Introduction
Face alignment aims to estimate a set of facial landmarks given a face image or video sequence. It is a classic computer vision problem that has attributed to many advanced machine learning algorithms
Fan et al. (2018); Bulat and Tzimiropoulos (2017); Trigeorgis et al. (2016); Peng et al. (2015, 2016); Kowalski, Naruniec, and Trzcinski (2017); Chen et al. (2017); Liu et al. (2017); Hu et al. (2018). Nowadays, with the rapid development of consumer hardwares (e.g., mobile phones, digital cameras), HighResolution (HR) video sequences can be easily collected. Estimating facial landmarks on such highresolution facial data has tremendous applications, e.g., face makeup Chen, Shen, and Jia (2017), editing with special effects Korshunova et al. (2017) in live broadcast videos. However, most existing face alinement methods work on faces with medium image resolutions Chen et al. (2017); Bulat and Tzimiropoulos (2017); Peng et al. (2016); Liu et al. (2017). Therefore, developing face alignment algorithms for highresolution videos is at the core of this paper.To this end, we propose an accurate and stable algorithm for highresolution videobased face alignment, named Fractional Heatmap Regression (FHR). It is well known that heatmap regression have shown its effectiveness in landmark estimation tasks Chen et al. (2017); Newell, Yang, and Deng (2016); Chen* et al. (2018). However, Conventional Heatmap Regression (CHR) is not accurate nor stable when dealing with highresolution facial images, since it finds the maximum activated location in heatmaps which are generated from rounding coordinates, and thus leads to quantization errors as the heatmap resolution is much lower than the input image resolution (e.g., vs. shown in Fig. 1) due to the scaling operation. To address this problem, we propose a novel transformation between heatmaps and coordinates, which not only preserves the fractional part when generating heatmaps from the coordinates, but also accurately estimate the fractional part according to the D Gaussian function by sampling three points in heatmaps.
Using our proposed FHR, we can estimate more accurate landmarks compared to the conventional heatmap regression model, and achieve stateoftheart performance on popular video benchmarks: Talking Face FGNET (2014) and VW datasets Shen et al. (2017) compared to recent videobased face alignment models Liu et al. (2017); Peng et al. (2016). However, realworld applications such as face makeup in videos often demands extremely high stability, since the makeup jumps if the estimated landmarks oscillate between consecutive frames, which negatively impacts the user’s experience. To make the sequential estimations as stable as possible, we further develop a novel stabilization algorithm on the landmarks estimated by FHR, which contains two terms, a regularization term () and a temporal coherence term (), to address two common difficulties: time delay and nonsmooth problems, respectively. Specifically, combines traditional Euclidean loss and a novel loss account for time delay; generalizes the temporal coherence loss in Cao, Hou, and Zhou (2014) to better handle nonlinear movement of facial landmark.
In summary, the main contributions of this paper are:

A novel Fractional Heatmap Regression method for highresolution video based face alignment that leverages D Gaussian prior to preserve the fractional part of points.

A novel stabilization algorithm that addresses time delay and nonsmooth problems among continuous video frames is proposed.
Related Work
Heatmap Regression
Heatmap regression is one of the most widely used approaches for landmark localization tasks, which estimates a set of heatmaps rather than coordinates. Stacked Hourglass Networks (SHN) are popular architectures in heatmap regression, which have symmetric topology that capture and consolidate information across all scales of the image. Newell et al. Newell, Yang, and Deng (2016) proposed SHN for D human pose estimation, which achieved remarkable results even for very challenging datasets Andriluka et al. (2014). With the hourglass structure, Chu et al. Chu et al. (2017)
introduced multicontext attention mechanism into Convolutional Neural Networks (CNN). Apart from applications in human pose estimation, there are also several heatmap regression based models for face alignment. Chen et al.
Chen et al. (2017) proposed a structureaware fully convolutional network to implicitly model the priors during training. Bulat et al. Bulat and Tzimiropoulos (2017) built a powerful CNN for face alignment based on the hourglass network and a hierarchical, parallel and multiscale block.However, all existing models drops the fractional part of coordinates during the transformation between heatmaps and points, which brings quantization errors to highresolution facial images. On the contrary, our proposed FHR can accurately estimate the fractional part according to the D Gaussian function by sampling three points in heatmaps, and thus achieves more accurate alignment.
