where the convolutional neural networks and long short term memory were combined to generate the edge probability feature map and their correlations across different views. Qi et al.
[46] combined the volumetric representations and multiview representations, with introducing of architectures of volumetric CNNs, as well as introducing of multiresolution filtering in 3D into multiview CNNs, to provide better methods available for object classification on 3D mesh data. However, transferring mesh to voxel grids of different view or to images may produce unnecessarily intermediate steps and raise issues. PointNet[41] was designed by Qi et al. for applications of 3D object classification, part segmentation, and scene semantic parsing, which directly consumes point clouds, and respects the permutation invariance of points in the input. PointNet was further developed to be PointNet++[38], which applies PointNet recursively on a nested partitioning of the input pointset, to learn deep point set features efficiently and robustly. Different from point clouds, the teeth mesh data have vertex connections represented by the edges of facets. If treated as point clouds, the geometric connections between mesh vertex, which is very important to retrieve the mesh topological structures, will lost. Thus, in this research, we proposed an architecture of a neural network named mesh CNN that exploits the edge connection information to compute local area features on mesh data, and finally implement the vertex classification.2 Proposed Approach
We constructed a neural network for vertex detection, where an expanding layer was proposed, which utilized the connection information of the triangle meshes to generate links between vertexes, concatenating features of one vertex to that of its adjacent vertexes. CNN was applied to the concatenated features, after serval repeat of the expanding layer and CNN layer architectures, probability of labels for each vertex was finally output. The details of expanding layer are described below. In a mesh with vertexes and edges, all vertexes () and edges () can be defined as:
(1) 
(2) 
For any vertex , the vertexes directly connected to are those on the same edge with , which can be defined as:
(3) 
Further, for ,
(4) 
In this research, we define ring 0 neighbor vertexes of as its self:
(5) 
Ring1 neighbor vertexes of as vertexes directly connected to :
(6) 
Ring2 neighbor vertexes of as vertexes directly connected to ring 1 neighbor vertexes of , excluding ring 1 neighbor vertexes and :
(7) 
For , ring neighbor vertexes of as vertexes directly connected to ring neighbor vertexes of , excluding ring neighbor vertexes and inner vertexes:
(8)  
It can be easily inferred that ring n neighbor vertexes will not directly connect to ring and inner vertexes, that is:
(9) 
So, the equation 8 can be simplified as:
(10) 
For a vertex , there might be several features, such as coordinates , curvatures, distances, which can be presented as:
(11) 
For , we define the mean features of vertexes in as:
(12) 
where is the number of vertexes in . Consider input of a mesh matrix with all vertexes, each vertex with features:
(13) 
We can calculate its mean ring neighbor features:
(14) 
where contain some context information of mesh connections, and it can be easily inferred that . The expanding layer, take mesh feature matrix as input, and output an augmented matrix with means of several ring neighbor features:
(15) 
Here is an instance of matrix computation algorithm to implement the function. Firstly, we constructed an adjacent matrix , which represent the connection relationship between mesh vertexes:
(16) 
Secondly, construct a new matrix to collect ring neighbor features:
(17) 
Finally, construct another new matrix computing means of ring neighbor features:
(18) 
Then, the constructed matrix will be equal to output of expanding layer, as defined in equation 15:
(19) 
The input features can also be expanded to noncontinuous ring number neighbor features, of which the result is a slice of , for example:
(20) 
(21) 
The operations of expanding layer and convolution layer were illustrated in Figure 2. All other vertices besides the red one will be set as center vertex and subjected to the same operation pipelines to update their feature matrices. The updated feature matrix contain learned information (by CNNs) from ring neighbor vertices, thus is able to extract local area features automatically. Once all vertices in a mesh updated their feature matrices, next iteration of expanding and convolution operations can be scheduled to extract deeper features.
Based on expanding layers and convolution layers, 6 kinds of networks were constructed, of which a network without expanding layer was set as baseline, as in Table 1.
