1 Introduction
Twenty years ago, Blanz and Vetter demonstrated a remarkable achievement [2]. They showed that it is possible to reconstruct 3D facial geometry from a single image. This was possible by solving a nonlinear optimization problem whose solution space was confined by a linear statistical model of the 3D facial shape and texture, the socalled 3D Morphable Model (3DMM). Methods based on 3DMMs are still among the stateoftheart for 3D face reconstruction, even from images captured inthewild [6, 4, 5].
During the past two years, a lot of works have been conducted on how to harness the power of Deep Convolutional Neural Networks (DCNNs) for 3D shape and texture estimation from 2D facial images. The first such methods either trained regression DCNNs from image to the parameters of a 3DMM
[36]or used a 3DMM to synthesize images and formulate an imagetoimage translation problem in order to estimate the depth, using DCNNs
[31]. The recent, more sophisticated, DCNNbased methods were trained using selfsupervised techniques [17, 37, 38] and made use of differentiable image formation architectures and differentiable renderers [17]. The most recent methods such as [37, 38] and [34] used selfsupervision to go beyond the standard 3DMMs in terms of texture and shape. In particular, [34] used both the 3DMMs model, as well as additional network structures (called correctives) that can capture information outside the space of 3DMMs, in order to represent the shape and texture. The method in [37, 38] tried to learn nonlinear spaces (i.e., decoders, which are called nonlinear 3DMMs) of shape and texture directly from the data. Nevertheless, in order to avoid poor training performance, these methods used 3DMMs fittings for the model pretraining.In all the above methods the 3DMMs, linear or nonlinear in a form of a decoder, were modelled with either fully connected nodes [36] or, especially in the texture space, with 2D convolutions on unwrapped UV space [37, 38]
. In this paper, we take a radically different direction. That is, motivated by the line of research on Geometric Deep Learning (GDL), a field that attempts to generalize DCNNs to nonEuclidean domains such as graphs/manifolds/meshes
[33, 12, 21, 7, 27], we make the first attempt to develop a nonlinear 3DMM, that describes both shape and texture, by using mesh convolutions. Apart from being more intuitive defining nonlinear 3DMMs using mesh convolutions, their major advantage is that they are defined by networks that have a very small number of parameters and hence can have very small computational complexity. In summary, the contributions of our paper are the following:
[leftmargin=20pt]

We present the first, to the best of our knowledge, nonlinear 3DMM using mesh convolutions. The proposed method decodes both shape and texture directly on the mesh domain with a compact model size (MB) and amazing efficiency (over 2500 FPS on CPU). This decoder is different from the recently proposed decoder in [27] which only decodes 3D shape information.

We propose an encoderdecoder structure that reconstructs the texture and shape directly from an inthewild 2D facial image. Due to the efficiency of the proposed Coloured Mesh Decoder (CMD), our method can estimate the 3D shape over FPS (for the entire system).
2 Related Work
In the following, we briefly touch upon related topics in the literature such as linear and nonlinear 3DMM representations.
Linear 3D Morphable Models. For the past two decades, the method of choice for representing and generating 3D faces was Principal Component Analysis (PCA). PCA was used for building statistical 3D shape models (i.e., 3D Morphable Models (3DMMs)) in many works [2, 3, 29]. Recently, PCA was adopted for building largescale statistical models of the 3D face [6] and head [11]. It is very convenient for representing and generating faces to decouple facial identity variations from expression variations. Hence, statistical blend shape models were introduced representing only the expression variations using PCA [22, 9]. The original 3DMM [2] used a PCA model for also describing the texture variations. Nevertheless, this is quite limited in describing the texture variability in image captured inthewild conditions.
Nonlinear 3D Morphable Models. In the past year, the first attempts for learning nonlinear 3DMMs were introduced [37, 38, 34]. These 3DMMs can be regarded as decoders that use DCNNs, coupled with an imageencoder. In particular, the method [34] used selfsupervision to learn a new decoder with fullyconnected layers that combined a linear 3DMM with new structures that can reconstruct arbitrary images. Similarly, the methods [37, 38] used either fully connected layers or 2D convolutions on a UV map for decoding the shape and texture.
All the above methods used either fully connected layers or 2D convolutions on unwrapped spaces to define the nonlinear 3DMM decoders. However, these methods lead to deep networks with a large number of parameters and do not exploit the local geometry of the 3D facial structure. Therefore, decoders that use convolutions directly in the nonEuclidean facial mesh domain should be built. The field of deep learning on nonEuclidean domains, also referred to as Geometric Deep Learning [7], has recently gained some popularity. The first works included [23] that proposed the socalled MeshVAE which trains a VariationalAutoEncoder (VAE) using convolutional operators from [39] and CoMA [27] that used a similar architecture with spectral Chebyshev filters [12] and additional spatial pooling to generate 3D facial meshes. The authors demonstrated that CoMA can represent better faces with expressions than PCA in a very small dimensional latent space of only eight dimensions.
