Towards Fast, Accurate and Stable3D Dense Face Alignment

Jianzhu Guo1,2?[0000−0002−8493−3689], Xiangyu Zhu1,2?[0000−0002−4636−9677],Yang Yang1,2[0000−0003−0559−5464], Fan Yang3[0000−0003−4348−3148],

Zhen Lei1,2†[0000−0002−0791−189X], and Stan Z. Li4[0000−0002−2961−8096]

1 CBSR&NLPR, Institute of Automation, Chinese Academy of Sciences2 School of Artificial Intelligence, University of Chinese Academy of Sciences

3 College of Software, Beihang University4 School of Engineering, Westlake University

{jianzhu.guo,xiangyu.zhu,yang.yang,zlei,szli}@nlpr.ia.ac.cn,fanyang@buaa.edu.cn

Abstract. Existing methods of 3D dense face alignment mainly concen-trate on accuracy, thus limiting the scope of their practical applications.In this paper, we propose a novel regression framework which makesa balance among speed, accuracy and stability. Firstly, on the basis ofa lightweight backbone, we propose a meta-joint optimization strategyto dynamically regress a small set of 3DMM parameters, which greatlyenhances speed and accuracy simultaneously. To further improve the sta-bility on videos, we present a virtual synthesis method to transform onestill image to a short-video which incorporates in-plane and out-of-planeface moving. On the premise of high accuracy and stability, our modelruns at over 50fps on a single CPU core and outperforms other state-of-the-art heavy models simultaneously. Experiments on several challengingdatasets validate the efficiency of our method. The code and models willbe available at https://github.com/cleardusk/3DDFA_V2.

Keywords: 3D Dense Face Alignment · 3D Face Reconstruction

1 Introduction

3D dense face alignment is essential for many face related tasks, e.g., recogni-tion [41,6,25,23,12], animation [9], avatar retargeting [8], tracking [46], attributeclassification [3,21,20], image restoration [47,11,10], anti-spoofing [45,50,37,49,24].Recent studies are mainly divided into two categories: 3D Morphable Model(3DMM) parameters regression [54,31,33,44,55,22,57] and dense vertices regres-sion [26,17]. Dense vertices regression methods directly regress the coordinatesof all the 3D points (usually more than 20,000) through a fully convolutionalnetwork [26,17], achieving the state-of-the-art performance. However, the res-olution of reconstructed faces relies on the size of the feature map and these

? Equal contribution.† Corresponding author.

2 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

methods rely on heavy networks like hourglass [35] or its variants, which areslow and memory-consuming in inference. The natural way of speeding it up isto prune channels. We try to prune 77.5% channels on the state-of-the-art PR-Net [17] to achieve real-time speed on CPU, but find the error greatly increases44.8% (3.62% vs. 5.24%). Besides, an obvious disadvantage is the presence ofcheckerboard artifacts due to the deconvolution operators, which is present inthe supplementary material. Another strategy is to regress a small set of 3DMM

Fig. 1: A few results from our MobileNet (M+R+S) model, which runs at over50fps on a single CPU core or over 130fps on multiple CPU cores.

parameters (usually less than 200). Compared with dense vertices, 3DMM pa-rameters have low dimensionality and low redundancy, which are appropriate toregress by a lightweight network. However, different 3DMM parameters influencethe reconstructed 3D face [54] differently, making the regression challenging sincewe have to dynamically re-weight each parameter according to their importanceduring training. Cascaded structures [54,33,55] are always adopted to progres-sively update the parameters but the computation cost is increased linearly withthe number of cascaded stages.

In this paper, we aim to accelerate the speed to CPU real time and achieve thestate-of-the-art performance simultaneously. To this end, we choose to regress3DMM parameters with a fast backbone, e.g. MobileNet. To handle the opti-mization problem of the parameters regression framework, we exploit two differ-ent loss terms WPDC and VDC [54] (see Sec. 2.2) and propose our meta-jointoptimization to combine the advantages of them. The meta-joint optimizationlooks ahead by k-steps with WPDC and VDC on the meta-train batches, thendynamically selects the better one according to the error on the meta-test batch.By doing so, the whole optimization converges faster and achieves better perfor-mance than the vanilla-joint optimization. Besides, a landmark-regression regu-larization is introduced to further alleviate the optimization problem to achievehigher accuracy. In addition to single image, 3D face applications on videos arebecoming more and more popular [9,8,28,27], where reconstructing stable re-sults across consecutive frames is important, but it is often ignored by recentmethods [54,26,17,55]. Video-based training [36,32,16,40] is always adopted toimprove the stability in 2D face alignment. However, no video databases are

Towards Fast, Accurate and Stable 3D Dense Face Alignment 3

publicly available for 3D dense face alignment. To address it, we propose a 3Daided short-video-synthesis method, which simulates both in-plane and out-of-plane face moving to transform one still image to a short video, so that ournetwork can adjust results of consecutive frames. Experiments show our short-video-synthesis method significantly improves the stability on videos.

In general, our proposed methods are (i) fast : It takes about 7.2ms with ansingle image as input (almost 24x faster than PRNet) and runs at over 50fps(19.2ms) on a single CPU core or over 130fps (7.2ms) on multiple CPU cores(i5-8259U processor), (ii) accurate: By dynamically optimizing 3DMM parame-ters through a novel meta-optimization strategy combining the fast WPDC andVDC, we surpass the state-of-the-art results [54,26,17,55] under a strict compu-tation burden in inference, and (iii) stable: In a mini-batch, one still image istransformed slightly and smoothly into a short synthetic video, involving bothin-plane and out-of-plane rotations, which provides temporal information of ad-jacent frames for training. Extensive experimental results on four datasets showthat the overall performance of our method is the best.

2 Methodology

This section details our proposed approach. We first discuss 3D MorphableModel (3DMM) [5]. Then, we introduce the proposed methods of the meta-joint optimization, landmark-regression regularization and 3D aided short-video-synthesis. The overall pipeline is illustrated in Fig. 2 and the algorithm is de-scribed in Algorithm 1.

Input images

ŏ

A lightweight backbone: e.g. MobileNet

ŏ

3D aided short-video-synthesis

FC-lmk

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FC-param

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Meta-joint optimization

Conv.+BN.+Relu

Depthwise block

Global pooling

FC

Loss

62-d param

136-d landmark

k+1 batches for

meta-train/test

Selector

Landmark-regression regularizationSampling

In-plane rotation

Out-of-plane rotation

Noise

Motion blur

Fig. 2: Overview of our method. Our architecture consists of four parts: thelightweight backbone like MobileNet for predicting 3DMM parameters, the meta-joint optimization of fWPDC and VDC, the landmark-regression regularizationand the short-video-synthesis for training. The landmark-regression branch isdiscarded in inference, thus not increasing any computation burden.

4 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

Algorithm 1: The overall algorithm of our proposed methods.

Input: Training data X = {(xl, pl)}Ml=1 .Init: Model parameters θ initialized randomly, the learning rate α, look-ahead step k,

length of 3D aided short-video-synthesis n, and batch-size of B.1 for i in max iterations do

2 Randomly sampling k batches {X lmtr}kl=1 for meta-train and one disjoint batch Xmtefor meta-test, each batch contains B pairs: {(xl, pl)}Bl=1.

// short-video-synthesis3 for each x ∈ Xmtr or Xmte do4 Synthesize a short-video with n adjacent frames: {(x0, p0|x0)|x0 =

x} ∪ {(x′j , p′j |x′j)|x′j = (M ◦ P )(xj), xj = (T ◦ F )(xj−1), 1 ≤ j ≤ n− 1}.5 end

// Meta-joint optimization with landmark-regression regularization

6 Let θfi , θvi ← θi;

7 for j = 1 · · · k do

8 θfi+j ← α∇θfi+j-1

(Lfwpdc(θfi+j-1,X

jmtr) +

|lfwpdc||llrr|

· Llrr(θfi+j-1,Xjmtr)

);

θvi+j ← α∇θvi+j-1

(Lvdc(θvi+j-1,X jmtr) +

|lvdc||llrr|

· Llrr(θvi+j-1,X jmtr))

;

9 end

10 Select θi+1 ← arg minθi+k

(Lvdc(θfi+k,Xmte),Lvdc(θvi+k,Xmte)

);

11 end

2.1 Preliminary of 3DMM

The original 3DMM can be described as:

S = S + Aidαid + Aexpαexp, (1)

where S is the 3D face mesh, S is the mean 3D shape, αid is the shape parametercorresponding to the 3D shape base Aid, Aexp is the expression base and αexp isthe expression parameter. After the 3D face is reconstructed, it can be projectedonto the image plane with the scale orthographic projection:

V2d(p) = f ∗Pr ∗R ∗(S + Aidαid + Aexpαexp

)+ t2d, (2)

where V2d(p) is the projection function generating the 2D positions of modelvertices, f is the scale factor, Pr is the orthographic projection matrix, Ris the rotation matrix constructed by Euler angles including pitch, yaw, rolland t2d is the translation vector. The complete parameters of 3DMM are p =[f, pitch, yaw, roll, t2d,αid,αexp].

