1 Introduction

As the most significant biometric trait of human beings, the human face has been widely used for human identification and verification in the scientific research and the real-world application. The facial data acquisition method is natural, non-intrusive and contactless, which is friendly to accept in social activities [11, 12]. With the development of 3D scanner, 3D printer, Virtual Reality (VR) and Augmented Reality (AR), capturing and reconstructing 3D samples become more convenient in daily-life [30]. Meanwhile, 2D and 3D face recognition have been applied widely in the criminal investigation, the access control, the frontier inspection and the bank service. Even though the techniques of the face recognition have been widely studied in biometrics research area [1, 2, 9, 19, 26, 36] and many state-of-the-arts have been reported in many publications, the spoofing attacks against face recognition systems is a potential threat to biometric application.

Spoofing attack is defined as an intrusive act of deceiving a biometric system by presenting a fake evidence or a copied biometric trait to obtain a valid authentication [28]. By using photographs or videos captured in distance or collected via internet, the attacker can easily achieve the facial information of a valid user registered in a face recognition system. Then the attacker shows the fake photograph printed on paper or the recorded video displayed on a tablet for attempting to get access in the system. Furthermore, since few yeas ago, a social public website “Thats My Face”Footnote 1 started to provide the wearable 3D mask manufacturing service with only one frontal photo (another side-view photo is asked as option). It further reduced the difficulty of attacker’s deception by wearing such a 3D printed mask. The simplicity and the convenience of the acquirement and the manufacture of the 2D/3D face data, which should be the advantages of the face recognition, become gradually the jeopardy and the calamity to the reliability and the stability of the face recognition system.

Due to the vulnerability of face recognition systems, many papers have been published on countermeasure studies, and the reported experimental results showed that the corresponding methods are sufficient and efficient. Among the published works, liveness detection [3, 15, 29, 39], motion detection [5, 14] and texture analysis [20, 25] are three principal categories of anti-spoofing methods [6, 7] against photo- and video-based spoofing attacks. However, with the help of the improvement of 3D manufacture technology, the easily obtained high-quality 3D masks introduce new challenge to anti-spoofing research. Morpho databaseFootnote 2 and 3DMAD database [6], including 2D, 2.5D and 3D face samples of genuine person and imposters wearing 3D mask, were constructed for simulating this mask intrusion. Kose et al. proposed the countermeasure based on the fusion of the information extracted from texture and depth images, and tested it on the Morpho database [16,17,18]. Erdogmus et al. evaluated various LBP based countermeasures on texture images in [6, 7]. All of their works reported that the texture information can be essential discriminative characters to distinguish real faces and masks. However, the shape analysis based approach, as an important kind of methods in 3D face recognition, has not been discussed and studied in their works. To fill this gap, in this paper, we aims to evaluate the anti-spoofing performance of this kind of method.

The general technique of the shape analysis based 3D face recognition is to utilize the geometric attributes to describe and characterize the facial surface precisely. Geometry attributes, including principal curvatures, Gaussian curvature, mean curvature and their variations (e.g. shape index), have been commonly used to 3D face representation [21, 22, 35], keypoints location [24, 26, 38] and 3D facial feature descriptor generation [8, 19, 31]. In this study, we exploit “curvature measures” developed by [27, 32, 33] based on concept of normal cycle [4] to extract shape information of discrete surface (e.g. 3D meshed face), and design the corresponding facial description and recognition framework. This triangle mesh based geometric feature can highlight the micro-shape dissimilarity between genuine faces and manufactured masks, which leaves us more opportunities to distinguish them.

Besides, in the real-life scenario, comparing to the verification cases with genuine face samples, the spoofing mask attacks appear more rarely, which are regarded as an exceptional testing case. In order to simulate such a scenario, we firstly propose to combine Morpho database and FRGCv2 database [1], the largest public 3D face database, to appraise the anti-spoofing performance. The gallery set and the genuine probe set are formed by the genuine face samples coming from both Morpho database and FRGCv2 database, while the testing fake face scans from Morpho database are 5% and 1% of the scale of the whole probe set. This performance evaluation scenario corresponds better to the real-life case. Meanwhile, the conventional discriminative power evaluation and anti-spoofing performance test on Morpho database are also reported in the experiment part.