Baseline  A  B  C  D  E  
Input matrix, of which  












































Softmax 
3 Experiment
We collected 1445 dental meshes, the count of vertex in each mesh was controlled to be less than 6000, due to the limitation of our computation resources on a 8GB graphics card. Meshes were saved as .obj format, recording the vertical coordinates and its RGB color values, as well as vertex connections that composition triangle facets. Meshes were annotated that the teeth margin vertexes and interteeth vertexes were brushed to be red (Figure 1), that is, labeled as positive . All other vertexes were labeled as negative .
Classifier  Network  Training Features  Validation  
Accuracy  Precision  Recall  
1  Baseline  Curvatures (),  0.766  0.670  0.455 
2  A  Curvatures (),  0.863  0.786  0.756 
3  B  Curvatures (),  0.870  0.784  
4  C  Curvatures (),  0.857  0.784  0.733 
5  E  Curvatures (),  0.874  0.794  0.787 
6  D  Curvatures (),  0.789  
7  D  0.807  0.695  0.649  
8  D  , Curvatures (),  0.866  0.783  0.773 
For each vertex, 8 features were collected, including coordinates (), curvatures () and the mean neighbor distance (). Curvatures () are essential for description of the 3dimensional morphology of meshes, and both curvatures and distance have shiftinvariance and rotateinvariance properties. The mean neighbor distance () of a vertex was defined here as mean distance between and its ring 1 neighbor vertexes:
(22) 
where are coordinates of vertex , and are coordinate of vertex . The dataset was divided to be train dataset of 1156 meshes, validation dataset of 145 meshes and test dataset of 144 meshes.
In this research, networks should be trained by labeled data before they have ability to do classifications. A trained network here was specified as a Classifier. Networks shown in Table 1 were trained with different input features, as shown in Table 2
, where classifier 1 to 6 were designed to test the performance of different network architectures, while 6 to 8 were designed to test the influence of different combination of input features on the outcomes. The loss function was defined upon weighted cross entropy with logits:
(23)  
where the was set to be 3.0, and was used to minimize the , with initial
of 0.01, and decent to be 0.003 after 5000 train iterations, and 0.001 after 10000. This work was implemented on Tensorflow, and a total of 11500 train steps were performed. During training, the network was evaluated on validation dataset after every 500 iterations. Accuracy, precision and recall were calculated:
(24) 
(25) 
(26) 
where represent positive vertexes, i.e. vertexes that were predicted to be teeth margin vertexes, present negative vertexes, i.e. background vertexes. represent true positive vertexes, true negative vertexes, false negative vertexes, and false negative vertexes. Validation results were illustrated in Figure 3. Accuracy, precision and recall were all increased rapidly and reached a high level after about 4000 iterations, except that the recall of classifier 1 with baseline architecture dropped during the initial increase of iterations.
The final results were also shown in Table 2, of which the network with best validation results was chosen to be tested on test dataset.
Predicted margin vertexes were shown in Figure 4. Classifier 1, baseline network without expanding layer, worked like threshold segmentation, where areas with high curvature value were predicted as margin vertexes, producing false positive annotations in pit and fissure areas on occlusal surfaces and palatal wrinkle areas on upper jaws. Classifier 7 received raw vertex coordinates as input, roughly outlined the margin areas with obvious patch areas (false positive predictions) and gaps (false negative predictions). Gingiva margin areas were well predicted by classifier 2, 3, 4, 5, 6, 8, although there tend to be few false negative predictions in flat margin areas, and false positive predictions in rugged nonmargin areas.