In this paper, we propose the first autoencoder that directly uses mesh convolutions for joint texture and shape representation. This brings forth a highly effective and efficient coloured mesh decoder which can be used for 3D face reconstruction for inthewild data.
3 Proposed Approach
3.1 Coloured Mesh AutoEncoder
Mesh Convolution. We define our mesh autoencoder based on the undirected and connected graphs , where is a set of vertices containing the joint shape (e.g. x, y, z) and texture (e.g. r, g, b) information, and is an adjacency matrix encoding the connection status between vertices.
Following [12, 26], the nonnormalized graph Laplacian is defined as where is the diagonal matrix with and the normalized definition is where
is the identity matrix. The Laplacian
can be diagonalized by the Fourier bases such that where. The graph Fourier transform of our face representation
is then defined as , and its inverse as .The operation of the convolution on a graph can be defined by formulating mesh filtering with a kernel using a recursive Chebyshev polynomial [12, 26]. The filter can be parameterized as a truncated Chebyshev polynomial expansion of order ,
(1) 
where
is a vector of Chebyshev coefficients and
is the Chebyshev polynomial of order evaluated at a scaled Laplacian . can be recursively computed by with and .The spectral convolution can be defined as
(2) 
where is the input and is the output. The entire filtering operation is very efficient and only costs operations.
Mesh Downsampling and Upsampling. We follow [26] to employ a binary transformation matrix to perform downsampling of a mesh with vertices and conduct upsampling using another transformation matrix .
is calculated by iteratively contracting vertex pairs under the constraint of minimizing quadric error [15]. During downsampling, we store the barycentric coordinates of the discarded vertices with regard to the downsampled mesh so that the upsampling step can add new vertices with the same barycentric locations information.
For upsampling, vertices directly retained during the downsampling step undergo convolutional transformations. Vertices discarded during downsampling are mapped into the downsampled mesh surface using recorded barycentric coordinates. The upsampled mesh with vertices is efficiently predicted by a sparse matrix multiplication, .
3.2 Coloured Mesh Decoder intheWild
The nonlinear 3DMM fitting inthewild is designed in an unsupervised/selfsupervised manner. As we are able to construct joint shape & texture bases with the coloured mesh autoencoder, the problem can be treated as a matrix multiplication between the bases and the optimal coefficients that reconstruct the 3D face. From the perspective of a neural network, this can be viewed as an image encoder that is trained to regress to the 3D shape and texture, noted as . As shown in Fig. 2, a 2D convolution network is used to encode inthewild images followed by a mesh decoder , whose weights are shared across the decoder [10] in the mesh autoencoder. However, the output of the joint shape & texture decoder is a coloured mesh within a unit sphere. Like linear 3DMM [4], a camera model is required to project the 3D mesh from the objectcentered Cartesian coordinates into an image plane in the same Cartesian coordinates.
Projection Model. We employ a pinhole camera model in this work, which utilizes a perspective transformation model. The parameters of the projection operation can be formulated as following:
(3) 
where represent camera position, orientation and upright direction, respectively, in Cartesian coordinates. is the field of view (FOV) that controls the perspective projection. We also concatenate lighting parameters together with camera parameters as rendering parameters that will be predicted by the image encoder. Three point light sources and constant ambient light are assumed, to a total of 12 parameters for lighting. For abbreviation, we represent the rendering parameter as a vector of size 22 and the projection model as the function .
Differentiable Renderer. To make the network endtoend trainable, we incorporated a differentiable renderer [17] to project the output mesh onto the image plane . The
norm is pixelwisely calculated as the loss function. The renderer, also known as rasterizer, generates barycentric coordinates and corresponding triangle IDs for each pixel at the image plane. The rendering procedure involves Phong shading
[25]and interpolating according to the barycentric coordinates. Also, camera and illumination parameters are computed in the same framework. The whole pipeline is able to be trained endtoend with the loss gradients backpropagated through the differentiable renderer.
Losses. We have formulated a loss function applied jointly to undercontrolled coloured mesh autoencoder and inthewild coloured mesh decoder, thus enabling supervised and selfsupervised endtoend training. It is formulated as below:
(4) 
Where the objective function:
(5) 
is applied to enforce shape and texture reconstruction of the coloured mesh autoencoder, in which and norms are applied on shape and texture , respectively. The term:
(6) 
represents the pixelwise reconstruction error for inthewild images when applying a mask to only visible facial pixels. We use and gradually increase to during training.