Videobased Face Alignment
Videobased face alignment estimates facial landmarks in video sequences Liu (2010). Early methods Black and Yacoob (1995); Shen et al. (2017) used incremental learning to predict landmarks on still frames in a trackingbydetection manner. To address the issue that generic methods are sensitive to initializations, Peng et al. Peng et al. (2015) exploited incremental learning for personalized ensemble alignment, which samples multiple initial shapes to achieve image congealing within one frame. To explicitly model the temporal dependency Oh et al. (2015) of landmarks across frames, the authors Peng et al. (2016) further incorporated a sequence of spatial and temporal recurrences for sequential face alignment in videos. Recently, Liu et al. Liu et al. (2017)
proposed a TwoStream Transformer Networks (TSTN) approach, which captures the complementary information of both the spatial appearance on still frames and the temporal consistency across frames. Different from
Peng et al. (2016); Liu et al. (2017) that require temporal landmark labels across frames, our proposed method achieves stateoftheart accuracy only by making full use of the spatial appearance on still frames, which is able to remedy the problem that labeled sequential data are very limited.Apart from the accuracy of landmarks, stabilization is also a key metric to videobased alignment. Typically, two terms Cao, Hou, and Zhou (2014) are adopted for stabilization, where a regularization term drives the optimized results to be more expressive, and a temporal coherence term drives the results to be more stable and smooth. However, the existing stabilization algorithm is sensitive to time delay and nonlinear movements. Our proposed algorithm takes these into account and thus are overall more robust.
The Proposed Approach
In this section, we introduce the details of the proposed approach based on heatmap regression. A key point of heatmap regression is the transformation between the heatmaps and coordinates. Specifically, before model training, a preprocess step is conducted to convert the coordinates to the heatmaps, which are used as the ground truth labels. After estimating the heatmaps, a postprocess step is conducted to obtain the coordinates from the estimated heatmaps. In this work, we propose a novel transformation between the heatmaps and coordinates that is different from the conventional heatmap regression, which is demonstrated to be simple yet effective.
Fractional Heatmap Regression
As shown in Fig. 1, conventional heatmap regression mainly generates the heatmaps from integral coordinates, despite that the ground truth coordinates are usually with fractions.As a result, it causes quantization errors when scaling back to the original image resolution, since the heatmaps are usually of much lower resolution than the input image. To address this problem, our proposed fractional heatmap regression generates ground truth heatmaps based on the intact ground truth coordinates (see Fig. 1) as follows:
(1) 
where represents the coordinate, is the domain of the heatmap ,
denotes the standard deviation and
is the center of the D Gaussian in the th heatmap.Denoting an input image and the deep alignment model, we can estimate the recovered heatmaps by
(2) 
where , and is the number of landmarks. Given Eq. (1) and , estimating the fractional coordinate amounts to solving a binary quadratic equation, which has a closedform solution as long as we can sample any three nonzero points from the heatmap. Specifically, we first obtain as the location with the maximum likelihood, as in conventional heatmap regression:
(3) 
Conventional heatmap regression directly takes as the output, which loses the fractional part. In our method, we further sample another two points, e.g., near ^{1}^{1}1In case is located at the edge of the heatmap, we would sample the points in the opposite directions.. Let , and , we then estimate the fractional coordinate as follows:
(4)  
It should be noted that our fractional heatmap regression is applicable to any heatmap based methods. In this paper, we focus on face alignment, and adopt the stacked hourglass network Newell, Yang, and Deng (2016) as the alignment model that minimizes the loss of across the entire training set, where .
Stabilization Algorithm
We now introduce our stabilization algorithm for videobased alignment, which takes the alignment results of in all the past frames as input, and outputs a more accurate and stable result for the current frame.
Stabilization Model
We denote as the output of at frame and the stabilization model as , which has parameters to be optimized. takes as input, and outputs the stabilized landmarks of frame , which is denoted as . Therefore we have
(5) 
Assume there are videos in the training set, and the th video has frames. For frame in the th video, we denote its ground truth landmarks as , the output of as , and the stabilized output as (). Here, we have .
