1 Introduction
Estimation of the 3D facial surface and other intrinsic components of the face from single images (e.g., albedo, etc.) is a very important problem at the intersection of computer vision and machine learning with countless applications (e.g., face recognition, face editing, virtual reality). It is now twenty years from the seminal work of Blanz and Vetter [4] which showed that it is possible to reconstruct shape and albedo by solving a nonlinear optimization problem that is constrained by linear statistical models of facial texture and shape. This statistical model of texture and shape is called a 3D Morphable Model (3DMM). Arguably the most popular publicly available 3DMM is the Basel model built from 200 people [21]. Recently, large scale statistical models of face and head shape have been made publicly available [7, 10].
For many years 3DMMs and its variants were the methods of choice for 3D face reconstruction [33, 46, 22]. Furthermore, with appropriate statistical texture models on image features such as Scale Invariant Feature Transform (SIFT) and Histogram Of Gradients (HOG), 3DMMbased methodologies can still achieve stateoftheart performance in 3D shape estimation on images captured under unconstrained conditions [6]. Nevertheless, those methods [6] can reconstruct only the shape and not the facial texture. Another line of research in [45, 34] decouples texture and shape reconstruction. A standard linear 3DMM fitting strategy [41] is used for face reconstruction followed by a number of steps for texture completion and refinement. In these papers [34, 45], the texture looks excellent when rendered under professional renderers (e.g., Arnold), nevertheless when the texture is overlaid on the images the quality significantly drops ^{2}^{2}2Please see the supplementary materials for a comparison with [34, 45]..
In the past two years, a lot of work has been conducted on how to harness Deep Convolutional Neural Networks (DCNNs) for 3D shape and texture reconstruction. The first such methods either trained regression DCNNs from image to the parameters of a 3DMM [42] or used a 3DMM to synthesize images [30, 18]
and formulate an imagetoimage translation problem using DCNNs to estimate the depth
^{3}^{3}3The depth was afterwards refined by fitting a 3DMM and then changing the normals by using image features. [36]. The more recent unsupervised DCNNbased methods are trained to regress 3DMM parameters from identity features by making use of differentiable image formation architectures [9] and differentiable renderers [16, 40, 31].The most recent methods such as [39, 43, 14] use both the 3DMM model, as well as additional network structures (called correctives) in order to extend the shape and texture representation. Even though the paper [39] shows that the reconstructed facial texture has indeed more details than a texture estimated from a 3DMM [42, 40], it is still unable to capture highfrequency details in texture and subsequently many identity characteristics (please see the Fig. 4
). Furthermore, because the method permits the reconstructions to be outside the 3DMM space, it is susceptible to outliers (e.g., glasses etc.) which are baked in shape and texture. Although rendering networks (
i.e. trained by VAE [26]) generates outstanding quality textures, each network is capable of storing up to few individuals whom should be placed in a controlled environment to collect millions of images.In this paper, we still propose to build upon the success of DCNNs but take a radically different approach for 3D shape and texture reconstruction from a single inthewild image. That is, instead of formulating regression methodologies or autoencoder structures that make use of selfsupervision [39, 16, 43], we revisit the optimizationbased 3DMM fitting approach by the supervision of deep identity features and by using Generative Adversarial Networks (GANs) as our statistical parametric representation of the facial texture.
In particular, the novelties that this paper brings are:

We show for the first time, to the best of our knowledge, that a largescale highresolution statistical reconstruction of the complete facial surface on an unwrapped UV space can be successfully used for reconstruction of arbitrary facial textures even captured in unconstrained recording conditions^{4}^{4}4In the very recent works, it was shown that it is feasible to reconstruct the nonvisible parts a UV space for facial texture completion[11] and that GANs can be used to generate novel highresolution faces[38]. Nevertheless, our work is the first one that demonstrates that a GAN can be used as powerful statistical texture prior and reconstruct the complete texture of arbitrary facial images..

We formulate a novel 3DMM fitting strategy which is based on GANs and a differentiable renderer.

We devise a novel cost function which combines various content losses on deep identity features from a face recognition network.

We demonstrate excellent facial shape and texture reconstructions in arbitrary recording conditions that are shown to be both photorealistic and identity preserving in qualitative and quantitative experiments.