The rest of this paper is organized as follows. Section 2 gives a brief review of related anti-spoofing face recognition works. Section 3 presents principal curvature measures estimation method and the related shape analysis based facial feature descriptor. Section 4 shows the experimental results in several scenarios, and Sect. 5 finally concludes the paper.

2 Related Work

In the history of mask anti-spoofing research, the work of Kim et al. [13] can be regarded as the first published one. Due to the difference of the reflectance between face skin and materials used to manufacture mask, their work aimed to analyse the distribution of albedo values for illumination at various wavelengths. Based on Fisher’s linear discriminant, they selected a 2D feature vector consisting of radiance measurement to be the classification criteria in visual and NIR spectrum (685 and 850 nm respectively). Similarly, Zhang et al. published their mask detection countermeasure based on multi-spectral analysis in [39]. They claimed to abandon visual face image, but to analyse multi-spectral images captured in two discriminative wavelengths of illumination (850 and 1450 nm). They measured the albedo curves of different materials and trained SVM classifier to distinguish real face and mask. Even though these two papers above are effective in mask distinguishing, they haven’t resolve the anti-spoofing problem. Besides an extra multi-spectral capturing device is obligatory in their defence system.

Lately, Kose et al. reported their anti-spoofing works based on texture and depth information in 2D and 2.5D images from Morpho Database. Three baseline face recognition algorithms are tested and an anti-spoofing mask attack related experimental strategy is mentioned in [17]. Furthermore, they extracted LBP features in color and depth image and trained the linear SVM classifier to determine whether the input sample is genuine or fake in [16]. Then in order to combine the advantage of color and depth image in anti-spoofing mask attack, two feature and score level fusions were proposed in [18]. They concluded that texture analysis is a effective method for developing a countermeasure. Similarly, Erdogmus et al. introduced their 3D Mask Attack Database (3DMAD) and anti-spoofing countermeasures based on three extended LBP algorithms in [6]. Some more comparative anti-spoofing experimental results on Morpho and 3DMAD databases were reported in [7]. Even though the works presented above introduced great countermeasures against 3D mask attacks, all of their main methods are designed based on 2D images projected from 3D face scans. Moreover their evaluation of the anti-spoofing capability of 3D face samples is constructed relying on Thin Plate Spline (TPS) warping parameters and Iterative Closest Point (ICP) methods. But 3D shape based geometric attributes, as potent characters in 3D shape analysis, haven’t been evaluated their anti-spoofing potentiality. Besides, all their experiments are limited into the mask attack related database which doesn’t includes not enough genuine face scans to test. In this paper, we have two main purposes to fill the gaps:

  • We take full advantage of principal curvature measures based on asymptotic cone theory to design a facial descriptor, and evaluate the potential anti-spoofing performance of the facial descriptor for 3D meshed face scans.

  • For simulating a real-life recognition case, we firstly attempt to combine a scale limited genuine-imposter combined database (i.e. Morpho database) to a large genuine face scans database (i.e. FRGCv2 database), and perform comprehensive experimental scenarios.

3 Principal Curvature Measures Based 3D Face Recognition Scheme

The shape analysis is regarded as an important kind of 3D face recognition methods. Among this branch of approaches, both a precise estimated geometric attribute and a proper related facial feature descriptor are significant to represent and describe the shape of facial surface. In this section, we will introduce our principal curvature measures estimation method and their related 3D face recognition scheme, which meet these two demands of this kind of method. Meanshile, because the surface of manufactured mask is smoother than real face which is stated in [17, 18], our proposed descriptor, which can highlight the dissimilarity of the minor shape between genuine faces and masks, is capable to verify the liveness of the testing samples. The pipeline of our proposed method includes 4 steps: principal curvature measures estimation, 3D keypoint detection, 3D keypoint feature description and 3D keypoint matching.