However, the three Euler angles will cause the gimbal lock [30] when facesare close to the profile view. This ambiguity will confuse the regressor to degradethe performance, so we choose to regress the similarity transformation matrix in-stead of [f, pitch, yaw, roll, t2d] to reduce the regression difficulty: T = f [R; t3d],where T ∈ R3×4 is constructed by a scale factor f , a rotation matrix R and a

translation vector t3d =

[t2d

0

]. Therefore, the scale orthographic projection in

Eqn. 2 can be simplified as:

V2d(p) = Pr ∗T ∗[S + Aα

1

], (3)

Towards Fast, Accurate and Stable 3D Dense Face Alignment 5

where A = [Aid,Aexp] and α = [αid,αexp]. Our regression objective is describedas p = [T,α].

The high-dimensional parameters αshp ∈ R199, αexp ∈ R29 are redundant,since 3DMM models the 3D face shape with PCA and the last parts of param-eters have little effect on the face shape. We choose only the first 40 dimen-sions of αshp and the first 10 dimensions of αexp as our regression target, sincethe NME increase is acceptable and the reconstruction can be greatly acceler-ated. The NME error heatmap caused by different size of shape and expressiondimensions is present in the supplementary material. Therefore, our completeregression target is simplified as p = [T3×4,α50], with 62 dimensions in total,where α = [α40

shp,α10exp]. To eliminate the negative impact of magnitude dif-

ferences between T and α, Z-score normalizing is adopted: p = (p − µp)/σp,where µp ∈ R62 is the mean of parameters and σp ∈ R62 indicates the standarddeviation of parameters.

2.2 Meta-joint Optimization

We first review the Vertex Distance Cost (VDC) and Weighted Parameter Dis-tance Cost (WPDC) in [54], then derivate the meta-joint optimization to facili-tate the parameters regression.

The VDC term Lvdc directly optimizes p by minimizing the vertex distancesbetween the fitted 3D face and the ground truth:

Lvdc = ‖V3d (p)− V3d (pg)‖2 , (4)

where pg is the ground truth parameter, p is the predicted parameter and V3d(·)is the 3D face reconstruction formulated as:

V3d(p) = T ∗[S + Aα

1

]. (5)

Different from VDC, the WPDC term [54] Lwpdc assigns different weights toeach parameter:

Lwpdc = ‖w · (p− pg)‖2 , (6)

where w indicates the importance weight as follows:

w = (w1, w2, . . . , wi, . . . , wn) ,

wi =∥∥V3d(p

de,i)− V3d(pg)∥∥ /Z,

pde,i = (pg1,pg2, . . . ,pi, . . . ,p

gn) ,

(7)

where n is the number of parameters (n = 62 in our regression framework), pde,i

is the i-degraded parameter whose i-th element is from the predicted p, Z is the

6 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

maximum of w for regularization. The term∥∥V3d(p

de,i)− V3d(pg)∥∥ models the

importance of i-th parameter.

Algorithm 2: fWPDC: Fast WPDC Algorithm.

Input : Shape and expression base: A = [Aid,Aexp] ∈ R3N×50

Mean shape: S ∈ R3×N

Predicted parameters: p = [T ∈ R3×4,α ∈ R50]

Ground truth parameters: pg = [Tg ∈ R3×4,αg ∈ R50]Scale factor scalar: f

Output: WPDC item1 Initialize the weights of the parameter T and α: wT ∈ R3×4, wα ∈ R50;

// Calculating the weight of transform matrix

2 Reconstruct the vertices without projection: S = S + Aα ∈ R3×N ;3 for i = 1, 2, 3 do4 wT (:, i) =

(T(:, i)−Tg(:, i)

)· ‖S (i, :)‖;

5 end

6 wT (:, 4) =(T(:, 4)−Tg(:, 4)

)·√N/3 and then flatten wT to the vector form in

row-major order;// Calculating the weight of shape and expression parameters

7 for i = 1 . . . 50 do8 wα(i) = f ·

(α(i)− αg(i)

)· ‖A (:, i)‖;

9 end// Calculating the fWPDC item

10 Get the maximum value Z of the weights (wT ,wα) and normalize them: wT = wT /Z,wα = wα/Z;

11 Calculate the WPDC item: Lfwpdc = ‖wT · (T−Tg)‖2 + ‖wα · (α− αg)‖2

fWPDC. The original calculation of w in WPDC is rather slow as the calcu-lation of each wi needs to reconstruct all the vertices once, which is a bottleneckfor fast training. We find that the vertices can be only reconstructed once bydecomposing the weight calculation into two parts: the similarity transformationmatrix T, and the combination of shape and expression parameters α. Therefore,we design a fast implementation of WPDC named fWPDC: (i) reconstructingthe vertices without projection S = S + Aα and calculating wT using the normof row vectors; (ii) calculating wα using the norm of column vectors of A andthe input scale f : wα(i) = f ·

(α(i)−αg(i)

)· ‖A (:, i)‖; (iii) Combining them to

calculate the final cost. The detailed algorithm of fWPDC is described in Algo-rithm 2. fWPDC only reconstructs dense vertices once, not 62 times as WPDC,thus greatly reducing the computation cost. With 128 samples as a batch input,the original WPDC takes 41.7ms while fWPDC only takes 3.6ms. fWPDC isover 10x faster than the original WPDC while preserving the same outputs.

Exploitation of VDC and fWPDC. Through Eqn. 4 and Eqn. 6, wefind: WPDC/fWPDC is suitable for parameters regression since each parameteris appropriately weighted, while VDC can directly reflect the goodness of the3D face reconstructed from parameters. In Fig. 3, we investigate how these twolosses converge as the training progresses. It is shown that the optimization isdifficult for VDC since the vertex error is still over 15 when training converges.The work in [55] also demonstrates that optimizing VDC with gradient descentconverges very slowly due to the ”zig-zagging” problem. In contrast, the con-vergence of fWPDC is much faster than VDC and the error is about 7 whentraining converges. Surprisingly, if the fWPDC-trained model is fine-tuned byVDC, we can get a much lower error than fWPDC. Based on the above observa-

Towards Fast, Accurate and Stable 3D Dense Face Alignment 7

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5Iter. (104)

0

5

10

15

20

25

30

35

Verte

x er

ror i

n tra

inin

g

VDC from scratchfWPDC from scratchVDC from fWPDCVanilla-joint from scratchMeta-joint from scratchMeta-joint from scratch w/ lrr.

Fig. 3: The vertex error in training on 300W-LP supervised by different lossterms. VDC from scratch has the highest error, fWPDC is lower than VDC, andVDC from fWPDC is better than both. When combining VDC and fWPDC,the proposed meta-joint optimization converges faster and reaches lower errorthan vanilla-joint, and achieves even better convergence when incorporating thelandmark-regression regularization.

tion, we conclude that: training from scratch with VDC is hard to converge andthe network is not fully trained by fWPDC in the late stage.

Meta-joint optimization. Based on above discussions, it is natural toweight two terms to perform a vanilla-joint optimization: Lvanilla-joint = βLfwpdc+(1−β)

|lfwpdc||lvdc| ·Lvdc, where β ∈ [0, 1] controls the importance between fWPDC

and VDC. However, the vanilla-joint optimization relies on the manually sethyper-parameter β and does not achieves satisfactory results in Fig. 3. Inspiredby Lookahead [52] and MAML [18], we propose a meta-joint optimization strat-egy to dynamically combine fWPDC and VDC. The overview of the meta-jointoptimization is shown in Fig. 4. In the training process, the model looks aheadby k-steps with the cost fWPDC or VDC on k meta-train batches Xmtr, thenselects the better one between fWPDC and VDC according to the vertex erroron the meta-test batch. Specifically, the whole meta-joint optimization consistsof four steps: (i) sampling k batches of training samples Xmtr for meta-train andone batch Xmte for meta-test; (ii) meta-train: updating the current model pa-rameters θi with fWPDC and VDC on Xmtr by k-steps, respectively, getting twoparameter states θfi+k and θvi+k; (iii) meta-test: evaluating the vertex error θfi+kand θvi+k on Xmte; (iv) selecting the parameters which have the lower error toupdate θi. The proposed meta-joint optimization can be directly embedded intothe standard training regime. From Fig. 3, we can observe that the meta-jointoptimization converges faster than vanilla-joint and has the lower error.