2 History of 3DMM Fitting
Our methodology naturally extends and generalizes the ideas of texture and shape 3DMM using modern methods for representing texture using GANs, as well as defines loss functions using differentiable renderers and very powerful publicly available face recognition networks
[12]. Before we define our cost function, we will briefly outline the history of 3DMM representation and fitting.2.1 3DMM representation
The first step is to establish dense correspondences between the training 3D facial meshes and a chosen template with fixed topology in terms of vertices and triangulation.
2.1.1 Texture
Traditionally 3DMMs use a UV map for representing texture. UV maps help us to assign 3D texture data into 2D planes with universal perpixel alignment for all textures. A commonly used UV map is built by cylindrical unwrapping the mean shape into a 2D flat space formulation, which we use to create an RGB image . Each vertex in the 3D space has a texture coordinate in the UV image plane in which the texture information is stored. A universal function exists, where for each vertex we can sample the texture information from the UV space as .
In order to define a statistical texture representation, all the training texture UV maps are vectorized and Principal Component Analysis (PCA) is applied. Under this model any test texture
is approximated as a linear combination of the mean texture and a set of bases as follows:(1) 
where is the texture parameters for the text sample . In the early 3DMM studies, the statistical model of the texture was built with few faces captured in strictly controlled conditions and was used to reconstruct the test albedo of the face. Since, such texture models can hardly represent faces captured in uncontrolled recording conditions (inthewild). Recently it was proposed to use statistical models of handcrafted features such as SIFT or HoG [6] directly from inthewild faces. The interested reader is referred to [5, 32] for more details on texture models used in 3DMM fitting algorithms.
2.1.2 Shape
The method of choice for building statistical models of facial or head 3D shapes is still PCA [23]. Assuming that the 3D shapes in correspondence comprise of vertexes, i.e. . In order to represent both variations in terms of identity and expression, generally two linear models are used. The first is learned from facial scans displaying the neutral expression (i.e., representing identity variations) and the second is learned from displacement vectors (i.e., representing expression variations). Then a test facial shape can be written as
(2) 
where in the mean shape vector, is where the are the bases that correspond to identity variations, and the bases that correspond to expression. Finally, are the shape parameters which can be split accordingly to the identity and expression bases: = [, ].
2.2 Fitting
3D face and texture reconstruction by fitting a 3DMM is performed by solving a nonlinear energy based cost optimization problem that recovers a set of parameters where are the parameters related to a camera model and are the parameters related to an illumination model. The optimization can be formulated as:
(3) 
where is the test image to be fitted and is a vector produced by a physical image formation process (i.e., rendering) controlled by . Finally, is the regularization term that is mainly related to texture and shape parameters.
Various methods have been proposed for numerical optimization of the above cost functions [19, 2]. A notable recent approach is [6] which uses handcrafted features (i.e., ) for texture representation simplified the cost function as:
(4) 
where , is the orthogonal space to the statistical model of the texture and is the set of reduced parameters . The optimization problem in Eq. 4 is solved by GaussNewton method. The main drawback of this method is that the facial texture in not reconstructed.
In this paper, we generalize the 3DMM fittings and introduce the following novelties:

We use a GAN on highresolution UV maps as our statistical representation of the facial texture. That way we can reconstruct textures with highfrequency details.

Instead of other cost functions used in the literature such as lowlevel or loss (e.g., RGB values [29], edges [33]) or handcrafted features (e.g., SIFT [6]), we propose a novel cost function that is based on feature loss from the various layers of publicly available face recognition embedding network [12]. Unlike others, deep identity features are very powerful at preserving identity characteristics of the input image.

We replace physical image formation stage with a differentiable renderer to make use of first order derivatives (i.e., gradient descent). Unlike its alternatives, gradient descent provides computationally cheaper and more reliable derivatives through such deep architectures (i.e., abovementioned texture GAN and identity DCNN).
3 Approach
We propose an optimizationbased 3D face reconstruction approach from a single image that employs a high fidelity texture generation network as statistical prior as illustrated in Fig. 2. To this end, the reconstruction mesh is formed by 3D morphable shape model; textured by the generator network’s output UV map; and projected into 2D image by a differentiable renderer. The distance between the rendered image and the input image is minimized in terms of a number of cost functions by updating the latent parameters of 3DMM and the texture network with gradient descent. We mainly formulate these functions based on rich features of face recognition network [12, 35, 28] for smoother convergence and landmark detection network [13] for alignment and rough shape estimation.
