Research article

Learning deep forest for face anti-spoofing: An alternative to the neural network against adversarial attacks

  • 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.



    The classical rational Ramanujan-type series for π1 (cf. [1,2,8,27] and a nice introduction by S. Cooper [10,Chapter 14]) have the form

    k=0bk+cmka(k)=λdπ,()

    where b,c,m are integers with bm0, d is a positive squarefree number, λ is a nonzero rational number, and a(k) is one of the products

    (2kk)3, (2kk)2(3kk), (2kk)2(4k2k), (2kk)(3kk)(6k3k).

    In 1997 Van Hamme [47] conjectured that such a series () has a p-adic analogue of the form

    p1k=0bk+cmka(k)cp(εddp) (mod p3),

    where p is any odd prime with pdm and λZp, ε1{±1} and εd=1 if d>1. (As usual, Zp denotes the ring of all p-adic integers, and (p) stands for the Legendre symbol.) W. Zudilin [53] followed Van Hamme's idea to provide more concrete examples. Sun [33] realized that many Ramanujan-type congruences are related to Bernoulli numbers or Euler numbers. In 2016 the author [44] thought that all classical Ramanujan-type congruences have their extensions like

    pn1k=0(21k+8)(2kk)3pn1k=0(21k+8)(2kk)3(pn)3(2nn)3Zp,

    where p is an odd prime, and nZ+={1,2,3,}. See Sun [45,Conjectures 21-24] for more such examples and further refinements involving Bernoulli or Euler numbers.

    During the period 2002–2010, some new Ramanujan-type series of the form () with a(k) not a product of three nontrivial parts were found (cf. [3,4,9,29]). For example, H. H. Chan, S. H. Chan and Z. Liu [3] proved that

    n=05n+164nDn=83π,

    where Dn denotes the Domb number nk=0(nk)2(2kk)(2(nk)nk); Zudilin [53] conjectured its p-adic analogue:

    p1k=05k+164kDkp(p3) (mod p3)for any prime p>3.

    The author [45,Conjecture 77] conjectured further that

    1(pn)3(pn1k=05k+164kDk(p3)pn1k=05k+164rDk)Zp

    for each odd prime p and positive integer n.

    Let b,cZ. For each nN={0,1,2,}, we denote the coefficient of xn in the expansion of (x2+bx+c)n by Tn(b,c), and call it a generalized central trinomial coefficient. In view of the multinomial theorm, we have

    Tn(b,c)=n/2k=0(n2k)(2kk)bn2kck=n/2k=0(nk)(nkk)bn2kck.

    Note also that

    T0(b,c)=1,  T1(b,c)=b,

    and

    (n+1)Tn+1(b,c)=(2n+1)bTn(b,c)n(b24c)Tn1(b,c)

    for all nZ+. Clearly, Tn(2,1)=(2nn) for all nN. Those Tn:=Tn(1,1) with nN are the usual central trinomial coefficients, and they play important roles in enumerative combinatorics. We view Tn(b,c) as a natural generalization of central binomial and central trinomial coefficients.

    For nN the Legendre polynomial of degree n is defined by

    Pn(x):=nk=0(nk)(n+kk)(x12)k.

    It is well-known that if b,cZ and b24c0 then

    Tn(b,c)=(b24c)nPn(bb24c)for all nN.

    Via the Laplace-Heine asymptotic formula for Legendre polynomials, for any positive real numbers b and c we have

    Tn(b,c)(b+2c)n+1/224cnπas n+

    (cf. [40]). For any real numbers b and c<0, S. Wagner [48] confirmed the author's conjecture that

    limnn|Tn(b,c)|=b24c.

    In 2011, the author posed over 60 conjectural series for 1/π of the following new types with a,b,c,d,m integers and mbcd(b24c) nonzero (cf. Sun [34,40]).

    Type Ⅰ. k=0a+dkmk(2kk)2Tk(b,c).