3.1 Principal Curvature Measures Estimation

In general speaking, principal curvatures are the most basic but fundamental geometrical attributes in differential geometry. They are defined regularly as below: Suppose a point p locating on a smooth oriented surface S, its principal curvatures \(\lambda _{1_p}\) and \(\lambda _{2_p}\) are estimated as the eigenvalue sets of the corresponding second fundamental form h (q is the quadratic form associated to h). \(\lambda _{1_p}\) and \(\lambda _{2_p}\) can describe the local bending information around p of S. Remark that this definition is coherent because of the smoothness of surface S. However, 3D face sample is commonly recorded as the triangle mesh, which is continuous but piece-wise smooth. It makes the conventional curvature estimation method unsuitable here. A possible solution proposed and demonstrated in [32, 33] is to generalize the definition of curvatures to the discrete surface and to replace functions by measures. Here we will present the generalization from the smooth surface case to the discrete surface case. Please refer [32, 33] for a comprehensive and detailed introduction of the generalization.

Principal Curvature Measures of Smooth Surfaces. Inspiered to [32, 33], the second fundamental form h and its associated quadratic form q can be generalized to a measure on a smooth surface S on \(\mathbb {E}^3\). Suppose that any Borel subset B of \(\mathbb {E}^3\) and any vector field X of \(\mathbb {E}^3\), the definition of \(\overline{h}_B\) and \(\overline{q}_B\) are:

$$\begin{aligned} \begin{aligned} \overline{h}_B(X,X)&= \int _{S \cap B} h_p(pr_{T_pS}X_p, pr_{T_pS}X_p) dp,\\ \overline{q}_B(X)&= \int _{S \cap B} h_p(pr_{T_pS}X_p, pr_{T_pS}X_p) dp \\&= \int _{S \cap B} q_p(pr_{T_pS}X_p) dp, \end{aligned} \end{aligned}$$
(1)

where \(pr_{T_pS}\) denotes the orthogonal projection over the tangent plane \(T_pS\) of S at p. If X is a constant vector fields in \(\mathbb {E}^3\), for any fixed Borel subset, \(\overline{q}_B(X)\) is a measure. {\(\lambda _{1_B},\lambda _{2_B},\lambda _{3_B}\)} is the associated eigenvalue set, and {\(e_{1_B},e_{2_B},e_{3_B}\)} is the eigenvector set of \(\overline{h}_B\). The map \(\lambda _i:B \rightarrow \lambda _{i_B},i\in \{1,2,3\}\) is a measure of \(\mathbb {E}^3\), named as principal curvature measure. Remark that, in piece-wise case, principal curvature measures have three components rather than two in the point-wise approach.

Principal Curvature Measures of Triangle Meshes. A triangle mesh is a discrete surface apparently, which means its shape and bending information can not be describe by point-wise approach. That’s why the measure theoretic method is coherent in triangle mesh case. According to the concept and theory of normal cycle [4, 27], suppose a triangle mesh \(\mathcal {T}\) in \(\mathbb {E}^3\), an explicit formula of \(\overline{h}\) and \(\overline{q}\) defined of constant vector field X as:

$$\begin{aligned} \begin{aligned} \overline{h}_{B}(X,X)&=\sum _{e\in E}{}{l(e\cap B)}\angle (e)\,{<}X,e{>}{<}X,e{>},\\ \overline{q}_{B}(X)&=\sum _{e\in E}{}{l(e\cap B)}\angle (e)\,{<}X,e{>}^2. \end{aligned} \end{aligned}$$
(2)

where E denotes the set of edge e of \(\mathcal {T}\), \(l(e \cap B)\) denotes length of e belongs to B, and \(\angle e\) denotes the signed angle between unit normals \(n_1\) and \(n_2\) of incident facets \(f_1\) and \(f_2\) to e. Meanwhile, \(\overline{h}_B\) associated matrix \(F_B\) is written as:

$$\begin{aligned} F_B= \sum _{e \in E}l(e \cap B)\angle (e) e\cdot e^{t}. \end{aligned}$$
(3)

We similarly name the set of eigenvalues {\(\lambda _{1_B},\lambda _{2_B},\lambda _{3_B}\)} of \(\overline{h}_B\) is the principal curvature measures of \(\mathcal {T}\) over B. The corresponding set of eigenvectors {\(e_{1_B},e_{2_B},e_{3_B}\)} of \(\overline{h}_B\) can also be estimated. Based upon the generalization of \(\overline{h}_B\), three eigenvectors are respectively two principal directions and one normal direction of X over B.

In summary, the set of principal curvature measures \(\lambda _{1_B},\lambda _{2_B},\lambda _{3_B}\) is coherent to the geometrical properties of 3D face scans in triangle mesh and comprehensively suitable to describe facial surface. These principal curvature measures are the essential geometric attributes (i.e. the geometric feature) of our 3D keypoint descriptor presented in following parts.