2.3 Landmark-regression Regularization

In 3D face reconstruction [15,14,43,42,19], the 2D sparse landmarks after pro-jecting are usually used as an extra regularization to facilitate the parameters

8 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

meta-train

k-step ahead/forward

current parameters

updated parameters by fWPDC

updated parameters by VDC

✓i

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meta-test

✓i+k<latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit>

✓i+k<latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit><latexit sha1_base64="FeA0/HFV7nJdO7QaK3ARWmS1SqQ=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBZBEEoigh4LXjxWsR/QhLLZTtqlm03YnQgl9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5YSqFQdf9dkpr6xubW+Xtys7u3v5B9fCobZJMc2jxRCa6GzIDUihooUAJ3VQDi0MJnXB8O/M7T6CNSNQjTlIIYjZUIhKcoZV8H0eArJ+Li/G0X625dXcOukq8gtRIgWa/+uUPEp7FoJBLZkzPc1MMcqZRcAnTip8ZSBkfsyH0LFUsBhPk85un9MwqAxol2pZCOld/T+QsNmYSh7YzZjgyy95M/M/rZRjdBLlQaYag+GJRlEmKCZ0FQAdCA0c5sYRxLeytlI+YZhxtTBUbgrf88ippX9Y9t+7dX9UaD0UcZXJCTsk58cg1aZA70iQtwklKnskreXMy58V5dz4WrSWnmDkmf+B8/gAli5HK</latexit>

Update arg min✓i+k

⇣Lvdc(✓

fi+k, Xmte), Lvdc(✓

vi+k, Xmte)

⌘

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sha1_base64="u+PvSRYEsqjGTF54AoSS6799c4A=">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</latexit><latexit sha1_base64="u+PvSRYEsqjGTF54AoSS6799c4A=">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</latexit><latexit sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit><latexit 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sha1_base64="u31xHZhgW1hpgAP1zCxn/mADK9c=">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</latexit><latexit 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r✓iLvdc(✓i, Xmtr)

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Selector

Fig. 4: Overview of the meta-joint optimization.

regression. In our regression framework, we find that treating 2D sparse land-marks as a auxiliary regression task benefits more. As shown in Fig. 2, we add anadditional landmark-regression task on the global pooling layer, trained by L2loss. The difference between the former landmark-regularization and the latterlandmark-regression regularization is that the latter introduces extra parametersto regress the landmarks. In other words, the landmark-regression regularizationis a task-level regularization. From the tomato curve in Fig. 3, we get a lowererror by incorporating the landmark-regression regularization. The compara-tive results in Table 3 show our proposed landmark-regression regularization isbetter than landmark-regularization (3.59% vs. 3.71% on AFLW2000-3D). The

landmark-regression regularization is formulated as: Llrr = 1N

∑Ni=1 ‖li − l

gi ‖

22,

where N is 136 here as we utilize 68 2D landmarks and flatten them into a 136-dvector.

2.4 3D Aided Short-video-synthesis

Video based 3D face applications have become more and more popular [9,8,28,27]recently. In these applications, 3D dense face alignment methods are required torun on videos and provide stable reconstruction results across adjacent frames.The stability means that the changing of the reconstructed 3D faces across ad-jacent frames should be consistent with the true face moving in a fine-grainedlevel. However, most of existing methods [54,55,26,17] omit this requirenmentand the predictions suffer from random jittering. In 2D face alignment, post-processing like temporal filtering is a common strategy to reduce the jittering,but it degrades the precision and causes the frame delay. Besides, since no pub-lic video databases for 3D dense face alignment are available, the video trainingstrategies [16,36,40,32] cannot work here. A challenge arises: can we improve thestability on videos with only still images available when training?

To address this challenge, we propose a batch-level 3D aided short-video-synthesis strategy, which expands one still image to several adjacent frames,forming a short synthetic video in a mini-batch. The common patterns in a videocan be modelled as: (i) Noise. We model noise as P (x) = x + N (0, Σ), whereΣ = σ2I. (ii) Motion Blur. Motion blur can be formulated as M(x) = K ∗ x,where K is the convolution kernel (the operator ∗ denotes a convolution). (iii)In-plane rotation. Given two adjacent frames xt and xt+1, the in-plane temporal

Towards Fast, Accurate and Stable 3D Dense Face Alignment 9

change from xt to xt+1 can be described as a similarity transform T (·):

T (·) = ∆s

[cos(∆θ) − sin(∆θ) ∆t1sin(∆θ) cos(∆θ) ∆t2

], (8)

where ∆s is the scale perturbation, ∆θ is the rotation perturbation, ∆t1 and ∆t2are translation perturbations. (iv) Since human faces share similar 3D structure,we are also able to synthesize the out-of-plane face moving. Face profiling [54]F (·), which is originally proposed to solving large-pose face alignment, is utilizedto progressively increase the yaw angle ∆φ and pitch angle ∆γ of the face.Specifically, we sample several still images in a mini-batch and for each still imagex0, we transform it slightly and smoothly to generate a synthetic video with nadjacent frames: {x′j |x′j = (M ◦P )(xj), xj = (T ◦F )(xj−1), 1 ≤ j ≤ n−1}∪{x0}.In Fig. 5, we give an illustration of how these transformations are applied on animage to generate several adjacent frames.

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Out-of-plane rotation In-plane rotation Noise Motion blur�� = 5�,�� = 2�

<latexit sha1_base64="hheZLZP8bAI4sdeXKx2mOuXp/N0=">AAACGXicbVDLSgNBEJyNrxhfUY9eBoPgQcJuiOglENCDxwjmAdk19E5mkyEzu8vMrBCW/IYXf8WLB0U86sm/cfICTSxoKKq66e7yY86Utu1vK7Oyura+kd3MbW3v7O7l9w8aKkokoXUS8Ui2fFCUs5DWNdOctmJJQficNv3B1dhvPlCpWBTe6WFMPQG9kAWMgDZSJ2+715RrwG7cZ7iCz+9TlzBJRmd4bvRACKiU5kYnX7CL9gR4mTgzUkAz1Dr5T7cbkUTQUBMOSrUdO9ZeClIzwuko5yaKxkAG0KNtQ0MQVHnp5LMRPjFKFweRNBVqPFF/T6QglBoK33QK0H216I3F/7x2ooNLL2VhnGgakumiIOFYR3gcE+4ySYnmQ0OASGZuxaQPEog2YeZMCM7iy8ukUSo6dtG5LReq5VkcWXSEjtEpctAFqqIbVEN1RNAjekav6M16sl6sd+tj2pqxZjOH6A+srx+D7J9Y</latexit><latexit sha1_base64="hheZLZP8bAI4sdeXKx2mOuXp/N0=">AAACGXicbVDLSgNBEJyNrxhfUY9eBoPgQcJuiOglENCDxwjmAdk19E5mkyEzu8vMrBCW/IYXf8WLB0U86sm/cfICTSxoKKq66e7yY86Utu1vK7Oyura+kd3MbW3v7O7l9w8aKkokoXUS8Ui2fFCUs5DWNdOctmJJQficNv3B1dhvPlCpWBTe6WFMPQG9kAWMgDZSJ2+715RrwG7cZ7iCz+9TlzBJRmd4bvRACKiU5kYnX7CL9gR4mTgzUkAz1Dr5T7cbkUTQUBMOSrUdO9ZeClIzwuko5yaKxkAG0KNtQ0MQVHnp5LMRPjFKFweRNBVqPFF/T6QglBoK33QK0H216I3F/7x2ooNLL2VhnGgakumiIOFYR3gcE+4ySYnmQ0OASGZuxaQPEog2YeZMCM7iy8ukUSo6dtG5LReq5VkcWXSEjtEpctAFqqIbVEN1RNAjekav6M16sl6sd+tj2pqxZjOH6A+srx+D7J9Y</latexit><latexit sha1_base64="hheZLZP8bAI4sdeXKx2mOuXp/N0=">AAACGXicbVDLSgNBEJyNrxhfUY9eBoPgQcJuiOglENCDxwjmAdk19E5mkyEzu8vMrBCW/IYXf8WLB0U86sm/cfICTSxoKKq66e7yY86Utu1vK7Oyura+kd3MbW3v7O7l9w8aKkokoXUS8Ui2fFCUs5DWNdOctmJJQficNv3B1dhvPlCpWBTe6WFMPQG9kAWMgDZSJ2+715RrwG7cZ7iCz+9TlzBJRmd4bvRACKiU5kYnX7CL9gR4mTgzUkAz1Dr5T7cbkUTQUBMOSrUdO9ZeClIzwuko5yaKxkAG0KNtQ0MQVHnp5LMRPjFKFweRNBVqPFF/T6QglBoK33QK0H216I3F/7x2ooNLL2VhnGgakumiIOFYR3gcE+4ySYnmQ0OASGZuxaQPEog2YeZMCM7iy8ukUSo6dtG5LReq5VkcWXSEjtEpctAFqqIbVEN1RNAjekav6M16sl6sd+tj2pqxZjOH6A+srx+D7J9Y</latexit><latexit sha1_base64="hheZLZP8bAI4sdeXKx2mOuXp/N0=">AAACGXicbVDLSgNBEJyNrxhfUY9eBoPgQcJuiOglENCDxwjmAdk19E5mkyEzu8vMrBCW/IYXf8WLB0U86sm/cfICTSxoKKq66e7yY86Utu1vK7Oyura+kd3MbW3v7O7l9w8aKkokoXUS8Ui2fFCUs5DWNdOctmJJQficNv3B1dhvPlCpWBTe6WFMPQG9kAWMgDZSJ2+715RrwG7cZ7iCz+9TlzBJRmd4bvRACKiU5kYnX7CL9gR4mTgzUkAz1Dr5T7cbkUTQUBMOSrUdO9ZeClIzwuko5yaKxkAG0KNtQ0MQVHnp5LMRPjFKFweRNBVqPFF/T6QglBoK33QK0H216I3F/7x2ooNLL2VhnGgakumiIOFYR3gcE+4ySYnmQ0OASGZuxaQPEog2YeZMCM7iy8ukUSo6dtG5LReq5VkcWXSEjtEpctAFqqIbVEN1RNAjekav6M16sl6sd+tj2pqxZjOH6A+srx+D7J9Y</latexit>