    Type Ⅱ. k=0a+dkmk(2kk)(3kk)Tk(b,c).

    Type Ⅲ. k=0a+dkmk(4k2k)(2kk)Tk(b,c).

    Type Ⅳ. k=0a+dkmk(2kk)2T2k(b,c).

    Type Ⅴ. k=0a+dkmk(2kk)(3kk)T3k(b,c).

    Type Ⅵ. k=0a+dkmkTk(b,c)3,

    Type Ⅶ. k=0a+dkmk(2kk)Tk(b,c)2,

    In general, the corresponding p-adic congruences of these seven-type series involve linear combinations of two Legendre symbols. The author's conjectural series of types Ⅰ-Ⅴ and Ⅶ were studied in [6,49,54]. The author's three conjectural series of type Ⅵ and two series of type Ⅶ remain open. For example, the author conjectured that

    k=03990k+1147(288)3kTk(62,952)3=43295π(942+19514)

    as well as its p-adic analogue

    p1k=03990k+1147(288)3kTk(62,952)3p19(4230(2p)+17563(14p)) (mod p2),

    where p is any prime greater than 3.

    In 1905, J. W. L. Glaisher [15] proved that

    k=0(4k1)(2kk)4(2k1)4256k=8π2.

    This actually follows from the following finite identity observed by the author [38]:

    nk=0(4k1)(2kk)4(2k1)4256k=(8n2+4n+1)(2nn)4256n for all nN.

    Motivated by Glaisher's identity and Ramanujan-type series for 1/π, we obtain the following theorem.

    Theorem 1.1. We have the following identities:

    k=0k(4k1)(2kk)3(2k1)2(64)k=1π, (1.1)
    k=0(4k1)(2kk)3(2k1)3(64)k=2π, (1.2)
    k=0(12k21)(2kk)3(2k1)2256k=2π, (1.3)
    k=0k(6k1)(2kk)3(2k1)3256k=12π, (1.4)
    k=0(28k24k1)(2kk)3(2k1)2(512)k=32π, (1.5)
    k=0(30k2+3k2)(2kk)3(2k1)3(512)k=2728π, (1.6)
    k=0(28k24k1)(2kk)3(2k1)24096k=3π, (1.7)
    k=0(42k23k1)(2kk)3(2k1)34096k=278π, (1.8)
    k=0(34k23k1)(2kk)2(3kk)(2k1)(3k1)(192)k=103π, (1.9)
    k=0(64k211k7)(2kk)2(3kk)(k+1)(2k1)(3k1)(192)k=12539π, (1.10)
    k=0(14k2+k1)(2kk)2(3kk)(2k1)(3k1)216k=3π, (1.11)
    k=0(90k2+7k+1)(2kk)2(3kk)(k+1)(2k1)(3k1)216k=932π, (1.12)
    k=0(34k23k1)(2kk)2(3kk)(2k1)(3k1)(12)3k=23π, (1.13)
    k=0(17k+5)(2kk)2(3kk)(k+1)(2k1)(3k1)(12)3k=93π, (1.14)
    k=0(111k27k4)(2kk)2(3kk)(2k1)(3k1)1458k=454π, (1.15)
    k=0(1524k2+899k+263)(2kk)2(3kk)(k+1)(2k1)(3k1)1458k=33754π, (1.16)
    k=0(522k255k13)(2kk)2(3kk)(2k1)(3k1)(8640)k=54155π, (1.17)
    k=0(1836k2+2725k+541)(2kk)2(3kk)(k+1)(2k1)(3k1)(8640)k=2187155π, (1.18)
    k=0(529k245k16)(2kk)2(3kk)(2k1)(3k1)153k=5532π, (1.19)
    k=0(77571k2+68545k+16366)(2kk)2(3kk)(k+1)(2k1)(3k1)153k=5989532π, (1.20)
    k=0(574k273k11)(2kk)2(3kk)(2k1)(3k1)(48)3k=203π, (1.21)
    k=0(8118k2+9443k+1241)(2kk)2(3kk)(k+1)(2k1)(3k1)(48)3k=22503π, (1.22)
    k=0(978k2131k17)(2kk)2(3kk)(2k1)(3k1)(326592)k=990749π, (1.23)
    k=0(592212k2+671387k2+77219)(2kk)2(3kk)(k+1)(2k1)(3k1)(326592)k=4492125749π, (1.24)
    k=0(116234k217695k1461)(2kk)2(3kk)(2k1)(3k1)(300)3k=26503π, (1.25)
    k=0(223664832k2+242140765k+18468097)(2kk)2(3kk)(k+1)(2k1)(3k1)(300)3k=334973253π, (1.26)
    k=0(122k2+3k5)(2kk)2(4k2k)(2k1)(4k1)648k=212π, (1.27)
    k=0(1903k2+114k+41)(2kk)2(4k2k)(k+1)(2k1)(4k1)648k=3432π, (1.28)
    k=0(40k22k1)(2kk)2(4k2k)(2k1)(4k1)(1024)k=4π, (1.29)
    k=0(8k22k1)(2kk)2(4k2k)(k+1)(2k1)(4k1)(1024)k=165π, (1.30)
    k=0(176k26k5)(2kk)2(4k2k)(2k1)(4k1)482k=83π, (1.31)
    k=0(208k2+66k+23)(2kk)2(4k2k)(k+1)(2k1)(4k1)482k=1283π, (1.32)
    k=0(6722k2411k152)(2kk)2(4k2k)(2k1)(4k1)(632)k=1957π, (1.33)
    k=0(281591k2757041k231992)(2kk)2(4k2k)(k+1)(2k1)(4k1)(632)k=2746257π, (1.34)
    k=0(560k242k11)(2kk)2(4k2k)(2k1)(4k1)124k=242π, (1.35)
    k=0(112k2+114k+23)(2kk)2(4k2k)(k+1)(2k1)(4k1)124k=25625π, (1.36)
    k=0(248k218k5)(2kk)2(4k2k)(2k1)(4k1)(3×212)k=283π, (1.37)
    k=0(680k2+1482k+337)(2kk)2(4k2k)(k+1)(2k1)(4k1)(3×212)k=548839π, (1.38)
    k=0(1144k2102k19)(2kk)2(4k2k)(2k1)(4k1)(21034)k=60π, (1.39)
    k=0(3224k2+4026k+637)(2kk)2(4k2k)(k+1)(2k1)(4k1)(21034)k=2000π, (1.40)
    k=0(7408k2754k103)(2kk)2(4k2k)(2k1)(4k1)284k=56033π, (1.41)
    k=0(3641424k2+4114526k+493937)(2kk)2(4k2k)(k+1)(2k1)(4k1)284k=8960003π, (1.42)
    k=0(4744k2534k55)(2kk)2(4k2k)(2k1)(4k1)(214345)k=1932525π, (1.43)
    k=0(18446264k2+20356230k+1901071)(2kk)2(4k2k)(k+1)(2k1)(4k1)(214345)k=66772496525π, (1.44)
    k=0(413512k250826k3877)(2kk)2(4k2k)(2k1)(4k1)(210214)k=12180π, (1.45)
    k=0(1424799848k2+1533506502k+108685699)(2kk)2(4k2k)(k+1)(2k1)(4k1)(210214)k=341446000π, (1.46)
    k=0(71312k27746k887)(2kk)2(4k2k)(2k1)(4k1)15842k=84011π, (1.47)
    k=0(50678512k2+56405238k+5793581)(2kk)2(4k2k)(k+1)(2k1)(4k1)15842k=548800011π, (1.48)
    k=0(7329808k2969294k54073)(2kk)2(4k2k)(2k1)(4k1)3964k=1201202π, (1.49)
    k=0(2140459883152k2+2259867244398k+119407598201)(2kk)2(4k2k)(k+1)(2k1)(4k1)3964k=44×182032π, (1.50)
    k=0(164k2k3)(2kk)(3kk)(6k3k)(2k1)(6k1)203k=752π, (1.51)
    k=0(2696k2+206k+93)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)203k=6865π, (1.52)
    k=0(220k28k3)(2kk)(3kk)(6k3k)(2k1)(6k1)(215)k=72π, (1.53)
    k=0(836k21048k309)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(215)k=6862π, (1.54)
    k=0(504k211k8)(2kk)(3kk)(6k3k)(2k1)(6k1)(15)3k=915π, (1.55)
    k=0(189k211k8)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(15)3k=2431535π, (1.56)
    k=0(516k219k7)(2kk)(3kk)(6k3k)(2k1)(6k1)(2×303)k=11152π, (1.57)
    k=0(3237k2+1922k+491)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(2×303)k=39931510π, (1.58)
    k=0(684k240k7)(2kk)(3kk)(6k3k)(2k1)(6k1)(96)3k=96π, (1.59)
    k=0(2052k2+2536k+379)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(96)3k=4866π, (1.60)
    k=0(2556k2131k29)(2kk)(3kk)(6k3k)(2k1)(6k1)663k=63334π, (1.61)
    k=0(203985k2+212248k+38083)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)663k=83349334π, (1.62)
    k=0(5812k2408k49)(2kk)(3kk)(6k3k)(2k1)(6k1)(3×1603)k=253309π, (1.63)
    k=0(3471628k2+3900088k+418289)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(3×1603)k=3238855430135π, (1.64)
    k=0(35604k22936k233)(2kk)(3kk)(6k3k)(2k1)(6k1)(960)3k=18915π, (1.65)
    k=0(13983084k2+15093304k+1109737)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(960)3k=4500846155π, (1.66)
    k=0(157752k211243k1304)(2kk)(3kk)(6k3k)(2k1)(6k1)2553k=5132552π, (1.67)
    k=0(28240947k2+31448587k+3267736)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)2553k=4500189925570π, (1.68)
    k=0(2187684k2200056k11293)(2kk)(3kk)(6k3k)(2k1)(6k1)(5280)3k=1953330π, (1.69)
    k=0(101740699836k2+107483900696k+5743181813)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(5280)3k=49661001183305π, (1.70)
    k=0(16444841148k21709536232k53241371)(2kk)(3kk)(6k3k)(2k1)(6k1)(640320)3k=167220910005π, (1.71)