3.2 \(\lambda _B\) Based 3D Keypoint Detection

In order to guarantee the scale invariance property of facial descriptor, our keypoint detection is inspired by Lowe’s SIFT [23] and related works [19, 31], but the difference of principal curvature measures are used to locate keypoints. We firstly construct the Gaussian scale space by smoothing the face scan in triangle mesh with a series of Gaussian kernel \(g_{\sigma _s}\) (\(\sigma _s\) denotes to different scales). Given a vertex \(v_i\) in a face scan, the facial surface adjacent becomes smoother by convolving \(g_{\sigma _s}\) over neighbour vertices \(v_j\) and \(v_i\) is updated to \(v_{i_{\sigma _s}}\) as (4).

$$\begin{aligned} v_{i_{\sigma _s}}=\frac{\sum _{v_j\in {N}(v_i,1)}{}g_{\sigma _s}(v_i,v_j)\cdot v_j}{\sum _{v_j\in { N}(v_i,1)}{}g_{\sigma _s}(v_i,v_j)} \end{aligned}$$
(4)

where \(N(v_i,1)\) denotes the set of vertices within 1-ring neighbourhood of \(v_i\) and Gaussian kernel \(g_{\sigma _s}\) is defined as

$$\begin{aligned} g_{\sigma _s}(v_i,v_j) =exp(-\left\| v_i-v_j \right\| ^2/2\sigma _s^2). \end{aligned}$$
(5)

We estimate principal curvature measures \(\lambda _{i_B}\) over each scale space of 3D facial surface as introduced in Sect. 3.1, and then compute the keypoint location criterion called Difference of Curvatutre (DoC), referring to Difference of Gaussian (DoG) in SIFT.

$$\begin{aligned} \delta (\lambda _{i}(B_{v_{\sigma _s}}) = \lambda _{i}(B_{v_{\sigma _s}})-\lambda _{i}(B_{v_{\sigma _{s-1}}}),i=1,2,3 \end{aligned}$$
(6)

where \(\delta \) denotes Difference of Curvature over B. If DoC associated to \(v_i\) is the extreme among 1-ring vertices around \(v_i\) in three scales \(\sigma _{s-1}\), \(\sigma _{s}\) and \(\sigma _{s+1}\), \(v_i\) is defined as a keypoint \(v_k\) and \(\sigma _s\) is its corresponding detection scale. The keypoints detected by three principal curvature measures separately are combined as one group for following 3D keypoint description.

3.3 \(\lambda _B\) Based 3D Keypoint Description

3D keypoint description can be divided into two parts. The first one is to assign a primary direction for improving the robustness to minor head pose change. The second part is to construct histograms of curvature measures based feature descriptor.

Primary Direction Determination. Let’s suppose one keypoint \(v_k\) and its proper scale \(\sigma _s\) of a face scan F as before. The primary direction \(\mathbf d _{v_k}\) associated to \(v_k\) is defined by the neighbour vertices \(\mathcal {N}(v_k)\) within a geodesic disc of \(\sigma _s\) related radius \(R_{\sigma _s}\):

$$\begin{aligned} \mathcal {N}(v_k)=\{ v_j \in F| Dist(v_k,v_j) \le R_{\sigma _s}\} \end{aligned}$$
(7)

We first determine a plane \(TS_{v_k}\) orthogonal to the unit normal vector \(\xi _{v_k}\) of \(v_k\), then project the unit normal vector \(\xi _{v_j}\) of \(v_j\) on \(TS_{v_k}\). The primary direction of \(v_k\) is created by computing a Gaussian weighted histogram of 360 bins (1 bin per degree) on \(TS_{v_k}\), and determined as the peak of the weighted direction histogram. Here the Gaussian weight is defined as:

$$\begin{aligned} \begin{aligned} w(v_k,v_j)=mag(v_j)\cdot g_{\sigma _s}(v_k,v_j), \\ mag(v_j)= \sqrt{\xi ^x(v_j)^2 + \xi ^y(v_j)^2}. \end{aligned} \end{aligned}$$
(8)