4 Experimental Results
4.1 Datasets
We train our method using both undercontrolled data (3DMD [13]) and inthewild data (300WLP [40] and CelebA [24]). The 3DMD dataset [13] contains around k raw scans of 3,564 unique identities with expression variations. The 300WLP dataset [40] consists of about k large pose facial data, which are synthetically generated by the profiling method of [40]. The CelebA dataset [24] is a largescale face attributes dataset with more than k celebrity images, which cover large pose variations and background clutter. Each training image is cropped to bounding boxes of indexed 68 facial landmarks with random perturbation to simulate a coarse face detector.
We perform extensive qualitative experiments on AFLW20003D [40], 300VW [30] and CelebA testset [24]. We also conducted quantitative comparisons with prior works on FaceWarehouse [8] and Florence [1], where accurate 3D meshes are available for evaluation. FaceWarehouse is a 3D facial expressions database collected by a Kinect RGBD camera. 150 candidates aged from 7 to 80 of various ethnic groups are involved. Florence is a 3D face dataset that contains 53 subjects with their ground truth 3D meshes acquired from a structuredlight scanning system.
4.2 Implementation Details
Network Architecture. Our architecture consists of four submodules as shown in Fig. 2, named Image Encoder [37, 38], Coloured Mesh Encoder [26], a shared Coloured Mesh Decoder [26] and a differentiable rendering module [17]. The image encoder part takes input images of shape followed by 10 convolution layers. It reduces the dimension of the input images to and applies a fully connected layer that constructs a
dimension embedding space. Every convolutional layer is followed by a batch normalization layer and a ReLU activation layer. The kernel size of all convolution layers is 3 and the stride is 2 for any downsampling convolution layer. The coloured mesh decoder takes an embedding of size
and decodes to a coloured mesh of size (3 shape and 3 texture channels). The encoder/decoder consists of 4 geometric convolutional filters [26], each one of which is followed by a down/upsampling layer that reduces/increases the number of vertices by 4 times. Every graph convolutional layer is followed by a ReLU activation function similar to those in the image encoder.
Training Details.
Both (1) the undercontrolled coloured mesh autoencoder and (2) the inthewild coloured mesh decoder are jointly trained endtoend although each one uses a different data source. Both models are trained with Adam optimizer with a start learning rate of 1e4. A learning rate decay is applied with the rate at 0.98 of each epoch. We train the model for 200 epochs. We perturb the training image with a random flipping, random rotation, random scaling and random cropping to the size of
from a input.4.3 Ablation Study on Coloured Mesh AutoEncoder
Reconstruction Capacity. We compare the power of linear and nonlinear 3DMMs in representing realworld 3D scans with different embedding dimensions to emphasize the compactness of our coloured mesh decoder. Here, we use of 3D face scans from the 3DMD dataset as the test set.
As illustrated in the top of Fig. 3, we compare the visual quality of reconstruction results produced by linear and nonlinear models. To quantify the results of shape modelling, we use the Normalized Mean Error (NME), which is the averaged pervertex errors between the groundtruth shapes and the reconstructed shapes normalized by interocular distances. For evaluation of texture modelling, we employ the pixelwise Mean Absolute Error (MAE) between the groundtruth and reconstructed texture.
As shown in Tab. 1, our nonlinear shape model has a significantly smaller shape reconstruction error than the linear model. Moreover, the joint nonlinear model notably reduces the reconstruction error even further, indicating that integrating texture information is helpful to constrain the deformation of vertices. For the comparison on the texture reconstruction, a slightly higher reconstruction error of texture is expected as the missing texture information between vertices was interpolated in our model, while a linear model has the full texture information.
Shape  Texture  

PCA =64  
PCA =128  
PCA =185  
=64    
=128    
=256    
=64  
=128  
=256 
Attribute Embedding. To get a better understanding of different faces embedded in our coloured mesh decoder, we investigate the semantic attribute embedding. For a given attribute, e.g., smile, we feed the face data (shape and texture) with that attribute into our coloured mesh encoder to obtain the embedding parameters , which represent corresponding distributions of the attribute in the low dimensional embedding space. Taking the mean parameters as input to the trained coloured mesh decoder, we can reconstruct the mean shape and texture with that attribute. Based on the principal component analysis on the embedding parameters , we can conveniently use one variable (principal component) to change the attribute. Fig. 3 shows some 3D shapes with texture sampled from the latent space. Here, we can observe that the power of our nonlinear coloured mesh decoder is excellent at modelling expressions, illuminations and even beards with a tight embedding dimension ().
