Face anti-spoofing (FAS) is significant for the security of face recognition systems. neural networks (NNs), including convolutional neural network (CNN) and vision transformer (ViT), have been dominating the field of the FAS. However, NN-based methods are vulnerable to adversarial attacks. Attackers could insert adversarial noise into spoofing examples to circumvent an NN-based face-liveness detector. Our experiments show that the CNN or ViT models could have at least an 8% equal error rate (EER) increment when encountering adversarial examples. Thus, developing methods other than NNs is worth exploring to improve security at the system level. In this paper, we have proposed a novel solution for FAS against adversarial attacks, leveraging a deep forest model. Our approach introduces a multi-scale texture representation based on local binary patterns (LBP) as the model input, replacing the grained-scanning mechanism (GSM) used in the traditional deep forest model. Unlike GSM, which scans raw pixels and lacks discriminative power, our LBP-based scheme is specifically designed to capture texture features relevant to spoofing detection. Additionally, transforming the input from the RGB space to the LBP space enhances robustness against adversarial noise. Our method achieved competitive results. When testing with adversarial examples, the increment of EER was less than 3%, more robust than CNN and ViT. On the benchmark database IDIAP REPLAY-ATTACK, a 0% EER was achieved. This work provides a competitive option in a fusing scheme for improving system-level security and offers important ideas to those who want to explore methods besides CNNs. To the best of our knowledge, this is the first attempt at exploiting the deep forest model in the problem of FAS, with the consideration of adversarial attacks.
Citation: Rizhao Cai, Liepiao Zhang, Changsheng Chen, Yongjian Hu, Alex Kot. Learning deep forest for face anti-spoofing: An alternative to the neural network against adversarial attacks[J]. Electronic Research Archive, 2024, 32(10): 5592-5614. doi: 10.3934/era.2024259
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Face anti-spoofing (FAS) is significant for the security of face recognition systems. neural networks (NNs), including convolutional neural network (CNN) and vision transformer (ViT), have been dominating the field of the FAS. However, NN-based methods are vulnerable to adversarial attacks. Attackers could insert adversarial noise into spoofing examples to circumvent an NN-based face-liveness detector. Our experiments show that the CNN or ViT models could have at least an 8% equal error rate (EER) increment when encountering adversarial examples. Thus, developing methods other than NNs is worth exploring to improve security at the system level. In this paper, we have proposed a novel solution for FAS against adversarial attacks, leveraging a deep forest model. Our approach introduces a multi-scale texture representation based on local binary patterns (LBP) as the model input, replacing the grained-scanning mechanism (GSM) used in the traditional deep forest model. Unlike GSM, which scans raw pixels and lacks discriminative power, our LBP-based scheme is specifically designed to capture texture features relevant to spoofing detection. Additionally, transforming the input from the RGB space to the LBP space enhances robustness against adversarial noise. Our method achieved competitive results. When testing with adversarial examples, the increment of EER was less than 3%, more robust than CNN and ViT. On the benchmark database IDIAP REPLAY-ATTACK, a 0% EER was achieved. This work provides a competitive option in a fusing scheme for improving system-level security and offers important ideas to those who want to explore methods besides CNNs. To the best of our knowledge, this is the first attempt at exploiting the deep forest model in the problem of FAS, with the consideration of adversarial attacks.
Let A indicate an analytic functions family, which is normalized under the condition f (0)= f′(0)−1=0 in U={z:z∈C and |z |<1} and given by the following Taylor-Maclaurin series:
f (z)=z+∞∑n=2anzn . | (1.1) |
Further, by S we shall denote the class of all functions in A which are univalent in U.