Next, we present the specific form of as well as its parameters . Our model follows a Bayesian framework. Specifically, we model the prior distribution of given as a component Gaussian mixture:
(6) 
where indicates the density function,
indicates a normal distribution with mean
and covariance . We then model the likelihood of given as Gaussian(7) 
and use the Bayesian rule to obtain the most probable value of
:(8)  
Combining (6),(7) and (8), we can obtain the closedform solution of (5). In practice, we fix since it already achieves satisfactory results and larger may cause both efficiency and overfitting issues. Moreover, to reflect the fact that has a decreasing correlation with when increases, we assume that
(9)  
where along with positive semidefinite matrices and are unknown model parameters. In practice Eq. (9) can be calculated recursively, whose computational complexity remains constant when increases.
To further reduce the number of parameters, we calculate the covariance matrix of all
in the training set, denote the matrix of eigenvectors of
as , and finally assume(10) 
where and are diagonal matrices. In summary, we have
, which are optimized with the loss function in the next subsection.
Methods  SDM (2013)  CFAN (2014)  CFSS (2015)  MDM (2016)  DAN (2017)  GAN (2017)  CHR (2016)  FHR 

NRMSE  
AUC  
Failure 
Loss Function Design
We now introduce and optimize a novel loss function so as to estimate the above stabilization model parameter . Throughout this section, we denote all as to emphasize that the stabilized landmarks are functions of model parameter . Our loss function is defined as follows:
(11) 
This loss has two terms. The first term, is called the regularization loss, which regularizes the stabilized output to be close to the ground truth, and is defined as follows:
(12) 
where
indicates the MoorePenrose general inverse of vector/matrix
. We can see that the first term of (12) is the average Euclidean distance of the ground truth and the model output. The second term aims to fit every in terms of , where is the coefficient, and estimate the expectation of .If this expectation is large, it means that 1) is more similar to than , and 2) the model output has a significant time delay, which is undesirable. Since our stabilization model uses the alignment results of the past frames, how to avoid time delay is a critical task. Therefore we emphasize the time delay loss as an individual term in (see Fig. 2).The second term in (11), is called the smooth loss which favors the stabilized output to be smooth, and is defined as follows:
(13) 
where and . can be seen as a tradeoff between two stability losses, controlled by . When , it is equivalent to
(14) 
which is the average distance from to the line .
On the other hand, when , is equivalent to
(15) 
which is the average distance from to the midpoint of the line . The tradeoff smooth loss will cause the loss contour of to have an ellipse with long axis (see Fig. 2), which we argue is a more reasonable indicator of the smoothness of the movement trajectory .
With the stabilization model and the loss function, we can estimate the model parameters using the standard optimization method Lagarias et al. (1998). The proposed stabilization algorithm is trained by various videos, which learns the statistics of different kinds of movements and thus is more robust than the traditional stabilization model Cao, Hou, and Zhou (2014). The optimization process converges within hours on a conventional laptop.
Experiments
Experimental Setup
Datasets
We conduct extensive experiments on both image and videobased alignment datasets, including W Sagonas et al. (2013), VW Shen et al. (2017) and Talking Face (TF) FGNET (2014). To test on images of W private set, we follow Chen et al. (2017) to use use training images from LFPW, HELEN and AFW datasets. To test on VW, we follow Shen et al. (2017) to use videos for training and the rest videos for testing. Specifically, the videos are divided to three categories: welllit (Scenario), mild unconstrained (Scenario) and challenging (Scenario) according to the difficulties. To test on frames of TF dataset, we follow Liu et al. (2017) to use the model trained by the training set of VW dataset.