�s = 1.05,�✓ = 3�, (�t1,�t2) = (3, 3)<latexit sha1_base64="nlQEkPoMMnISzCp+fJfI+rRyJgw=">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</latexit><latexit sha1_base64="nlQEkPoMMnISzCp+fJfI+rRyJgw=">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</latexit><latexit sha1_base64="nlQEkPoMMnISzCp+fJfI+rRyJgw=">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</latexit><latexit sha1_base64="nlQEkPoMMnISzCp+fJfI+rRyJgw=">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</latexit>

� = 3<latexit sha1_base64="GBQt+XVuIpCPRFOuda3V0os/dNw=">AAAB8XicbVBNSwMxEJ3Ur1q/qh69BIvgqexqQS9CwYvHCvYD26Vk02wbmmSXJCuUpf/CiwdFvPpvvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHuwkYYEkQ8UjTol10mPP8KEk+AZf9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTa/eIrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz97HA64ZtWLiCKGau1sxHRFNqHUhlVwI/vLLq6R1UfW9qn9fq9RreRxFOIFTOAcfrqAOd9CAJlBQ8Ayv8IYMekHv6GPRWkD5zDH8Afr8AUF5j+0=</latexit><latexit sha1_base64="GBQt+XVuIpCPRFOuda3V0os/dNw=">AAAB8XicbVBNSwMxEJ3Ur1q/qh69BIvgqexqQS9CwYvHCvYD26Vk02wbmmSXJCuUpf/CiwdFvPpvvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHuwkYYEkQ8UjTol10mPP8KEk+AZf9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTa/eIrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz97HA64ZtWLiCKGau1sxHRFNqHUhlVwI/vLLq6R1UfW9qn9fq9RreRxFOIFTOAcfrqAOd9CAJlBQ8Ayv8IYMekHv6GPRWkD5zDH8Afr8AUF5j+0=</latexit><latexit sha1_base64="GBQt+XVuIpCPRFOuda3V0os/dNw=">AAAB8XicbVBNSwMxEJ3Ur1q/qh69BIvgqexqQS9CwYvHCvYD26Vk02wbmmSXJCuUpf/CiwdFvPpvvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHuwkYYEkQ8UjTol10mPP8KEk+AZf9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTa/eIrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz97HA64ZtWLiCKGau1sxHRFNqHUhlVwI/vLLq6R1UfW9qn9fq9RreRxFOIFTOAcfrqAOd9CAJlBQ8Ayv8IYMekHv6GPRWkD5zDH8Afr8AUF5j+0=</latexit><latexit sha1_base64="GBQt+XVuIpCPRFOuda3V0os/dNw=">AAAB8XicbVBNSwMxEJ3Ur1q/qh69BIvgqexqQS9CwYvHCvYD26Vk02wbmmSXJCuUpf/CiwdFvPpvvPlvTNs9aOuDgcd7M8zMCxPBjfW8b1RYW9/Y3Cpul3Z29/YPyodHLROnmrImjUWsOyExTHDFmpZbwTqJZkSGgrXD8e3Mbz8xbXisHuwkYYEkQ8UjTol10mPP8KEk+AZf9ssVr+rNgVeJn5MK5Gj0y1+9QUxTyZSlghjT9b3EBhnRllPBpqVealhC6JgMWddRRSQzQTa/eIrPnDLAUaxdKYvn6u+JjEhjJjJ0nZLYkVn2ZuJ/Xje10XWQcZWklim6WBSlAtsYz97HA64ZtWLiCKGau1sxHRFNqHUhlVwI/vLLq6R1UfW9qn9fq9RreRxFOIFTOAcfrqAOd9CAJlBQ8Ayv8IYMekHv6GPRWkD5zDH8Afr8AUF5j+0=</latexit> kernel size = 5<latexit sha1_base64="3o1O2K+UWd8WzgRFcxQGonYGSio=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0YtQ8OKxgv2ANpTNdtou3WzC7qZQQ/+JFw+KePWfePPfuG1z0NYHA4/3ZpiZFyaCa+N5305hbX1jc6u4XdrZ3ds/cA+PGjpOFcM6i0WsWiHVKLjEuuFGYCtRSKNQYDMc3c385hiV5rF8NJMEg4gOJO9zRo2Vuq47QiVRdLqaPyG5JVddt+xVvDnIKvFzUoYcta771enFLI1QGiao1m3fS0yQUWU4EzgtdVKNCWUjOsC2pZJGqINsfvmUnFmlR/qxsiUNmau/JzIaaT2JQtsZUTPUy95M/M9rp6Z/E2RcJqlByRaL+qkgJiazGEiPK2RGTCyhTHF7K2FDqigzNqySDcFffnmVNC4qvlfxHy7L1cs8jiKcwCmcgw/XUIV7qEEdGIzhGV7hzcmcF+fd+Vi0Fpx85hj+wPn8AZL5kuY=</latexit><latexit sha1_base64="3o1O2K+UWd8WzgRFcxQGonYGSio=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0YtQ8OKxgv2ANpTNdtou3WzC7qZQQ/+JFw+KePWfePPfuG1z0NYHA4/3ZpiZFyaCa+N5305hbX1jc6u4XdrZ3ds/cA+PGjpOFcM6i0WsWiHVKLjEuuFGYCtRSKNQYDMc3c385hiV5rF8NJMEg4gOJO9zRo2Vuq47QiVRdLqaPyG5JVddt+xVvDnIKvFzUoYcta771enFLI1QGiao1m3fS0yQUWU4EzgtdVKNCWUjOsC2pZJGqINsfvmUnFmlR/qxsiUNmau/JzIaaT2JQtsZUTPUy95M/M9rp6Z/E2RcJqlByRaL+qkgJiazGEiPK2RGTCyhTHF7K2FDqigzNqySDcFffnmVNC4qvlfxHy7L1cs8jiKcwCmcgw/XUIV7qEEdGIzhGV7hzcmcF+fd+Vi0Fpx85hj+wPn8AZL5kuY=</latexit><latexit sha1_base64="3o1O2K+UWd8WzgRFcxQGonYGSio=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0YtQ8OKxgv2ANpTNdtou3WzC7qZQQ/+JFw+KePWfePPfuG1z0NYHA4/3ZpiZFyaCa+N5305hbX1jc6u4XdrZ3ds/cA+PGjpOFcM6i0WsWiHVKLjEuuFGYCtRSKNQYDMc3c385hiV5rF8NJMEg4gOJO9zRo2Vuq47QiVRdLqaPyG5JVddt+xVvDnIKvFzUoYcta771enFLI1QGiao1m3fS0yQUWU4EzgtdVKNCWUjOsC2pZJGqINsfvmUnFmlR/qxsiUNmau/JzIaaT2JQtsZUTPUy95M/M9rp6Z/E2RcJqlByRaL+qkgJiazGEiPK2RGTCyhTHF7K2FDqigzNqySDcFffnmVNC4qvlfxHy7L1cs8jiKcwCmcgw/XUIV7qEEdGIzhGV7hzcmcF+fd+Vi0Fpx85hj+wPn8AZL5kuY=</latexit><latexit sha1_base64="3o1O2K+UWd8WzgRFcxQGonYGSio=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0YtQ8OKxgv2ANpTNdtou3WzC7qZQQ/+JFw+KePWfePPfuG1z0NYHA4/3ZpiZFyaCa+N5305hbX1jc6u4XdrZ3ds/cA+PGjpOFcM6i0WsWiHVKLjEuuFGYCtRSKNQYDMc3c385hiV5rF8NJMEg4gOJO9zRo2Vuq47QiVRdLqaPyG5JVddt+xVvDnIKvFzUoYcta771enFLI1QGiao1m3fS0yQUWU4EzgtdVKNCWUjOsC2pZJGqINsfvmUnFmlR/qxsiUNmau/JzIaaT2JQtsZUTPUy95M/M9rp6Z/E2RcJqlByRaL+qkgJiazGEiPK2RGTCyhTHF7K2FDqigzNqySDcFffnmVNC4qvlfxHy7L1cs8jiKcwCmcgw/XUIV7qEEdGIzhGV7hzcmcF+fd+Vi0Fpx85hj+wPn8AZL5kuY=</latexit>