The following sections introduce firstly our novel texture model that employs a generator network trained by progressive growing GAN framework. After describing the procedure for image formation with differentiable renderer, we formulate our cost functions and the procedure for fitting our shape and texture models onto a test image.
3.1 GAN Texture Model
Although conventional PCA is powerful enough to build a decent shape and texture model, it is often unable to capture high frequency details and ends up having blurry textures due to its Gaussian nature. This becomes more apparent in texture modelling which is a key component in 3D reconstruction to preserve identity as well as photorealism.
GANs are shown to be very effective at capturing such details. However, they suffer from preserving 3D coherency [17] of the target distribution when the training images are semialigned. We found that a GAN trained with UV representation of real textures with per pixel alignment avoids this problem and is able to generate realistic and coherent UVs from of its latent space while at the same time generalizing well to unseen data.
In order to take advantage of this perfect harmony, we train a progressive growing GAN [24] to model distribution of UV representations of 10,000 high resolution textures and use the trained generator network
(5) 
as texture model that replaces 3DMM texture model in Eq. 1.
While fitting with linear models, i.e. 3DMM, is as simple as linear transformation, fitting with a generator network can be formulated as an optimization that minimizes perpixel Manhattan distance between target texture in UV space
and the network output with respect to the latent parameter , i.e. .3.2 Differentiable Renderer
Following [16]
, we employ a differentiable renderer to project 3D reconstruction into a 2D image plane based on deferred shading model with given camera and illumination parameters. Since color and normal attributes at each vertex are interpolated at the corresponding pixels with barycentric coordinates, gradients can be easily backpropagated through the renderer to the latent parameters.
A 3D textured mesh at the center of Cartesian origin is projected onto 2D image plane by a pinhole camera model with the camera standing at , directed towards and with the focal length . The illumination is modelled by phong shading given 1) direct light source at 3D coordinates with color values , and 2) color of ambient lighting .
Finally, we denote the rendered image given geometry (), texture (), camera () and lighting parameters ( by the following:
(6) 
where we construct shape mesh by 3DMM as given in Eq. 2 and texture by GAN generator network as in Eq. 5. Since our differentiable renderer supports only color vectors, we sample from our generated UV map to get vectorized color representation as explained in Sec. 2.1.1.
Additionally, we render a secondary image with random expression, pose and illumination in order to generalize identity related parameters well with those variations. We sample expression parameters from a normal distribution as
and sample camera and illumination parameters from the Gaussian distribution of 300W3D dataset as
and . This rendered image of the same identity as (i.e., with same and parameters) is expressed by the following:(7) 
3.3 Cost Functions
Given an input image , we optimize all of the aforementioned parameters simultaneously with gradient descent updates. In each iteration, we simply calculate the forthcoming cost terms for the current state of the 3D reconstruction, and take the derivative of the weighted error with respect to the parameters using backpropagation.
3.3.1 Identity Loss
With the availability of large scale datasets, CNNs have shown incredible performance on many face recognition benchmarks. Their strong identity features are robust to many variations including pose, expression, illumination, age etc. These features are shown to be quite effective at many other tasks including novel identity synthesizing [15], face normalization [9] and 3D face reconstruction [16]. In our approach, we take advantage of an offtheshelf stateoftheart face recognition network [12]^{5}^{5}5We empirically deduced that other face recognition networks work almost equally well and this choice is orthogonal to the proposed approach. in order to capture identity related features of an input face image and optimize the latent parameters accordingly. More specifically, given a pretrained face recognition network consisting of convolutional filters, we calculate the cosine distance between the identity features (i.e., embeddings) of the real target image and our rendered images as following:
(8) 
We formulate an additional identity loss on the rendered image that is rendered with random pose, expression and lighting. This loss ensures that our reconstruction resembles the target identity under different conditions. We formulate it by replacing by in Eq. 8 and it is denoted as .