    and

    k=0P(k)(2kk)(3kk)(6k3k)(k+1)(2k1)(6k1)(640320)3k=18×5574033100055π, (1.72)

    where

    P(k):=637379600041024803108k2+657229991696087780968k+19850391655004126179.

    Recall that the Catalan numbers are given by

    Cn:=(2nn)n+1=(2nn)(2nn+1)  (nN).

    For kN it is easy to see that

    (2kk)2k1={1if k=0,2Ck1if k>0.

    Thus, for any a,b,c,mZ with |m|64, we have

    k=0(ak2+bk+c)(2kk)3(2k1)3mk=c+k=1(ak2+bk+c)(2Ck1)3mk=c+8mk=0a(k+1)2+b(k+1)+cmkC3k.

    For example, (1.2) has the equivalent form

    k=04k+3(64)kC3k=816π.(1.2)

    For any odd prime p, the congruence (1.4) of V.J.W. Guo and J.-C. Liu [19] has the equivalent form

    (p+1)/2k=0(4k1)(2kk)3(2k1)3(64)kp(1p)+p3(Ep32) (mod p4)

    (where E0,E1, are the Euler numbers), and we note that this is also equivalent to the congruence

    (p1)/2k=04k+3(64)kC3k8(1p(1p)p3(Ep32)) (mod p4).