x1 = (T � F )(x0)<latexit sha1_base64="YI2jCanabh7NsVJ0P85FzhPzr/A=">AAACAHicbVBNS8NAEJ3Ur1q/oh48eFksQnspiQh6UCgI4rFCP4Q2hM122y7dbMLuRlpCL/4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4g5U9pxvq3cyura+kZ+s7C1vbO7Z+8fNFWUSEIbJOKRfAiwopwJ2tBMc/oQS4rDgNNWMLyZ+q1HKhWLRF2PY+qFuC9YjxGsjeTbRyPfRdeoVEcdwiRBt2VUGvlO2beLTsWZAS0TNyNFyFDz7a9ONyJJSIUmHCvVdp1YeymWmhFOJ4VOomiMyRD3adtQgUOqvHT2wASdGqWLepE0JTSaqb8nUhwqNQ4D0xliPVCL3lT8z2snunfppUzEiaaCzBf1Eo50hKZpoC6TlGg+NgQTycytiAywxESbzAomBHfx5WXSPKu4TsW9Py9Wr7I48nAMJ1ACFy6gCndQgwYQmMAzvMKb9WS9WO/Wx7w1Z2Uzh/AH1ucPapKTsA==</latexit><latexit sha1_base64="YI2jCanabh7NsVJ0P85FzhPzr/A=">AAACAHicbVBNS8NAEJ3Ur1q/oh48eFksQnspiQh6UCgI4rFCP4Q2hM122y7dbMLuRlpCL/4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4g5U9pxvq3cyura+kZ+s7C1vbO7Z+8fNFWUSEIbJOKRfAiwopwJ2tBMc/oQS4rDgNNWMLyZ+q1HKhWLRF2PY+qFuC9YjxGsjeTbRyPfRdeoVEcdwiRBt2VUGvlO2beLTsWZAS0TNyNFyFDz7a9ONyJJSIUmHCvVdp1YeymWmhFOJ4VOomiMyRD3adtQgUOqvHT2wASdGqWLepE0JTSaqb8nUhwqNQ4D0xliPVCL3lT8z2snunfppUzEiaaCzBf1Eo50hKZpoC6TlGg+NgQTycytiAywxESbzAomBHfx5WXSPKu4TsW9Py9Wr7I48nAMJ1ACFy6gCndQgwYQmMAzvMKb9WS9WO/Wx7w1Z2Uzh/AH1ucPapKTsA==</latexit><latexit sha1_base64="YI2jCanabh7NsVJ0P85FzhPzr/A=">AAACAHicbVBNS8NAEJ3Ur1q/oh48eFksQnspiQh6UCgI4rFCP4Q2hM122y7dbMLuRlpCL/4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4g5U9pxvq3cyura+kZ+s7C1vbO7Z+8fNFWUSEIbJOKRfAiwopwJ2tBMc/oQS4rDgNNWMLyZ+q1HKhWLRF2PY+qFuC9YjxGsjeTbRyPfRdeoVEcdwiRBt2VUGvlO2beLTsWZAS0TNyNFyFDz7a9ONyJJSIUmHCvVdp1YeymWmhFOJ4VOomiMyRD3adtQgUOqvHT2wASdGqWLepE0JTSaqb8nUhwqNQ4D0xliPVCL3lT8z2snunfppUzEiaaCzBf1Eo50hKZpoC6TlGg+NgQTycytiAywxESbzAomBHfx5WXSPKu4TsW9Py9Wr7I48nAMJ1ACFy6gCndQgwYQmMAzvMKb9WS9WO/Wx7w1Z2Uzh/AH1ucPapKTsA==</latexit><latexit sha1_base64="YI2jCanabh7NsVJ0P85FzhPzr/A=">AAACAHicbVBNS8NAEJ3Ur1q/oh48eFksQnspiQh6UCgI4rFCP4Q2hM122y7dbMLuRlpCL/4VLx4U8erP8Oa/cdvmoK0PBh7vzTAzL4g5U9pxvq3cyura+kZ+s7C1vbO7Z+8fNFWUSEIbJOKRfAiwopwJ2tBMc/oQS4rDgNNWMLyZ+q1HKhWLRF2PY+qFuC9YjxGsjeTbRyPfRdeoVEcdwiRBt2VUGvlO2beLTsWZAS0TNyNFyFDz7a9ONyJJSIUmHCvVdp1YeymWmhFOJ4VOomiMyRD3adtQgUOqvHT2wASdGqWLepE0JTSaqb8nUhwqNQ4D0xliPVCL3lT8z2snunfppUzEiaaCzBf1Eo50hKZpoC6TlGg+NgQTycytiAywxESbzAomBHfx5WXSPKu4TsW9Py9Wr7I48nAMJ1ACFy6gCndQgwYQmMAzvMKb9WS9WO/Wx7w1Z2Uzh/AH1ucPapKTsA==</latexit>

F (x1)<latexit sha1_base64="ys9OuwzeuyjOO+XNXPMvVoW/G9s=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquCHrwUBDEYwW7LbRLyabZNjabLElWLEv/gxcPinj1/3jz35i2e9DWBwOP92aYmRcmnGnjut9OYWV1bX2juFna2t7Z3SvvH/haporQJpFcqnaINeVM0KZhhtN2oiiOQ05b4eh66rceqdJMinszTmgQ44FgESPYWMm/qT71vNNeueLW3BnQMvFyUoEcjV75q9uXJI2pMIRjrTuem5ggw8owwumk1E01TTAZ4QHtWCpwTHWQza6doBOr9FEklS1h0Ez9PZHhWOtxHNrOGJuhXvSm4n9eJzXRZZAxkaSGCjJfFKUcGYmmr6M+U5QYPrYEE8XsrYgMscLE2IBKNgRv8eVl4p/VPLfm3Z1X6ld5HEU4gmOoggcXUIdbaEATCDzAM7zCmyOdF+fd+Zi3Fpx85hD+wPn8AWCRjk8=</latexit><latexit sha1_base64="ys9OuwzeuyjOO+XNXPMvVoW/G9s=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquCHrwUBDEYwW7LbRLyabZNjabLElWLEv/gxcPinj1/3jz35i2e9DWBwOP92aYmRcmnGnjut9OYWV1bX2juFna2t7Z3SvvH/haporQJpFcqnaINeVM0KZhhtN2oiiOQ05b4eh66rceqdJMinszTmgQ44FgESPYWMm/qT71vNNeueLW3BnQMvFyUoEcjV75q9uXJI2pMIRjrTuem5ggw8owwumk1E01TTAZ4QHtWCpwTHWQza6doBOr9FEklS1h0Ez9PZHhWOtxHNrOGJuhXvSm4n9eJzXRZZAxkaSGCjJfFKUcGYmmr6M+U5QYPrYEE8XsrYgMscLE2IBKNgRv8eVl4p/VPLfm3Z1X6ld5HEU4gmOoggcXUIdbaEATCDzAM7zCmyOdF+fd+Zi3Fpx85hD+wPn8AWCRjk8=</latexit><latexit sha1_base64="ys9OuwzeuyjOO+XNXPMvVoW/G9s=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquCHrwUBDEYwW7LbRLyabZNjabLElWLEv/gxcPinj1/3jz35i2e9DWBwOP92aYmRcmnGnjut9OYWV1bX2juFna2t7Z3SvvH/haporQJpFcqnaINeVM0KZhhtN2oiiOQ05b4eh66rceqdJMinszTmgQ44FgESPYWMm/qT71vNNeueLW3BnQMvFyUoEcjV75q9uXJI2pMIRjrTuem5ggw8owwumk1E01TTAZ4QHtWCpwTHWQza6doBOr9FEklS1h0Ez9PZHhWOtxHNrOGJuhXvSm4n9eJzXRZZAxkaSGCjJfFKUcGYmmr6M+U5QYPrYEE8XsrYgMscLE2IBKNgRv8eVl4p/VPLfm3Z1X6ld5HEU4gmOoggcXUIdbaEATCDzAM7zCmyOdF+fd+Zi3Fpx85hD+wPn8AWCRjk8=</latexit><latexit sha1_base64="ys9OuwzeuyjOO+XNXPMvVoW/G9s=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRahXsquCHrwUBDEYwW7LbRLyabZNjabLElWLEv/gxcPinj1/3jz35i2e9DWBwOP92aYmRcmnGnjut9OYWV1bX2juFna2t7Z3SvvH/haporQJpFcqnaINeVM0KZhhtN2oiiOQ05b4eh66rceqdJMinszTmgQ44FgESPYWMm/qT71vNNeueLW3BnQMvFyUoEcjV75q9uXJI2pMIRjrTuem5ggw8owwumk1E01TTAZ4QHtWCpwTHWQza6doBOr9FEklS1h0Ez9PZHhWOtxHNrOGJuhXvSm4n9eJzXRZZAxkaSGCjJfFKUcGYmmr6M+U5QYPrYEE8XsrYgMscLE2IBKNgRv8eVl4p/VPLfm3Z1X6ld5HEU4gmOoggcXUIdbaEATCDzAM7zCmyOdF+fd+Zi3Fpx85hD+wPn8AWCRjk8=</latexit>