The accuracy, precision and recall of classifier 18 on validation dataset was shown in Table 2. Without expanding layers, the baseline reached the lowest accuracy of 0.766, as well as the lowest precision of 0.670 and recall of 0.455. The order of accuracy between networks were: , and between features were: . Classifier 6, which is network trained with curvatures and mean neighbor distances (), achieved highest accuracy and precision among all the 8 classifiers. The recall of classifier 6 also reach 0.789, which is only 0.001 lower than the highest one achieved by classifier 3. The classifier 7, taking raw coordinates as train features, got lowest accuracy among classifier 28, but still higher than baseline.
Since classifier 6 performed best among classifier 18, it was selected to predict vertex labels on test dataset. Results shown that the accuracy, precision and recall were 0.877, 0.782 and 0.804, respectively, which was close to the values predicted on the validation dataset (by classifier 6).
4 Discussion
Expanding layer here align vertex features with its neighbor vertex features, and then subjected to convolution layer, multiplied by convolution kernels, to expand the receptive field. The linkage between vertexes defined by the mesh edges and facets were rarely considered in previous studies, which is very important to revel the topological structure of mesh. However, the number of adjacent vertexes varied from point to point, which could not be presented by a matrix as in 2 dimensional images, thus convolution was not available. In this work, adjacent relationship was presented as ring neighbor vertexes linkages, and each ring neighbor vertexes’ features were reduced to mean, equivalent to features of one vertex in number. After expanded the input features to specified number of ring neighbors, a new feature matrix with adjacent feature information was constructed. Then the expanded feature matrix passed through a conv layer to learn the reduced ring neighbor feature pattern, and output feature matrix with the same shape as initial input, which could pass though expanding layer plus conv layer iteratively to enlarge its receptive field.
Expand to further ring neighbors will expand receptive field efficiently, but may miss some nearby information. Classifier 2 with expanding to ring 0, 2 and 4 performed better than classifier 1 (expanding to ring 0, 1 and 2) and classifier 3 (expanding to ring 0, 4 and 8), implies moderate expanding is preferred. What’s more, a combination of different expanding distances in classifier 6 and 7 gained further improvement. After the last 3 expanding plus conv layer blocks replaced by fully connected layers, classifier 7 performed slightly inferior than classifier 6. This might be explained that fully connected layers didn’t concern local mesh vertex topological structures as expanding plus conv layers did. With regarding to the input mesh vertex features, it seems that the curvature and distance features are better than coordinates feature to train our mesh CNN. This may because curvatures and distances have shiftinvariance and rotateinvariance properties, which will not be affected by the inconsistent of pose or position of the teeth scanned.
In this work, negative samples were much more than positive samples (teeth margin vertexes), although we brushed a thick rough line at teeth margin area to mark more vertexes as positive samples, the ratio was still about 1:3 for positive: negative. The loss function was defined upon weighted cross entropy with logits to balance the samples. If not weighted, this unbalanced distribution may affect the performances of our networks, with potential to assign all vertices with negative labels to achieve a local optimum.
There are ways to shrinkage the thick rough teeth margin strip areas to be lines, such as grassfire algorithm, where a continuous gingival strip area without gaps is required. With classifier 6 in this study, there is still few gaps in flat areas. However, this kind of gaps can be easily filled interactivity with little manual labor.
Although convolution of mesh vertex features was implemented in this work, subsampling and upsampling were not covered, which is very important for semantic segmentation of mesh. The subsampling and upsampling will come down to remeshing, which will be investigated in our further work.
5 Conclusion
In this research, we constructed an endtoend neural network which is capable to directly take mesh feature data and vertex connection information as input, and achieved good performance on teeth mesh vertex classification. The expanding layers designed to extract local vertex features can significantly improve the classification result, and the strategy of assembly of expanding layers with different reach sizes can be properly investigated to achieve better performance. Mesh vertex features with shiftinvariance and rotateinvariance properties are preferred to train our mesh CNN.
acknowledgements
This study was funded by the National Natural Science Foundation of China (No. 51705006), and open fund of Shanxi Province Key Laboratory of Oral Diseases Prevention and New Materials (KF202004).
Conflict of interest
The authors declare that they have no conflict of interest.
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