With a view to recalling the principle of subordination between analytic functions, let the functions f and g be analytic in U. Then we say that the function f is subordinate to g if there exists a Schwarz function w(z), analytic in U with
ω(0)=0, |ω(z)|<1, (z∈U) |
such that
f (z)=g (ω(z)). |
We denote this subordination by
f≺g or f (z)≺g (z). |
In particular, if the function g is univalent in U, the above subordination is equivalent to
f (0)=g (0), f (U)⊂g (U). |
The Koebe-One Quarter Theorem [11] asserts that image of U under every univalent function f∈A contains a disc of radius 14. thus every univalent function f has an inverse f−1 satisfying f−1(f(z))=z and f ( f−1 (w))=w (|w|<r 0(f ),r 0(f ) >14 ), where
f−1(w)=w−a2w2+(2a22−a3)w3−(5a32−5a2a3+a4)w4+⋯. | (1.2) |
A function f∈A is said to be bi-univalent functions in U if both f and f−1 are univalent in U. A function f∈S is said to be bi-univalent in U if there exists a function g∈S such that g(z) is an univalent extension of f−1 to U. Let Λ denote the class of bi-univalent functions in U. The functions z1−z, −log(1−z), 12log(1+z1−z) are in the class Λ (see details in [20]). However, the familiar Koebe function is not bi-univalent. Lewin [17] investigated the class of bi-univalent functions Λ and obtained a bound |a2|≤1.51. Motivated by the work of Lewin [17], Brannan and Clunie [9] conjectured that |a2|≤√2. The coefficient estimate problem for |an|(n∈N,n≥3) is still open ([20]). Brannan and Taha [10] also worked on certain subclasses of the bi-univalent function class Λ and obtained estimates for their initial coefficients. Various classes of bi-univalent functions were introduced and studied in recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al. [20]. Motivated by this, many researchers [1], [4,5,6,7,8], [13,14,15], [20], [21], and [27,28,29], also the references cited there in) recently investigated several interesting subclasses of the class Λ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. Recently, many researchers have been exploring bi-univalent functions, few to mention Fibonacci polynomials, Lucas polynomials, Chebyshev polynomials, Pell polynomials, Lucas–Lehmer polynomials, orthogonal polynomials and the other special polynomials and their generalizations are of great importance in a variety of branches such as physics, engineering, architecture, nature, art, number theory, combinatorics and numerical analysis. These polynomials have been studied in several papers from a theoretical point of view (see, for example, [23,24,25,26,27,28,29,30] also see references therein).
We recall the following results relevant for our study as stated in [3].
Let p(x) and q(x) be polynomials with real coefficients. The (p,q)− Lucas polynomials Lp,q,n(x) are defined by the recurrence relation
Lp,q,n(x)=p(x)Lp,q,n−1(x)+q(x)Lp,q,n−2(x)(n≥2), |
from which the first few Lucas polynomials can be found as
Lp,q,0(x)=2,Lp,q,1(x)=p(x),Lp,q,2(x)=p2(x)+2q(x),Lp,q,3(x)=p3(x)+3p(x)q(x),.... | (1.3) |
For the special cases of p(x) and q(x), we can get the polynomials given Lx,1,n(x)≡Ln(x) Lucas polynomials, L2x,1,n(x)≡Dn(x) Pell–Lucas polynomials, L1,2x,n(x)≡jn(x) Jacobsthal–Lucas polynomials, L3x,−2,n(x)≡Fn(x) Fermat–Lucas polynomials, L2x,−1,n(x)≡Tn(x) Chebyshev polynomials first kind.
Lemma 1.1. [16] Let G{L(x)}(z)be the generating function of the (p,q)−Lucas polynomial sequence Lp,q,n(x).Then,
G{L(x)}(z)=∞∑n=0Lp,q,n(x)zn=2−p(x)z1−p(x)z−q(x)z2 |
and
G{L(x)}(z)=G{L(x)}(z)−1=1+∞∑n=1Lp,q,n(x)zn=1+q(x)z21−p(x)z−q(x)z2. |
Definition 1.2. [22] For ϑ≥0, δ∈R, ϑ+iδ≠0 and f∈A, let B(ϑ,δ) denote the class of Bazilevič function if and only if
Re[(zf′(z)f(z))(f(z)z)ϑ+iδ]>0. |
Several authors have researched different subfamilies of the well-known Bazilevič functions of type ϑ from various viewpoints (see [3] and [19]). For Bazilevič functions of order ϑ+iδ, there is no much work associated with Lucas polynomials in the literature. Initiating an exploration of properties of Lucas polynomials associated with Bazilevič functions of order ϑ+iδ is the main goal of this paper. To do so, we take into account the following definitions. In this paper motivated by the very recent work of Altinkaya and Yalcin [3] (also see [18]) we define a new class B(ϑ,δ), bi-Bazilevič function of Λ based on (p,q)− Lucas polynomials as below:
Definition 1.3. For f∈Λ, ϑ≥0, δ∈R, ϑ+iδ≠0 and let B(ϑ,δ) denote the class of Bi-Bazilevič functions of order t and type ϑ+iδ if only if
[(zf′(z)f(z))(f(z)z)ϑ+iδ]≺G{L(x)}(z)(z∈U) | (1.4) |
and
[(zg′(w)g(w))(g(w)w)ϑ+iδ]≺G{L(x)}(w)(w∈U), | (1.5) |
where GLp,q,n(z)∈Φ and the function g is described as g(w)=f−1(w).