x2 = (T � F )(x1)<latexit sha1_base64="9RK2HK5IbrlIDq29T1BLnammzTY=">AAACAHicbVDLSgNBEOyNrxhfqx48eBkMQnIJu0HQg0JAEI8R8oJkWWYns8mQ2Qczs5Kw5OKvePGgiFc/w5t/4yTZgyYWNBRV3XR3eTFnUlnWt5FbW9/Y3MpvF3Z29/YPzMOjlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG17o9uZ336kQrIobKhJTJ0AD0LmM4KVllzzZOxW0Q0qNVCPMEHQXRmVxq5dds2iVbHmQKvEzkgRMtRd86vXj0gS0FARjqXs2lasnBQLxQin00IvkTTGZIQHtKtpiAMqnXT+wBSda6WP/EjoChWaq78nUhxIOQk83RlgNZTL3kz8z+smyr9yUhbGiaIhWSzyE45UhGZpoD4TlCg+0QQTwfStiAyxwETpzAo6BHv55VXSqlZsq2I/XBRr11kceTiFMyiBDZdQg3uoQxMITOEZXuHNeDJejHfjY9GaM7KZY/gD4/MHba+Tsg==</latexit><latexit sha1_base64="9RK2HK5IbrlIDq29T1BLnammzTY=">AAACAHicbVDLSgNBEOyNrxhfqx48eBkMQnIJu0HQg0JAEI8R8oJkWWYns8mQ2Qczs5Kw5OKvePGgiFc/w5t/4yTZgyYWNBRV3XR3eTFnUlnWt5FbW9/Y3MpvF3Z29/YPzMOjlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG17o9uZ336kQrIobKhJTJ0AD0LmM4KVllzzZOxW0Q0qNVCPMEHQXRmVxq5dds2iVbHmQKvEzkgRMtRd86vXj0gS0FARjqXs2lasnBQLxQin00IvkTTGZIQHtKtpiAMqnXT+wBSda6WP/EjoChWaq78nUhxIOQk83RlgNZTL3kz8z+smyr9yUhbGiaIhWSzyE45UhGZpoD4TlCg+0QQTwfStiAyxwETpzAo6BHv55VXSqlZsq2I/XBRr11kceTiFMyiBDZdQg3uoQxMITOEZXuHNeDJejHfjY9GaM7KZY/gD4/MHba+Tsg==</latexit><latexit sha1_base64="9RK2HK5IbrlIDq29T1BLnammzTY=">AAACAHicbVDLSgNBEOyNrxhfqx48eBkMQnIJu0HQg0JAEI8R8oJkWWYns8mQ2Qczs5Kw5OKvePGgiFc/w5t/4yTZgyYWNBRV3XR3eTFnUlnWt5FbW9/Y3MpvF3Z29/YPzMOjlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG17o9uZ336kQrIobKhJTJ0AD0LmM4KVllzzZOxW0Q0qNVCPMEHQXRmVxq5dds2iVbHmQKvEzkgRMtRd86vXj0gS0FARjqXs2lasnBQLxQin00IvkTTGZIQHtKtpiAMqnXT+wBSda6WP/EjoChWaq78nUhxIOQk83RlgNZTL3kz8z+smyr9yUhbGiaIhWSzyE45UhGZpoD4TlCg+0QQTwfStiAyxwETpzAo6BHv55VXSqlZsq2I/XBRr11kceTiFMyiBDZdQg3uoQxMITOEZXuHNeDJejHfjY9GaM7KZY/gD4/MHba+Tsg==</latexit><latexit sha1_base64="9RK2HK5IbrlIDq29T1BLnammzTY=">AAACAHicbVDLSgNBEOyNrxhfqx48eBkMQnIJu0HQg0JAEI8R8oJkWWYns8mQ2Qczs5Kw5OKvePGgiFc/w5t/4yTZgyYWNBRV3XR3eTFnUlnWt5FbW9/Y3MpvF3Z29/YPzMOjlowSQWiTRDwSHQ9LyllIm4opTjuxoDjwOG17o9uZ336kQrIobKhJTJ0AD0LmM4KVllzzZOxW0Q0qNVCPMEHQXRmVxq5dds2iVbHmQKvEzkgRMtRd86vXj0gS0FARjqXs2lasnBQLxQin00IvkTTGZIQHtKtpiAMqnXT+wBSda6WP/EjoChWaq78nUhxIOQk83RlgNZTL3kz8z+smyr9yUhbGiaIhWSzyE45UhGZpoD4TlCg+0QQTwfStiAyxwETpzAo6BHv55VXSqlZsq2I/XBRr11kceTiFMyiBDZdQg3uoQxMITOEZXuHNeDJejHfjY9GaM7KZY/gD4/MHba+Tsg==</latexit>

P (x1)<latexit sha1_base64="WNPuuBbhiyEUF7nMtjbP2nMqjIM=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AnghYWARvLCOYDkiPsbeaSJXt7x+6eGI78CBsLRWz9PXb+GzfJFZr4YODx3gwz84JEcG1c99sprK1vbG4Vt0s7u3v7B+XDo5aOU8WwyWIRq05ANQousWm4EdhJFNIoENgOxrczv/2ISvNYPphJgn5Eh5KHnFFjpXaDVJ/63nm/XHFr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfu6UnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+xmXSWpQssWiMBXExGT2OxlwhcyIiSWUKW5vJWxEFWXGJlSyIXjLL6+S1kXNc2ve/WWlfpPHUYQTOIUqeHAFdbiDBjSBwRie4RXenMR5cd6dj0VrwclnjuEPnM8fxkmOgw==</latexit><latexit sha1_base64="WNPuuBbhiyEUF7nMtjbP2nMqjIM=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AnghYWARvLCOYDkiPsbeaSJXt7x+6eGI78CBsLRWz9PXb+GzfJFZr4YODx3gwz84JEcG1c99sprK1vbG4Vt0s7u3v7B+XDo5aOU8WwyWIRq05ANQousWm4EdhJFNIoENgOxrczv/2ISvNYPphJgn5Eh5KHnFFjpXaDVJ/63nm/XHFr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfu6UnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+xmXSWpQssWiMBXExGT2OxlwhcyIiSWUKW5vJWxEFWXGJlSyIXjLL6+S1kXNc2ve/WWlfpPHUYQTOIUqeHAFdbiDBjSBwRie4RXenMR5cd6dj0VrwclnjuEPnM8fxkmOgw==</latexit><latexit sha1_base64="WNPuuBbhiyEUF7nMtjbP2nMqjIM=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AnghYWARvLCOYDkiPsbeaSJXt7x+6eGI78CBsLRWz9PXb+GzfJFZr4YODx3gwz84JEcG1c99sprK1vbG4Vt0s7u3v7B+XDo5aOU8WwyWIRq05ANQousWm4EdhJFNIoENgOxrczv/2ISvNYPphJgn5Eh5KHnFFjpXaDVJ/63nm/XHFr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfu6UnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+xmXSWpQssWiMBXExGT2OxlwhcyIiSWUKW5vJWxEFWXGJlSyIXjLL6+S1kXNc2ve/WWlfpPHUYQTOIUqeHAFdbiDBjSBwRie4RXenMR5cd6dj0VrwclnjuEPnM8fxkmOgw==</latexit><latexit sha1_base64="WNPuuBbhiyEUF7nMtjbP2nMqjIM=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AnghYWARvLCOYDkiPsbeaSJXt7x+6eGI78CBsLRWz9PXb+GzfJFZr4YODx3gwz84JEcG1c99sprK1vbG4Vt0s7u3v7B+XDo5aOU8WwyWIRq05ANQousWm4EdhJFNIoENgOxrczv/2ISvNYPphJgn5Eh5KHnFFjpXaDVJ/63nm/XHFr7hxklXg5qUCORr/81RvELI1QGiao1l3PTYyfUWU4Ezgt9VKNCWVjOsSupZJGqP1sfu6UnFllQMJY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UxNe+xmXSWpQssWiMBXExGT2OxlwhcyIiSWUKW5vJWxEFWXGJlSyIXjLL6+S1kXNc2ve/WWlfpPHUYQTOIUqeHAFdbiDBjSBwRie4RXenMR5cd6dj0VrwclnjuEPnM8fxkmOgw==</latexit>