HoC Based Feature Descriptor Representation. We construct the feature descriptor of a keypoint by using principal curvature measures estimated on a set of neighbor vertices. Following 2D daisy descriptor [37], all the neighbor vertices locating in 9 ovelapping circles \(r_1, r_2, \cdots , r_9\) with a radius of \(3.75\sigma _s\) around the keypoint support one keypoint feature descriptor. \(r_1\) locates in the central part and its center is the keypoint here. Starting form the primary direction, other 8 circles around range along as clock-wise order (as shown in Fig. 1), and the distance from their center to the keypoint is \(4.5\sigma _s\). This kind of daisy flower pattern descriptor simulates the functioning of human complex cells in visual cortex [10], and tends to be invariant to minor face transformation.

Then we build three histograms of three principal curvature measures (\(hoc_i\)) respectively in each circular region of \(r_1, r_2, \cdots , r_9\). In each circular region, the value of \(i^{th}\) principal curvature measure are quantized equally to 8 bins and weighted by Gaussian kernel, and the standard deviation is assumed as the Euclidean distance between current point to corresponding center of circle. After that, we normalize and concatenate all three principal curvature measures related 27 histograms (3 principal curvature measures \(\times \) 9 regions) following this rule:

$$\begin{aligned} \begin{aligned} HOC = \{ hoc_1^{r_1}, hoc_1^{r_2} \cdots hoc_1^{r_9}, hoc_2^{r_1}, hoc_2^{r_2} \cdots hoc_2^{r_9}, hoc_3^{r_1}, hoc_3^{r_2} \cdots hoc_3^{r_9} \} \end{aligned} \end{aligned}$$
(9)

HOC, denoting to Histogram Of principal Curvature measures, is the keypoint feature descriptor.

Fig. 1.
figure 1

Keypoint descriptor configuration in 9 overlapping circles for generating HoC.

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3.4 3D Keypoint Matching

For keypoint matching, we aim to find matched keypoint pairs based on HOC feature descriptor. Assume a keypoint \(v_{k_i}^1\) belongs to first facial surface and the set of all keypoints \(\{v_{k_j}^2\}\) in second facial surface. We estimate the angle set \(\{\alpha _j^i\}\) between feature vectors of \(v_{k_i}^1\) and \(\{v_{k_j}^2\}\) inspired to [31]. Each angle is defined as:

$$\begin{aligned} \alpha _j^i = cos^{-1}\left( \frac{<HOC_i^1,HOC^2_j>}{||HOC^1_i|| \cdot || HOC_j^2 ||}\right) . \end{aligned}$$
(10)

The angles \({\alpha _j^i}\) are then ranked in ascending order. If the ratio between the first and second angle is smaller than predefined threshold \(r_{\alpha }\), the match is accept. Otherwise it is rejected. Finally the number of matching keypoints is set as the similarity measurement \(\mu \) between two facial surfaces.

4 Experiments

4.1 Database

In the experiment part, Morpho database and FRGC database are both involved for evaluation in different scenarios. We will introduce them briefly as follow.

Morpho Database. In Morpho databse, 16 masks were manufactured according to the facial information of 16 persons. Their faces are captured by 3D scanner with the structured light, and the mask is manufactured with 3D printer by Sculpteo 3D Printing [16]. Morpho database consists of two parts: (a) 20 subjects with 10 genuine face samples; (b) 20 subjects wearing their own or other’s mask are captured around 10 times. In part (b), a person wearing his/her own mask is marked as a type \(A_A\) mask sample. Otherwise it’s marked as a type \(A_B\) mask sample. In the following experiments, both \(A_A\) and \(A_B\) samples are both regarded as spoofing mask attacks. Some examples in Morpho database are shown in Fig. 2.

Fig. 2.
figure 2

2D texture image and corresponding 3D mesh sample in Morpho database. (a) genuine face scan of person A; (b) genuine face scan of person B; (c) person A wears his own mask (type \(A_A\) fake sample); (d) person A wears person B’s mask (type \(A_B\) fake sample).

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FRGC V2.0 Database. FRGCv2 database is the largest published 3D face database, which is built with 4,007 3D face scans of 466 subjects with various facial expressions, genders and ages. All the face samples recorded are genuine faces. The face samples are captured in controlled pose and lighting condition by Minolta Vivid 900 scanner. After preprocessing step following [34], ROI contains about 30,000 vertices and 40,000 facets.