Remark 1.4. We note that for δ=0 the class R(ϑ,0)=R(ϑ) is defined by Altinkaya and Yalcin [2].
The class B(0,0)=S∗Λ is defined as follows:
Definition 1.5. A function f∈Λ is said to be in the class S∗Λ, if the following subordinations hold
zf′(z)f(z)≺G{L(x)}(z)(z∈U) |
and
wg′(w)g(w)≺G{L(x)}(w)(w∈U) |
where g(w)=f−1(w).
We begin this section by finding the estimates of the coefficients |a2| and |a3| for functions in the class B(ϑ,δ).
Theorem 2.1. Let the function f(z) given by 1.1 be in the class B(ϑ,δ). Then
|a2|≤p(x)√2p(x)√|{((ϑ+iδ)2+3(ϑ+iδ)+2)−2(ϑ+iδ+1)2}p2(x)−4q(x)(ϑ+iδ+1)2|. |
and
|a3|≤p2(x)(ϑ+1)2+δ2+p(x)√(ϑ+2)2+δ2. |
Proof. Let f∈B(ϑ,δ,x) there exist two analytic functions u,v:U→U with u(0)=0=v(0), such that |u(z)|<1, |v(w)|<1, we can write from (1.4) and (1.5), we have
[(zf′(z)f(z))(f(z)z)ϑ+iδ]=G{L(x)}(z)(z∈U) | (2.1) |
and
[(zg′(w)g(w))(g(w)w)ϑ+iδ]=G{L(x)}(w)(w∈U), | (2.2) |
It is fairly well known that if
|u(z)|=|u1z+u2z2+⋯|<1 |
and
|v(w)|=|v1w+v2w2+⋯|<1. |
then
|uk|≤1and|vk|≤1(k∈N) |
It follows that, so we have
G{L(x)}(u(z))=1+Lp,q,1(x)u(z)+Lp,q,2(x)u2(z)+…=1+Lp,q,1(x)u1z+[Lp,q,1(x)u2+Lp,q,2(x)u21]z2+… | (2.3) |
and
G{L(x)}(v(w))=1+Lp,q,1(x)v(w)+Lp,q,2(x)v2(w)+…=1+Lp,q,1(x)v1w+[Lp,q,1(x)v2+Lp,q,2(x)v21]w2+… | (2.4) |
From the equalities (2.1) and (2.2), we obtain that
[(zf′(z)f(z))(f(z)z)ϑ+iδ]=1+Lp,q,1(x)u1z+[Lp,q,1(x)u2+Lp,q,2(x)u21]z2+…, | (2.5) |
and
[(zg′(w)g(w))(g(w)w)ϑ+iδ]=1+Lp,q,1(x)v1w+[Lp,q,1(x)v2+Lp,q,2(x)v21]w2+…, | (2.6) |
It follows from (2.5) and (2.6) that
(ϑ+iδ+1)a2=Lp,q,1(x)u1,, | (2.7) |
(ϑ+iδ−1)(ϑ+iδ+2)2a22−(ϑ+iδ+2)a3=Lp,q,1(x)u2+Lp,q,2(x)u21, | (2.8) |
and
−(ϑ+iδ+1)a2=Lp,q,1(x)v1, | (2.9) |
(ϑ+iδ+2)(ϑ+iδ+3)2a22+(ϑ+iδ+2)a3=Lp,q,1(x)v2+Lp,q,2(x)v21, | (2.10) |
From (2.7) and (2.9)
u1=−v1 | (2.11) |
and
2(ϑ+iδ+1)2a22=L2p,q,1(x)(u21+v21)., | (2.12) |
by adding (2.8) to (2.10), we get
((ϑ+iδ)2+3(ϑ+iδ)+2)a22=Lp,q,1(x)(u2+v2)+Lp,q,2(x)(u21+v21), | (2.13) |
by using (2.12) in equality (2.13), we have
[((ϑ+iδ)2+3(ϑ+iδ)+2)−2Lp,q,2(x)(ϑ+iδ+1)2L2p,q,1(x)]a22=Lp,q,1(x)(u2+v2), |
a22=L3p,q,1(x)(u2+v2)[((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)−2Lp,q,2(x)(ϑ+iδ+1)2]. | (2.14) |
Thus, from (1.3) and (2.14) we get
|a2|≤p(x)√2p(x)√|{((ϑ+iδ)2+3(ϑ+iδ)+2)−2(ϑ+iδ+1)2}p2(x)−4q(x)(ϑ+iδ+1)2|. |
Next, in order to find the bound on |a3|, by subtracting (2.10) from (2.8), we obtain
2(ϑ+iδ+2)a3−2(ϑ+iδ+2)a22=Lp,q,1(x)(u2−v2)+Lp,q,2(x)(u21−v21)2(ϑ+iδ+2)a3=Lp,q,1(x)(u2−v2)+2(ϑ+iδ+2)a22a3=Lp,q,1(x)(u2−v2)2(ϑ+iδ+2)+a22 | (2.15) |
Then, in view of (2.11) and (2.12), we have from (2.15)
a3=L2p,q,1(x)2(ϑ+iδ+2)2(u21+v21)+Lp,q,1(x)2(ϑ+iδ+2)(u2−v2). |
|a3|≤p2(x)|ϑ+iδ+1|2+p(x)|ϑ+iδ+2|=p2(x)(ϑ+1)2+δ2+p(x)√(ϑ+2)2+δ2 |
This completes the proof.