4.4 Coloured Mesh Decoder Applied Inthewild
4.4.1 3D Face Alignment
Since our method can model shape and texture simultaneously, we apply it for 3D morphable fitting in the wild and test the performance on the task of sparse 3D face alignment. We compare our model with the most recent stateoftheart methods, e.g. 3DDFA [40], N3DMM [37] and PRNet [14] on the AFLW20003D [40] dataset. The accuracy is evaluated by the Normalized Mean Error (NME), that is the average of landmark error normalized by the bounding box size on three pose subsets [40].
Method  3DDFA[40]  N3DMM [38]  PRNet [14]  CMD 
NME  5.42  4.12  3.62  3.98 
3DDFA [40] is a cascade of CNNs that iteratively refines its estimation in multiple steps. N3DMM [38] utilizes the 2D deep convolutional neural networks to build a nonlinear 3DMM on the UV position and texture maps, and fits the unconstrained 2D inthewild face images in a weakly supervised way. By contrast, our method employs the coloured mesh decoder to build the nonlinear 3DMM. Our model not only has better performance but also has a more compact model size and a more efficient running time. PRNet [38] employs an encoderdecoder neural network to directly regress the UV position map. The performance of our method is slightly worse than PRNet majorly due to the complexity of the network.



In Fig. 4, we give some exemplary alignment results, which demonstrate successful sparse 3D face alignment results under extreme poses, exaggerated expressions, heavy occlusions and variable illuminations. We also see that the dense shape (vertices) predictions are also very robust in the wild, which means that for any kind of facial landmark configuration our method is able to give accurate localization results if the landmark correspondence with our shape configuration is given.
4.4.2 3D Face Reconstruction
We first qualitatively compare our approach with five recent stateoftheart 3D face reconstruction methods: (1) 3DMM fitting networks learned in a supervised way (Sela et al. [31]), (2) 3DMM fitting networks learned in an unsupervised way named MoFA (Tewari et al. [35]), (3) a direct volumetric CNN regression approach called VRN (Jackson et al. [19]), (4) a direct UV position map regression method named PRNet (Feng et al. [14]), (5) a nonlinear 3DMM fitting networks learned in weakly supervised fashion named N3DMM (Tran et al. [38]). As PRNet and N3DMM both employ 2D convolution networks on the UV position map to learn the shape model, we view PRNet and N3DMM as the closest baselines to our method.
Input 
Sela [31]  PRNet [14]  N3DMM [38]  CMD  













Comparison to Sela et al. [31]. Their elementary imagetoimage network is trained on synthetic data generated by the linear model. Due to the domain gap between synthetic and real images, the network output tends to be unstable on some occluded regions for the inthewild testing (Fig. 5), which leads to failure in later steps. By contrast, our coloured mash decoder is trained on the realworld unconstrained dataset in an endtoend selfsupervised fashion, thus our model is robust in handling the inthewild variations. In addition, the method of Sela et al. [31] requires a slow offline nonrigid registration step (s) to obtain a holefree reconstruction from the predicted depth map. Nevertheless, the proposed coloured mesh decoder can run extremely fast. Furthermore, our method is complementary to Sela et al. [31]’s fine detail reconstruction module. Employing Shape from Shading (SFS) [20] to refine our fitting results could lead to better results with details.
Input 
MoFA [35]  PRNet [14]  N3DMM [38]  CMD  













Comparison to MoFA [35]. The monocular 3D face reconstruction method, MoFA, proposed by Tewari et al. [35], employs an unsupervised fashion to learn 3DMM fitting in the wild. However, their reconstruction space is still limited to the linear bases. Hence, their reconstructions suffer from unnatural surface deformations when dealing with very challenging texture,i.e. beard, as shown in Fig. 6. By contrast, our method employs a nonlinear coloured mesh decoder to jointly reconstruct shape and texture. Therefore, our method can achieve highquality reconstruction results even under hairy texture.