3.3.2 Content Loss
Face recognition networks are trained to remove all kinds of attributes (e.g. expression, illumination, age, pose) other than abstract identity information throughout the convolutional layers. Despite their strength, the activations in the very last layer discard some of the midlevel features that are useful for 3D reconstruction, e.g. variations that depend on age. Therefore we found it effective to accompany identity loss by leveraging intermediate representations in the face recognition network that are still robust to pixellevel deformations and not too abstract to miss some details. To this end, normalized euclidean distance of intermediate activations, namely content loss, is minimized between input and rendered image with the following loss term:
(9) 
3.3.3 Pixel Loss
While identity and content loss terms optimize albedo of the visible texture, lighting conditions are optimized based on pixel value difference directly. While this cost function is relatively primitive, it is sufficient to optimize lighting parameters such as ambient colors, direction, distance and color of a light source. We found that optimizing illumination parameters jointly with others helped to improve albedo of the recovered texture. Furthermore, pixel loss support identity and content loss with finegrained texture as it supports highest available resolution while images needs to be downscaled to before identity and content loss. The pixel loss is defined by pixel level loss function as:
(10) 
3.3.4 Landmark Loss
The face recognition network is pretrained by the images that are aligned by similarity transformation to a fixed landmark template. To be compatible with the network, we align the input and rendered images under the same settings. However, this process disregards the aspect ratio and scale of the reconstruction. Therefore, we employ a deep face alignment network [13] to detect landmark locations of the input image and align the rendered geometry onto it by updating the shape, expression and camera parameters. That is, camera parameters are optimized to align with the pose of image and geometry parameters are optimized for the rough shape estimation. As a natural consequence, this alignment drastically improves the effectiveness of the pixel and content loss, which are sensitive to misalignment between the two images.
The alignment error is achieved by pointtopoint euclidean distances between detected landmark locations of the input image and 2D projection of the 3D reconstruction landmark locations that is available as metadata of the shape model. Since landmark locations of the reconstruction heavily depend on camera parameters, this loss is great a source of information the alignment of the reconstruction onto input image and is formulated as following:
(11) 
3.4 Model Fitting
We first roughly align our reconstruction to the input image by optimizing shape, expression and camera parameters by: . We then simultaneously optimize all of our parameters with gradient descent and backpropagation so as to minimize weighted combination of above loss terms in the following:
(12)  
where we weight each of our loss terms with parameters. In order to prevent our shape and expression models and lighting parameters from exaggeration to arbitrarily bias our loss terms, we regularize those parameters by .
Input Images  
Ours  
Genova  
[16]  
A.T.Tran et al.  
[42]  
Tewari et al.  
[39]  
Ours  
Geometry  
Tewari et al.  
[39]  
L. Tran et al.  
[43] 
Fitting with Multiple Images (i.e. Video):
While the proposed approach can fit a 3D reconstruction from a single image, one can take advantage of more images effectively when available, e.g. from a video recording. This often helps to improve reconstruction quality under challenging conditions, e.g. outdoor, low resolution. While stateoftheart methods follow naive approaches by averaging either the reconstruction [42] or featurestoberegressed [16] before making a reconstruction, we utilize the power of iterative optimization by averaging identity reconstruction parameters () after every iteration. For an image set , we reformulate our parameters as in which we average shape and texture parameters by the following:
(13) 
4 Experiments
This section demonstrates the excellent performance of the proposed approach for 3D face reconstruction and shape recovery. We verify this by qualitative results in Figures 1, 3, qualitative comparisons with the stateoftheart in Sec. 4.2 and quantitative shape reconstruction experiment on a database with ground truth in Sec. 4.3.
4.1 Implementation Details
For all of our experiments, a given face image is aligned to our fixed template using 68 landmark locations detected by an hourglass 2D landmark detection [13]. For the identity features, we employ ArcFace [12] network’s pretrained models. For the generator network , we train a progressive growing GAN [24] with around 10,000 UV maps from [7] at the resolution of . We use the Large Scale Face Model [7] for 3DMM shape model with and the expression model learned from 4DFAB database [8] with . During fitting process, we optimize parameters using Adam Solver [25] with 0.01 learning rate. And we set our balancing factors as the following: . The Fitting converges in around 30 seconds on an Nvidia GTX 1080 TI GPU for a single image.