4.2 Experiment Scenarios

The basic purpose of our 3D face verification system is to guarantee the verification accuracy for genuine face, and the complementary purpose is to distinguish faces and masks. We hence firstly define a series of threshold of the similarity measurement \(t_\mu ^i\) in a baseline estimation scenario, and then we apply the same series of thresholds to evaluate the anti-spoofing performance of our facial feature descriptor. Furthermore, we also try to control the quantity ratio of genuine faces to fake faces in probe set for simulating a real-life case. Based on this idea, two series of experiment scenarios (Scenario A and B) are designed. For introducing clearly in following part, \(G_M\) and \(S_M\) denotes respectively to the set of all genuine face scans and the set of all spoofing mask scans (including type \(A_A\) and \(A_B\) mask scans) in Morpho database. \(G_F\) denotes to the group of all genuine face scans in FRGC database. \({G_M}_1\) and \({G_M}_i\) represents respectively the group of the first scan of individuals, and other samples of individuals in \(G_M\) (similarly for \(G_F\)).

  • Scenario A-1: Baseline evaluation with Morpho database.

    \({G_M}_1\) forms gallery set, and \({G_M}_i\) builds probe set.

  • Scenario A-2: Anti-spoofing performance evaluation with Morpho database.

    \({G_M}_1\) forms gallery set, and \({G_M}_i\) and \(S_M\) build respectively probe set and spoofing probe set.

  • Scenario B-1: Baseline evaluation simulating real-life case.

    \({G_M}_1\) and \({G_F}_1\) build gallery set, and \({G_M}_i\) and \({G_F}_i\) build probe set.

  • Scenario B-2: Simulating real-life case including 5% fake scans against 95% genuine face scans.

    \({G_M}_1\) and \({G_F}_1\) build gallery set, \({G_M}_i\) and \({G_F}_i\) build genuine face probe set. \(S_M\) build spoofing probe set.

  • Scenario B-3: Simulating real-life case including 1% fake scans against 99% genuine face scans.

    Gallery set and genuine face probe set is same as B-2, but only 20% scans of \(S_M\) enrolled in spoofing probe set. The quantity ratio between fake scans and genuine face scans is 1:99.

4.3 Experimental Results and Analysis

In scenarios A-1 and B-1, we evaluate the baseline of verification performance with True Accept Rate (TAR). Because TAR varies along with \(t_\mu ^{i}\), TAR is shown in Table 1 under several False Accept Rate (FAR) cases. For scenarios A-2, B-2 and B-3 in Table 1, we first take only spoofing probe set to evaluate the anti-spoofing performance. Recall it that all scans in spoofing probe set are type \(A_A\) or \(A_B\) mask scans which are treated as illegal samples and should be rejected by system. True Accept Rate (TAR) and False Reject Rate (FRR), which are commonly used as criterion, can’t be estimated in this case, because there are only two verification results for the spoofing probe sample, which are accepted falsely (FAR) or rejected correctly (TRR). Therefore we evaluate the performance with Spoofing True Reject Rate (STRR) as criterion for spoofing scans in scenarios A-2, B-2 and B-3 with FAR predefined in scenario A-1 and B-2 respectively. STRR is a special criteria for the spoofing samples, so as to show the distinction to TRR for the genuine face samples.

As shown in Table 1, the verification rate in baseline evaluation with Morpho Database is above 92% except the case that FAR is 0.01. And in real-life simulating case, we extend the scale of database by adding FRGC database and the verification rate is 91.98% even FAR equals only 0.001. If FAR is 0.01, the TAR increases from 84.75% to 94.68% which means the real-life simulating scenario with more samples can evaluate more properly the performance. Here, we can conclude that, in the baseline evaluation scenarios, our 3D face feature guarantees the verification performance for only genuine faces. Even though TAR is 84.75% in A-1, it can’t deny the above conclusion. Because the scale of genuine face scans in A-1 is 180, which means there are only 1 or 2 samples accepted falsely if FAR is 0.01. And that’s why we don’t show the results of A-1 and A-2 when FAR is 0.001. By the way, based on the contrary thought, the results when FAR is 0.1 are blank in last three scenarios. Because there are 3,721 genuine face scans in scenario B-1, and if FAR is 0.1, there are too many samples accepted falsely which can’t show the performance correctly.