Input  VRN [19]  PRNet [14]  N3DMM [38]  CMD 








Comparison to VRN [19]. We also compare our approach with a direct volumetric regression method proposed by Jackson et al. [19]. VRN directly regresses a 3D shape volume via an encoderdecoder network with skip connection (i.e. Hourglass structure) to avoid explicitly using a linear 3DMM prior. This strategy potentially helps the network to explore a larger solution space than the linear model. However, this method discards the correspondence between facial meshes and the regression target is very large in size. Fig. 7 shows a visual comparison of 3D face reconstructions between VRN and our method. In general, VRN can robustly handle inthewild texture variations. However, due to the volumetric shape representation, the surface is not smooth and does not preserve details. By contrast, our method directly models shape and texture of vertices, thus the model size is more compact and the output results are more smooth.
Besides qualitative comparisons with stateoftheart 3D face reconstruction methods, we also conducted quantitative comparisons on the FaceWarehouse dataset [8] and the Florence dataset [1] to show the superiority of the proposed coloured mesh decoder.
FaceWarehouse. Following the same setting in [35, 38], we also quantitatively compared our method with prior works on 9 subjects from the FaceWarehouse dataset [8]. Visual and quantitative comparisons are illustrated in Fig. 8. We achieved comparable results with Garrido et al. [16] and N3DMM [38], while surpassing all other regression methods [36, 28, 35]. As shown on the right side of Fig. 8, we can easily infer the expression of these three samples from their coloured vertices.
Florence. Following the same setting in [19, 14], we also quantitatively compared our approach with stateoftheart methods (e.g. VRN [19] and PRNet [14]) on the Florence dataset [1]. The face bounding boxes were calculated from the ground truth point cloud and the face images were cropped and used as the network input. Each subject was rendered with different poses as in [19, 14]: pitch rotations of , and and raw rotations between and . We only chose the common face region to compare the performance. For evaluation, we first used the Iterative Closest Points (ICP) algorithm to find the corresponding nearest points between our model output and ground truth point cloud and then calculated Mean Squared Error (MSE) normalized by the interocular distance of 3D coordinates.
Fig. 9(a) shows that our method obtained comparable results with PRNet. To better evaluate the reconstruction performance of our method across different poses, we calculated the NME under different yaw angles. As shown in Fig. 9(b), all the methods obtain good performance under the near frontal view. However, 3DDFA and VRN fail to keep low error as the yaw angle increases. The performance of our method is relatively stable under pose variations and comparable with the performance of PRNet under profile views.
4.5 Running Time and Model Size Comparisons
Time  Size  
Method  E  D  E  D 
Sela et al. [31]  ms  G  
VRN [19]  ms  G  
PRNet [14]  ms  M  
MoFA [35]  ms  ms  M  M 
N3DMM [38]  ms  ms  M  M 
PCA Shape  ms  ms  M  
PCA Texture  ms  ms  M  
CMD (=256)  ms  ms  M  M 
In Tab. 3, we compare the running time and the model size for multiple 3D reconstruction approaches. Since some methods were not publicly available [31, 35, 38], we only provide an approximate estimation for them. Sela et al. [31], VRN [19] and PRNet [14] all use an encoderdecoder network with similar running time. However, Sela et al. [31] requires an expensive nonrigid registration step as well as a refinement module.
Our method gets a comparable encoder running time with N3DMM [38] and MoFA [35]. However, N3DMM [38] requires decoding features via two CNNs for shape and texture, respectively. MoFA [35] directly uses liner bases, and the decoding step is a single multiplication around ms for 28K points. By contrast, the proposed coloured mesh decoder only needs one efficient mesh convolution network. On CPU (Intel i97900X@3.30GHz), our method can complete coloured mesh decoding within 0.367 ms (2500FPS), which is even faster than using linear shape bases. The model size of our nonlinear coloured mesh decoder (M) is almost oneseventh of the liner shape bases (MB) employed in MoFA. Most importantly, the capacity of our nonlinear mesh decoder is much higher than that of the linear bases as proved in the above experiments.
5 Conclusions
In this paper, we presented a novel nonlinear 3DMM method using mesh convolutions. Our method decodes both shape and texture directly on the mesh domain with compact model size (MB) and very low computational complexity (over 2500 FPS on CPU). Based on the mesh decoder, we propose an image encoder plus a coloured mesh decoder structure that reconstruct the texture and shape directly from an inthewild 2D facial image. Extensive qualitative visualization and quantitative reconstruction results confirm the effectiveness of the proposed method.
6 Acknowledgements
Stefanos Zafeiriou acknowledges support from EPSRC Fellowship DEFORM (EP/S010203/1) and a Google Faculty Fellowship. Jiankang Deng acknowledges insightful advice from friends (e.g. Sarah Parisot, Yao Feng, Luan Tran and Grigorios Chrysos), financial support from the Imperial President’s PhD Scholarship, and GPU donations from NVIDIA.
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