4.2 Qualitative Comparison to the Stateoftheart
Fig. 4 compares our results with the most recent face reconstruction studies [40, 39, 16, 42, 43] on a subset of MoFA testset. The first four rows after input images show a comparison of our shape and texture reconstructions to [16, 42, 39] and the last three rows show our reconstructed geometries without texture compared to [39, 43]. All in all, our method outshines all others with its high fidelity photorealistic texture reconstructions. Both of our texture and shape reconstructions manifest strong identity characteristics of the corresponding input images from the thickness and shape of the eyebrows to wrinkles around the mouth and forehead.
Cooperative  Indoor  Outdoor  

Method  Mean Std.  Mean Std.  Mean Std.  
Tran et al. [42]  1.93  0.27  2.02  0.25  1.86  0.23 
Booth et al. [6]  1.82  0.29  1.85  0.22  1.63  0.16 
Genova et al. [16]  1.50  0.13  1.50  0.11  1.48  0.11 
Ours  0.95  0.107  0.94  0.106  0.94  0.106 
Accuracy results for the meshes on the MICC Dataset using pointtoplane distance. The table reports the mean error (Mean), the standard deviation (Std.).
4.3 3D shape recovery on MICC dataset
We evaluate the shape reconstruction performance of our method on MICC Florence 3D Faces dataset (MICC) [1] in Table 1. The dataset provides 3D scans of 53 subjects as well as their short video footages under three difficulty settings: ’cooperative’, ’indoor’ and ’outdoor’. Unlike [16, 42] which processes all the frames in a video, we uniformly sample only 5 frames from each video regardless of their zoom level. And, we run our method with multiimage support for these 5 frames for each video separately as shown in Eq. 13. Each test mesh is cropped at a radius of mm around the tip of the nose according to [42] in order to evaluate the shape recovery of the inner facial mesh. We perform dense alignment between each predicted mesh and its corresponding ground truth mesh, by implementing an iterative closest point (ICP) method [3]
. As evaluation metric, we follow
[16] to measure the error by average symetric pointtoplane distance.Table 1 reports the normalized pointtoplain errors in millimeters. It is evident that we have improved the absolute error compared to the other two stateoftheart methods by . Our results are shown to be consistent across all different settings with minimal standard deviation from the mean error.
4.4 Ablation Study
Fig. 5 shows an ablation study on our method where the full model reconstructs the input face better than its variants, something that suggests that each of our components significantly contributes towards a good reconstruction. Fig. 5(c) indicates albedo is well disentangled from illumination and our model capture the light direction accurately.
While Fig. 5(df) shows each of the identity terms contributes to preserve identity, Fig. 5(h) demonstrates the significance identity features altogether. Still, overall reconstruction utilizes pixel intensities to capture better albedo and illumination as shown in Fig. 5(g). Finally, Fig. 5(i) shows the superiority of our textures over PCAbased ones.
5 Conclusion
In this paper, we revisit optimizationbased 3D face reconstruction under a new perspective, that is, we utilize the power of recent machine learning techniques such as GANs and face recognition network as statistical texture model and as energy function respectively.
To the best of our knowledge, this is the first time that GANs are used for model fitting and they have shown excellent results for high quality texture reconstruction. The proposed approach shows identity preserving high fidelity 3D reconstructions in qualitative and quantitative experiments.
Acknowledgements:
Baris Gecer is funded by the Turkish Ministry of National Education. Stefanos Zafeiriou acknowledges support by EPSRC Fellowship DEFORM (EP/S010203/1) and a Google Faculty Award.
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Appendix A Experiments on LFW
In order to evaluate identity preservation capacity of the proposed method, we run two face recognition experiments on Labelled Faces in the Wild (LFW) dataset [20]. Following [16], we feed real LFW images and rendered images of their 3D reconstruction by our method to a pretrained face recognition network, namely VGGFace[27]
. We then compute the activations at the embedding layer and measure cosine similarity between 1) real and rendered images and 2) renderings of same/different pairs.
Appendix B More Qualitative Results
Figures 8, 9, 10, and 11 illustrate the reconstructions of our method under different settings in comparison to the other stateoftheart methods. Please see figure captions for detailed explanation.
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