P (x2)<latexit sha1_base64="i5LE/egUHQeIZDY9QNdgLDcmsEg=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AXBC0sAjaWEUwiJEfY28wlS/b2jt09MRz5ETYWitj6e+z8N26SKzTxwcDjvRlm5gWJ4Nq47rdTWFvf2Nwqbpd2dvf2D8qHR20dp4phi8UiVg8B1Si4xJbhRuBDopBGgcBOML6Z+Z1HVJrH8t5MEvQjOpQ85IwaK3WapPrUr5/3yxW35s5BVomXkwrkaPbLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5udOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhlZ9xmaQGJVssClNBTExmv5MBV8iMmFhCmeL2VsJGVFFmbEIlG4K3/PIqaddrnlvz7i4qjes8jiKcwClUwYNLaMAtNKEFDMbwDK/w5iTOi/PufCxaC04+cwx/4Hz+AMfOjoQ=</latexit><latexit sha1_base64="i5LE/egUHQeIZDY9QNdgLDcmsEg=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AXBC0sAjaWEUwiJEfY28wlS/b2jt09MRz5ETYWitj6e+z8N26SKzTxwcDjvRlm5gWJ4Nq47rdTWFvf2Nwqbpd2dvf2D8qHR20dp4phi8UiVg8B1Si4xJbhRuBDopBGgcBOML6Z+Z1HVJrH8t5MEvQjOpQ85IwaK3WapPrUr5/3yxW35s5BVomXkwrkaPbLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5udOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhlZ9xmaQGJVssClNBTExmv5MBV8iMmFhCmeL2VsJGVFFmbEIlG4K3/PIqaddrnlvz7i4qjes8jiKcwClUwYNLaMAtNKEFDMbwDK/w5iTOi/PufCxaC04+cwx/4Hz+AMfOjoQ=</latexit><latexit sha1_base64="i5LE/egUHQeIZDY9QNdgLDcmsEg=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AXBC0sAjaWEUwiJEfY28wlS/b2jt09MRz5ETYWitj6e+z8N26SKzTxwcDjvRlm5gWJ4Nq47rdTWFvf2Nwqbpd2dvf2D8qHR20dp4phi8UiVg8B1Si4xJbhRuBDopBGgcBOML6Z+Z1HVJrH8t5MEvQjOpQ85IwaK3WapPrUr5/3yxW35s5BVomXkwrkaPbLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5udOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhlZ9xmaQGJVssClNBTExmv5MBV8iMmFhCmeL2VsJGVFFmbEIlG4K3/PIqaddrnlvz7i4qjes8jiKcwClUwYNLaMAtNKEFDMbwDK/w5iTOi/PufCxaC04+cwx/4Hz+AMfOjoQ=</latexit><latexit sha1_base64="i5LE/egUHQeIZDY9QNdgLDcmsEg=">AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoMQm3AXBC0sAjaWEUwiJEfY28wlS/b2jt09MRz5ETYWitj6e+z8N26SKzTxwcDjvRlm5gWJ4Nq47rdTWFvf2Nwqbpd2dvf2D8qHR20dp4phi8UiVg8B1Si4xJbhRuBDopBGgcBOML6Z+Z1HVJrH8t5MEvQjOpQ85IwaK3WapPrUr5/3yxW35s5BVomXkwrkaPbLX71BzNIIpWGCat313MT4GVWGM4HTUi/VmFA2pkPsWipphNrP5udOyZlVBiSMlS1pyFz9PZHRSOtJFNjOiJqRXvZm4n9eNzXhlZ9xmaQGJVssClNBTExmv5MBV8iMmFhCmeL2VsJGVFFmbEIlG4K3/PIqaddrnlvz7i4qjes8jiKcwClUwYNLaMAtNKEFDMbwDK/w5iTOi/PufCxaC04+cwx/4Hz+AMfOjoQ=</latexit>

Synthetic adjacent frames

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Fig. 5: An illustration of how two adjacent frames are synthesized in our 3Daided short-video-synthesis.

3 Experiments

In this section, we first introduce the datasets and protocols; then, we givecomparison experiments on the accuracy and stability; thirdly, the complexityand running speed are evaluated; extensive discussions are finally made. Theimplementation details, generalization and scaling-up ability of our proposedmethod are in the supplementary material.

3.1 Datasets and Evaluation Protocols

Five datasets are used in our experiments: 300W-LP [54] (300W Across LargePoses) is composed of the synthesized large-pose face images from 300W [38],including AFW [56], LFPW [2], HELEN [53], IBUG [38], and XM2VTS [34].Specifically, the face profiling method [54] is adopted to generate 122,450 sam-ples across large poses. AFLW [29] consists of 21,080 in-the-wild faces (following

10 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.93.6

3.7

3.8

3.9

NME

(%)

AFLW2000-3D

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (vanilla-joint)

4.6

4.7

4.8

4.9NM

E (%

)AFLW

5 10 50 100 200 300 9003.6

3.7

3.8

3.9 3.88

3.81 3.80

3.733.77 3.76 3.78

AFLW2000-3D

5 10 50 100 200 300 900k (meta-joint)

4.6

4.7

4.8

4.9

4.754.71 4.70

4.64 4.654.68 4.70

AFLW

Fig. 6: Ablative results of the vanilla-joint optimization with different β andmeta-joint optimization with different k. Lower NME (%) is better.

Table 1: The NME (%) of different methods on AFLW2000-3D and AFLW.The first and the second best results are highlighted. M, R, S denote the meta-joint optimization, landmark-regression regularization and short-video-synthesis,respectively.

MethodAFLW2000-3D (68 pts) AFLW (21 pts)

[0, 30] [30, 60] [60, 90] Mean [0, 30] [30, 60] [60, 90] Mean

ESR [13] 4.60 6.70 12.67 7.99 5.66 7.12 11.94 8.24SDM [46] 3.67 4.94 9.67 6.12 4.75 5.55 9.34 6.55

3DDFA [54] 3.78 4.54 7.93 5.42 5.00 5.06 6.74 5.603DDFA+SDM [54] 3.43 4.24 7.17 4.94 4.75 4.83 6.38 5.32

Yu et al. [48] 3.62 6.06 9.56 6.41 - - - -DeFA [33] - - - 4.50 - - - -3DSTN [4] 3.15 4.33 5.98 4.49 3.55 3.92 5.21 4.233D-FAN [7] 3.15 3.53 4.60 3.76 4.40 4.52 5.17 4.69

3DDFA-TPAMI [55] 2.84 3.57 4.96 3.79 4.11 4.38 5.16 4.55PRNet [17] 2.75 3.51 4.61 3.62 4.19 4.69 5.45 4.77

MobileNet (M+R) 2.75 3.49 4.53 3.59 4.06 4.41 5.02 4.50MobileNet (M+R+S) 2.63 3.42 4.48 3.51 3.98 4.31 4.99 4.43

[55,22]) with large poses (yaw from -90◦ to 90◦). Each image is annotated up to 21visible landmarks. AFLW2000-3D [54] is constructed by [54] for evaluating 3Dface alignment performance, which contains the ground truth 3D faces and thecorresponding 68 landmarks of the first 2,000 AFLW samples. Florence [1] is a3D face dataset containing 53 subjects with its ground truth 3D mesh acquiredfrom a structured-light scanning system. For evaluation, we generate render-ings with different poses for each subject following VRN [26] and PRNet [17].Menpo-3D [51] provides a benchmark for evaluating 3D facial landmark local-ization algorithms in the wild in arbitrary poses. Specifically, Menpo-3D provides3D facial landmarks for 55 videos from 300-VW [39] competition.

Towards Fast, Accurate and Stable 3D Dense Face Alignment 11

Table 2: The NME (%) on Florence, AFLW2000-3D (Dense), NME (%) / Stabil-ity (%) on Menpo-3D, running complexity and time with different methods. Ourmethod outputs 3D dense vertices with only 2.1ms (2ms for parameters predic-tion and 0.1ms for vertices reconstruction) in GPU or 7.2ms in CPU (6.2ms forparameters prediction and 1ms for vertices reconstruction). The first and secondbest results are highlighted.

Methods FlorenceAFLW2000-3D

(Dense)Menpo-3D Params MACs Run Time (ms)

3DDFA [54] 6.38 6.56 - - - 75.7=23.2(GPU)+52.5(CPU)VRN [26] 5.27 - - - - 69.0(GPU)DeFA [33] - 6.04 - - 1426M 35.4=11.8(GPU)+23.6(CPU)PRNet [17] 3.76 4.41 1.90 / 0.54 13.4M 6190M 9.8(GPU) / 175.0(CPU)

MobileNet (M+R) 3.59 4.20 1.86 / 0.523.27M 183M 2.1(GPU) / 7.2(CPU)

MobileNet (M+R+S) 3.56 4.18 1.71 / 0.48

Protocols. The protocol on AFLW follows [54] and Normalized Mean Error(NME) by bounding box size is reported. Two protocols on AFLW2000-3D areapplied: the first one follows AFLW, and the other one follows [17] to evaluatethe NME of 3D face reconstruction normalized by the bounding box size. ForFlorence, we follow [26,17] to evaluate the NME of 3D face reconstruction nor-malized by outer interocular distance. As for Menpo-3D, we evaluate the NMEon still frames and the stability across adjacent frames. We calculate the sta-bility following [40] by measuring the NME between the predicted offsets andthe ground-truth offsets of adjacent frames. Specifically, at frame t − 1 and t,the ground-truth landmark offset is ∆p = pt − pt−1, the prediction offset is∆q = qt − qt−1, the error ∆p−∆q normalized by the bounding box size repre-sents the stability. Since 300W-LP only has the indices of 68 landmarks, we use68 landmarks of Menpo-3D for consistency.