In scenario A-2, STRR is above 62.82% and arrives 75.64% when FAR is 0.01. It’s obvious that our algorithm can distinguish the genuine face and fraud mask from Morpho database. STRR arises along with decrease of FAR because threshold \(t_\mu ^i\) is more restricted. Moreover, in real-life simulating case, STRR is higher than 62.97% and achieves 81.28% and 84.59% including different spoofing probe set. In this case, STRR is lower a bit with same FAR than A-2 because the scale of gallery set is larger and \(t_\mu ^i\) is more rough. However, this experimental results also demonstrate our algorithm can complete the anti-spoofing mission even the mask manufactured in high quality.

Table 1. Verification and anti-spoofing performance evaluation in scenarios
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After that, for scenarios A-2, B-2 and B-3, we combine genuine face related probe set (\({G_M}_i\) and \({G_F}_i\)) and spoofing probe set (\(S_M\)) to evaluate the performance. Here a Detection Error Trade-off graph (DET) is given to show the experimental results (as shown in Fig. 3). Remark that in order to present clearly, the figure only shows the part when FAR and FRR is lower than 35% in scenarios B. In Fig. 3, the lower Equal Error Rate (ERR) is, with same gallery set, the better verification performance is. In this DET graph, when the scale of gallery set is limited in Morpho database, EER of A-2 is 9.3% higher than the baseline. It’s obvious to study that the involvement of spoofing mask samples decreases the verification rate in scenarios A. In similar, when we extend the scale of gallery set enrolling FRGC database, EER of B-2 and B-3 is higher than corresponding baseline experiment. But comparing to A-2, EER declines to 3.1% and 2.8%. A similar conclusion obtained as before that two goals have been achieved: (1) the verification system guarantees a high verification ability and (2) it possesses the distinguishable power against spoofing attack in real-life simulating case.

Table 2. Comparison of verification performance with spoofing attacks in Morpho database. (1) Results reported in [17], (2) results reported in [7], (3) results with our proposed method.
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Fig. 3.
figure 3

Detection error trade-off graph of experimental scenarios: Scenarios A-1 and A-2 are shown in left graph, Scenarios B-1, B-2 and B-3 are shown in right graph.

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4.4 Comparison with the State-of-the-Art Approaches

In this subsection the comparison with the state-of-the-art approaches using Morpho database is also given in Table 2. According to the experimental configuration assigned in [7, 17], here we use scenario A-1 to compute the EER. Then adopt the same threshold in A-2 to compute the SFAR, that is, Spoofing False Accept Rate. Spoofing False Accept indicates to the case that the samples with mask is false accepted by the sysem. In Table 2, we only report our experimental results using 3D mesh samples. Comparing to 2D texture image and 2.5D depth image, the face recognition based on 3D face samples generally achieve higher verification performance with all genuine face samples in the test. The warping parameters related method in [17] achieves the lowest EER of 3.85%, which is better than 6.91% EER obtained by our method. However, the WP-related FR system is the most vulnerable one among the reported systems. Our PCM-meshSIFT-based method is the most robust system when replace all the probe samples by the samples with masks. We achieve the lowest SFAR of 33.10% in such experiment. It proves that the minor shape difference between genuine faces and manufactured masks can be detected and highlighted by our principal curvature measure based 3D facial feature, which is effective to enhance the security level of FR system.

5 Conclusion

In this paper, we first propose to using 3D shape description related method to distinguish the genuine faces and the spoofing masks stored in 3D triangle meshes. Due to the estimation process relying on a integral form, principal curvature measures are suitable to present the shape of triangle mesh directly. Furthermore, principal curvature measures related feature descriptor can characterize properly the shape information of facial surface, and highlight the minor dissimilarity of shape between manufactured masks and genuine faces. Thereby our system can guarantee both high verification rate for genuine face and satisfactory anti-spoofing performance against mask attack.

Moreover, in a real-life case, the spoofing mask attack is a small proportion of testing samples. In experiment part, we hence propose to extend the probe set by combining mask samples in Morpho database and genuine faces in FRGCv2 database for simulating a real-life verification environment. The experimental results show that our method is effective in verification scenario and anti-spoofing performance during this simulating case.