Face scale  Ave. interocular dis.  CHR/FHR (RMSE ) 

Very small  / ( )  
Small  / ( )  
Medium  / ( )  
Large  / ( )  
Very large  / ( ) 
Training Setting
Training faces are cropped using the detection bounding boxes, and scaled to pixels. Following Chen et al. (2017), we augment the data (e.g., scaling, rotation) for more robustness to different face boxes. We use the stacked hourglass network Newell, Yang, and Deng (2016); Chen et al. (2017) as the alignment model. The network starts with a
convolutional layer with stride
to reduce the resolution to , followed by stacking hourglass modules. For evaluation, we adopt the standard Normalized Root Mean Squared Error (NRMSE), AreaundertheCurve (AUC) and the failure rate (at NRMSE) to measure the accuracy, and use the consistency between the movement of landmarks and ground truth as the metric to measure the stability. We train the network with the Torch7 toolbox Collobert, Kavukcuoglu, and Farabet (2011), using the RMSprop algorithm with an initial learning rate of
, a minibatch size of and . Training a fractional heatmap based hourglass model on W takes hours on a P GPU.During the stabilization training, we set and to make all terms in the stabilization loss (11
) on the same order of magnitude. We estimate the average variance
of across all training videos and all landmarks, and empirically set the initial value of as . Also, we initializeas a zero matrix
, as , and .Methods  FHR  FHR+STA  FHR+STA()  FHR+STA()  FHR+STA() 

Scenario  /  /  /  /  / 
Scenario  /  /  /  /  / 
Scenario  /  /  /  /  / 
Ablation Study
Fractional vs. Conventional Heatmap Regression
We first compare our fractional heatmap regression with the conventional version Newell, Yang, and Deng (2016) and other stateoftheart models Xiong and De la Torre (2013); Zhang et al. (2014); Zhu et al. (2015); Trigeorgis et al. (2016); Kowalski, Naruniec, and Trzcinski (2017); Chen et al. (2017) on W test set. The training set includes images, and the test set contains images. Note that the stacked hourglass networks for our fractional method and the conventional one are the same. The only difference is the transformation between the heatmaps and the coordinates, where our method preserves the fractional part. As shown in Tab. 1, our method significantly outperforms the conventional version with an improvement of on NRMSE, and is also better than the stateoftheart model Chen et al. (2017).
What’s more, the standard NRMSE cannot fully reflect the advantage of our FHR compared to CHR, since it eliminates the scaling effects by dividing the interocular distance. To further demonstrate the effects of FHR, we calculate RMSE without normalization. Specifically, we collect the interocular distances of all images and evenly divide the distances to five groups w.r.t. face scales. Tab. 2 shows that with larger scales, the gap between FHR and CHR is bigger. Especially in the largest scale, FHR achieves pixel promotion for each landmark on average.
Stabilization Loss for Time Delay
Here we demonstrate the effectiveness of our proposed time delay term (i.e., the right part of Eq. 12). As in Fig. 3, compared to the fractional heatmap regression’s output, the stabilized output is not only more smooth, but also closer to ground truth landmarks. Besides, when the time delay term is removed (), the stabilized outputs behave lagged behind the ground truth while this phenomena is largely suppressed by using our proposed loss (Eq. 12).
Methods  REDnet (2016)  FHR  FHR+STA 

Scenario  /  /  / 
Scenario  /  /  / 
Scenario  /  /  / 
Methods  CFAN (2014)  CFSS (2015)  IFA (2014)  REDnet (2016)  TSTN (2017)  CHR (2016)  FHR  FHR+STA 

NRMSE 
Stabilization Loss for Smooth
Next, we evaluate the impact of each term in our loss function (11) on NRMSE and stability. We use the consistency between the crosstime movement of the landmarks and the ground truth as an indicator of stability. Specifically, we calculate and for every video in the test set, and calculate the average NRMSE between and . Assuming that the ground truth is stable, a lower value indicates higher stability.
Comparisons with State of the Arts
We now compare our methods FHR and FHR+STA (i.e., the stabilized version) with stateoftheart models Peng et al. (2016); Liu et al. (2017); Kowalski, Naruniec, and Trzcinski (2017); Zhang et al. (2017) on two video datasets: VW and Talking Face. The comparison adopts two popular settings (i.e., and landmarks) used in prior works.
Methods  TSCN (2016)  CFSS (2015)  TCDCN (2016)  TSTN (2017)  CHR (2016)  FHR  FHR+STA 

Scenario  
Scenario  
Scenario 
Comparison with Landmarks
First, we evaluate our method on the VW Shen et al. (2017) dataset and compare with REDnet Peng et al. (2016), using the code released by the authors. Tab. 4 shows the results on the three test sets. Our proposed method achieves much better performance than REDnet in all cases. Especially in the hardest Scenario, our method achieves a large improvement of , which shows the robustness of our method to large facial variations.
Then, we evaluate our method on the Talking Face FGNET (2014) dataset compared with stateoftheart models, such as CFAN Zhang et al. (2014), CFSS Zhu et al. (2015), IFA Asthana et al. (2014), REDnet Peng et al. (2016) and TSTN Liu et al. (2017). Although the annotations of the TF dataset has the same landmark number as VW dataset, the definitions of landmarks are different. Therefore, following the setting in Liu et al. (2017); Peng et al. (2016), we use landmarks for fair comparisons. The results are shown in Tab. 5, in which the performance of Zhang et al. (2014); Zhu et al. (2015); Asthana et al. (2014) are directly cited from Peng et al. (2016); Liu et al. (2017). Since the images in TF set are collected in controlled environment and with small facial variations, all of the methods achieve relatively small errors, and our proposed method is still the best.