Taking δ=0, in Theorem 2.1, we get the following corollary.
Corollary 2.2. Let the function f(z) given by (1.1) be in the class B(ϑ). Then
|a2|≤p(x)√2p(x)√|{(ϑ2+3ϑ+2)−2(ϑ+1)2}p2(x)−4q(x)(ϑ+1)2| |
and
|a3|≤p2(x)(ϑ+2)2+p(x)ϑ+2 |
Also, taking ϑ=0 and δ=0, in Theorem 2.1, we get the results given in [18].
Fekete-Szegö inequality is one of the famous problems related to coefficients of univalent analytic functions. It was first given by [12], the classical Fekete-Szegö inequality for the coefficients of f∈S is
|a3−μa22|≤1+2exp(−2μ/(1−μ)) for μ∈[0,1). |
As μ→1−, we have the elementary inequality |a3−a22|≤1. Moreover, the coefficient functional
ςμ(f)=a3−μa22 |
on the normalized analytic functions f in the unit disk U plays an important role in function theory. The problem of maximizing the absolute value of the functional ςμ(f) is called the Fekete-Szegö problem.
In this section, we are ready to find the sharp bounds of Fekete-Szegö functional ςμ(f) defined for f∈B(ϑ,δ) given by (1.1).
Theorem 3.1. Let f given by (1.1) be in the class B(ϑ,δ) and μ∈R. Then
|a3−μa22|≤{p(x)√(ϑ+2)2+δ2, 0≤|h(μ)|≤12√(ϑ+2)2+δ22p(x)|h(μ)|, |h(μ)|≥12√(ϑ+2)2+δ2 |
where
h(μ)=L2p,q,1(x)(1−μ)((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)−2Lp,q,2(x)(ϑ+iδ+1)2. |
Proof. From (2.14) and (2.15), we conclude that
a3−μa22=(1−μ)L3p,q,1(x)(u2+v2)[((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)−2Lp,q,2(x)(ϑ+iδ+1)2]+Lp,q,1(x)2(ϑ+iδ+2)(u2−v2) |
=Lp,q,1(x)[(h(μ)+12(ϑ+iδ+2))u2+(h(μ)−12(ϑ+iδ+2))v2] |
where
h(μ)=L2p,q,1(x)(1−μ)((ϑ+iδ)2+3(ϑ+iδ)+2)L2p,q,1(x)−2Lp,q,2(x)(ϑ+iδ+1)2. |
Then, in view of (1.3), we obtain
|a3−μa22|≤{p(x)√(ϑ+2)2+δ2, 0≤|h(μ)|≤12√(ϑ+2)2+δ22p(x)|h(μ)|, |h(μ)|≥12√(ϑ+2)2+δ2 |
We end this section with some corollaries.
Taking μ=1 in Theorem 3.1, we get the following corollary.
Corollary 3.2. If f∈B(ϑ,δ), then
|a3−a22|≤p(x)√(ϑ+2)2+δ2. |
Taking δ=0 in Theorem 3.1, we get the following corollary.
Corollary 3.3. Let f given by (1.1) be in the class B(ϑ,0). Then
|a3−μa22|≤{p(x)ϑ+2, 0≤|h(μ)|≤12(ϑ+2)2p(x)|h(μ)|, |h(μ)|≥12(ϑ+2) |
Also, taking ϑ=0, δ=0 and μ=1 in Theorem 3.1, we get the following corollary.
Corollary 3.4. Let f given by (1.1) be in the class B. Then
|a3−a22|≤p(x)2. |
All authors declare no conflicts of interest in this paper.
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