3.2 Ablation Study

Table 3: The comparative and ablative results on AFLW2000-3D and AFLW.The mean NMEs (%) across small, medium and large poses on AFLW2000-3Dand AFLW are reported. lmk. indicates landmark constraint on the parameterregression like [43] and lrr. is the proposed landmark-regression regularization.

Method AFLW2000-3D AFLW

BaselineVDC 5.23 6.37

fWDPC 4.04 5.10

Joint-optimization OptionsVDC from fWPDC 3.88 4.83

Vanilla-joint 3.80 4.80Meta-joint 3.73 4.64

Utilization of 2D landmarks

VDC w/ lrr. 3.92 4.92fWPDC w/ lrr. 3.89 4.84

Meta-joint w/ lmk. 3.71 4.80Meta-joint w/ lrr. 3.59 4.50

12 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

Table 4: Comparisons of NME (%) / Stability (%) on Menpo-3D. svs. indicatesshort-video-synthesis, rnd. indicates applying in-plane and out-of-plane rotationsrandomly in one mini-batch.

Method Menpo-3D

fWPDC w/o svs. 1.96 / 0.54fWPDC w/ svs. 1.84 / 0.51

Meta-joint+lrr. w/o svs. 1.86 / 0.52Meta-joint+lrr. w/ rnd. 1.76 / 0.50Meta-joint+lrr. w/ svs. 1.71 / 0.48

To evaluate the effectiveness of the meta-joint optimization and the landmark-regression regularization, we carry out comparative experiments including ourtwo baselines: VDC and fWPDC, three joint options: (i) VDC from fWPDC :fine-tune the model with VDC loss from the pre-trained model by fWPDC; (ii)Vanilla-joint : weight VDC and fWPDC by the best scalar β = 0.5; (iii) Meta-joint : the proposed meta-joint optimization with best k = 100 and four optionsof how the 2D landmarks are utilized. From Table 3, Table 4, Fig. 3 and Fig. 6,we can draw the following conclusions:

Meta-joint optimization performs better. Comparing with two base-lines VDC and fWPDC, all three joint optimization methods perform better.Among three joint optimization methods, the proposed meta-joint performs bet-ter than VDC from fWPDC and vanilla-joint : the mean NME drops from 4.04%to 3.73% on AFLW2000-3D and 5.10% to 4.64% on AFLW when compared withthe baseline fWPDC. Furthermore, we conduct ablative experiments with differ-ent β for vanilla-joint and different look-ahead step k in Fig. 6. We can observethat β = 0.5 is the best setting for vanilla-joint, but meta-joint still outper-forms it and k = 100 performs best on both AFLW2000-3D and AFLW. Overall,the proposed meta-joint optimization is effective in alleviate the training andpromoting the performance.

Landmark-regression regularization benefits. Another contribution isthe landmark-regression regularization, which can also be regarded as an auxil-iary task to parameters regression. From Table 3, the improvements from fW-PDC to fWPDC w/ lrr. on AFLW2000-3D and AFLW are 0.15% and 0.26%, andthe improvements from Meta-joint to Meta-joint w/ lrr. on AFLW2000-3D andAFLW are 0.14% and 0.14%. We also compare the proposed landmark-regressionregularization with prior methods [43,42] which directly impose landmark con-straint on the parameter regression, the results show ours is significantly better:3.59% vs. 3.71% on AFLW2000-3D. We further evaluate the performance of thelandmark-regression branch on AFLW2000-3D and AFLW. The performancesare 3.58% and 4.52% respectively, which are close to the parameter branch. It in-dicates that these two tasks are highly related. Overall, the landmark-regressionregularization benefits the training and promotes the performance.

Short-video-synthesis improves stability. The last contribution is 3Daided short-video-synthesis, which is designed to enhance stability on videosby augmenting one still image to a short video in a mini-batch. The results in

Towards Fast, Accurate and Stable 3D Dense Face Alignment 13

Table 4 indicate that short-video-synthesis works for both the fWPDC and meta-joint optimization. With short-video-synthesis and landmark-regression regular-ization, the performance on still frames improves from 1.86% to 1.71% and thestability improves from 0.52% to 0.48%. We also evaluate the performance byrandomly applying in-plane and out-of-plane rotations in each mini-batch andfind it is worse than short-video-synthesis: 1.76% / 0.50% v.s. 1.71% / 0.48%.These results validate the effectiveness of the 3D aided short-video-synthesis.

3.3 Evaluations of Accuracy and Stability

Sparse Face Alignment. We use AFLW2000-3D and AFLW to evaluate sparseface alignment performance with small, medium and large yaw angles. The re-sults in Table 1 indicate that our method performs better than PRNet (3.51%vs. 3.62%) in AFLW2000-3D and better than 3DDFA-TPAMI [55] in AFLW(4.43% vs. 4.55%). Note that these results are achieved with only 3.27M param-eters (24% of PRNet) and it takes 6.2ms (3.5% of PRNet) in CPU. The samplingof 68 / 21 landmarks from 3DMM is extremely fast, only 0.01ms (CPU), whichcan be ignored.

Dense Face Alignment. Dense face alignment is evaluated on Florenceand AFLW2000-3D. Our evaluation settings follow [17] to keep consistency. Theresults in Table 2 show that the proposed method significantly outperformsothers. As for 3D dense vertices reconstruction, 45K vertices only takes 1msin CPU (0.1ms in GPU) with our regression framework.

Video-based 3D Face Alignment. We use Menpo-3D to evaluate both theaccuracy and stability. Table 4 has already shown the superiority of short-video-synthesis. We choose to compare our method with recent PRNet [17] in Table 2.The results indicate that our method significantly surpasses PRNet in both theaccuracy and stability on videos of Menpo-3D with a much lower computationcost.

3.4 Evaluations of Speed

We compare parameter numbers, MACs (Multiply-Accumulates) measuring thenumber of fused Multiplication and Addition operations, and the running timeof our method with others in Table 2. As for the running speed, 3DDFA [54]takes 23.2ms (GPU) for predicting parameters and 52.5ms (CPU) for PNCCconstruction, DeFA [17] needs 11.8ms (GPU) to predict 3DMM parameters and23.6ms (CPU) for post-processing, VRN [26] detects 68 2D landmarks with28.4ms (GPU) and regresses the 3D dense vertices with 40.6ms (GPU), PR-Net [17] predicts the 3D dense vertices with 9.8ms (GPU) or 175ms (CPU).Compared with them, our method takes only 2ms (GPU) or 6.2ms (CPU) topredict 3DMM parameters and 0.1ms (GPU) or 1ms (CPU) to reconstruct 3Ddense vertices.

Specifically, compared with the recent PRNet [17], the parameters of ourmodel (3.27M) are less than one-quarter of PRNet (13.4M), and the MACsare less than 1/30 (183.5M vs. 6190M). We measure the overall running time on

14 J. Guo, X. Zhu, Y. Yang, F. Yang, Z. Lei and S. Li

GeForce GTX 1080 GPU and i5-8259U CPU with 4 cores. Note that our methodtakes only 7.2ms, which is almost 24x faster than PRNet (175ms). Besides, webenchmark our method on a single CPU core (using only one thread) and therunning speed of our method is about 19.2ms (over 50fps), including the recon-struction time. The specific CPU configuration is i5-8259U CPU @ 2.30GHz ona 13-inch MacBook Pro.

3.5 Analysis of Meta-joint Optimization

We visualize the auto-selection result of fWPDC and VDC in the meta-jointoptimization, as shown in Fig. 7. We can observe that both k = 100 and k = 200show the same trend: fPWDC dominates in the early stage and VDC guides inthe late stage. This trend is consistent with the previous observations and givesa clear description of why our proposed meta-joint optimization works.

0 100 200 300 400 500Iter. (×400)

k = 100

fWPDCVDC

0 100 200 300 400 500Iter. (×400)

k = 200

fWPDCVDC

Fig. 7: Auto-selection result of the selector in the meta-joint optimization.

4 Conclusion

In this paper, we have successfully pursued the fast, accurate and stable 3D denseface alignment simultaneously. Towards this target, we make three main efforts:(i) proposing a fast WPDC named fWPDC and the meta-joint optimization tocombine fWPDC and VDC to alleviate the problem of optimization; (ii) imposingan extra landmark-regression regularization to promote the performance to state-of-the-art; (iii) proposing the 3D aided short-video-synthesis method to improvethe stability on videos. The experimental results demonstrate the effectivenessand efficiency of our proposed methods. Our promising results pave the way forreal-time 3D dense face alignment in practical use and the proposed methodsmay improve the environment by reducing the amount of carbon dioxide releasedby the huge amounts of energy consumed by GPUs.

Acknowledgement

This work was supported in part by the National Key Research & DevelopmentProgram (No. 2020YFC2003901), Chinese National Natural Science FoundationProjects #61872367, #61876178, #61806196, #61976229.

Towards Fast, Accurate and Stable 3D Dense Face Alignment 15

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