Comparison with Landmarks
Next, we evaluate our method on the VW Shen et al. (2017) dataset under the setting with landmarks. The comparison methods include TSCN Simonyan and Zisserman (2014), CFSS Zhu et al. (2015), TCDCN Zhang et al. (2017) and TSTN Liu et al. (2017). We cite the results of Simonyan and Zisserman (2014); Zhu et al. (2015); Zhang et al. (2017) from Liu et al. (2017), and list the performance in Tab. 6. As we can see, our proposed FHR achieves the best NRMSEs in all scenarios, and our stabilized version FHR+STA further improves the performance, especially in Scenario.
We then illustrate the Cumulative Errors Distribution (CED) curves of FHR, CHR and some SOTA methods on W in Fig. 5, where the gap between FHR and CHR is competitive to those gaps in prior toptier works (e.g., CFSSMDM). Fig. 5 also shows that our FHR contributes more to those relatively easy samples, which makes sense since the insight of FHR is to find a more precise location near a coarse but correct coordinate, whose heatmap output may accurately model the distribution of Ground Truth (GT). To demonstrate this, we collect some predicted heatmaps from those hard and easy samples, and show their averaged heatmap distributions in Fig. 4
by fixing the centers at the same position respectively. The easy samples’ heatmaps better resemble the Gaussian distribution in GT where FHR can improve the most, while hard samples resemble less and thus FHR contributes little.
Comparison on Stabilization
We further compare stabilization between our method and REDnet Peng et al. (2016) on VW Shen et al. (2017) with landmarks. As shown in Tab. 4, our FHR is much more stable than REDnet according to the metric mentioned in Ablation Study. To visualize stabilization improvement, we compute for each landmark estimated by FHR and our proposed FHR+STA, and plot them in Fig. 3. Fig. 3 plots the difference of the stability orientation of two methods, where indicates the orientation of a vector. We provide videos in our project website, which can effectively demonstrate the stability and superiority of our method. In some continuous frames, our stabilized landmarks are even more stable than the ground truth annotations.
In addition, Fig. 6 shows that our stabilization model (i.e., STA) significantly outperforms other four baselines in both of NRMSE and stability, where all methods take the same from FHR as the input. Especially in the most challenging set Scenario where lots of complex movements exist, our method is much better than the closest competitor (i.e., moving average), with the improvements of NRMSE and Stability to be and , respectively. The reasons include: ) moving average filter and first order smoother may cause serious time delay problem; ) although second order and constant speed methods can handle time delay, it cannot handle multiple movement types (e.g., blinking and turning head). In contrast, our algorithm can effectively address time delay issue and multiple movement types through the Gaussian mixture setting, and hence is more precise and stable. Fig. 7 shows the stability comparisons with the second order stabilization method, which is chosen as a good baseline considering its stability performance. As we can see, our method significantly outperforms the second order method when handling complex movements, and also shows better ability for time delay issue which is very close to the GT.
Time Complexity
Note that our fractional heatmap regression does not impose any additional complexity burden during training compared to the conventional heatmap regression. For inference, our method provides a closedform solution to estimate the coordinates from the heatmaps as in Eq. (4), whose runtime is negligible. Besides, after the parameter of our stabilization model is learnt, our stabilization algorithm costs s to process the entire test videos of VW, which costs s per image, and can also be ignored.
Conclusions
In this paper, a novel Fractional Heatmap Regression (FHR) is proposed for highresolution videobased face alignment. The main contribution in FHR is that we leverage D Gaussian generation prior to accurately estimate the fraction part of coordinates, which is ignored in conventional heatmap regression based methods. To further stabilize the landmarks among video frames, we propose a novel stabilization model that addresses the timedelay and nonsmooth issues. Extensive experiments on popular benchmarks demonstrate our proposed method is more accurate and stable than the state of the arts. Except for the facial landmark estimation task, the proposed FHR has the potential to be plugged into any existing heatmap based system (e.g., human pose estimation task) and boost the accuracy.
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