    Recently, C. Wang [50] proved that for any prime p>3 we have

    (p+1)/2k=0(3k1)(2kk)3(2k1)216kp+2p3(1p)(Ep33) (mod p4)

    and

    p1k=0(3k1)(2kk)3(2k1)216kp2p3 (mod p4).

    (Actually, Wang stated his results only in the language of hypergeometric series.) These two congruences extend a conjecture of Guo and M. J. Schlosser [21].

    We are also able to prove some other variants of Ramanujan-type series such as

    k=0(56k2+118k+61)(2kk)3(k+1)24096k=192π

    and

    k=0(420k2+992k+551)(2kk)3(k+1)2(2k1)4096k=1728π.

    Now we state our second theorem.

    Theorem 1.2. We have the identities

    k=128k2+31k+8(2k+1)2k3(2kk)3=π282, (1.73)
    k=142k2+39k+8(2k+1)3k3(2kk)3=9π2882, (1.74)
    k=1(8k2+5k+1)(8)k(2k+1)2k3(2kk)3=46G, (1.75)
    k=1(30k2+33k+7)(8)k(2k+1)3k3(2kk)3=54G52, (1.76)
    k=1(3k+1)16k(2k+1)2k3(2kk)3=π282, (1.77)
    k=1(4k+1)(64)k(2k+1)2k2(2kk)3=48G, (1.78)
    k=1(4k+1)(64)k(2k+1)3k3(2kk)3=16G16, (1.79)
    k=1(2k211k3)8k(2k+1)(3k+1)k3(2kk)2(3kk)=485π22, (1.80)
    k=2(178k2103k39)8k(k1)(2k+1)(3k+1)k3(2kk)2(3kk)=1125π21109636, (1.81)
    k=1(5k+1)(27)k(2k+1)(3k+1)k2(2kk)2(3kk)=69K, (1.82)
    k=2(45k2+5k2)(27)k1(k1)(2k+1)(3k+1)k3(2kk)2(3kk)=3748K16, (1.83)
    k=1(98k221k8)81k(2k+1)(4k+1)k3(2kk)2(4k2k)=21620π2, (1.84)
    k=2(1967k2183k104)81k(k1)(2k+1)(4k+1)k3(2kk)2(4k2k)=20000π2190269120, (1.85)
    k=1(46k2+3k1)(144)k(2k+1)(4k+1)k3(2kk)2(4k2k)=722252K, (1.86)
    k=2(343k2+18k16)(144)k(k1)(2k+1)(4k+1)k3(2kk)2(4k2k)=9375K704810, (1.87)

    where

    G:=k=0(1)k(2k+1)2  and  K:=k=0(k3)k2.

    For k=j+1Z+, it is easy to see that

    (k1)k(2kk)=2(2j+1)j(2jj).

    Thus, for any a,b,c,mZ with 0<|m|64, we have

    j=1(aj2+bj+c)mj(2j+1)3j3(2jj)3=8mk=2(a(k1)2+b(k1)+c)mk(k1)3k3(2kk)3.

    For example, (1.77) has the following equivalent form

    k=2(2k1)(3k2)16k(k1)3k3(2kk)3=π28.(1.77)

    In contrast with the Domb numbers, we introduce a new kind of numbers

    Sn:=nk=0(nk)2TkTnk  (n=0,1,2,).

    The values of Sn (n=0,,10) are

    1,2,10,68,586,5252,49204,475400,4723786,47937812,494786260

    respectively. We may extend the numbers Sn (nN) further. For b,cZ, we define

    Sn(b,c):=nk=0(nk)2Tk(b,c)Tnk(b,c)  (n=0,1,2,).

    Note that Sn(1,1)=Sn and Sn(2,1)=Dn for all nN.

    Now we state our third theorem.

    Theorem 1.3. We have

    k=07k+324kSk(1,6)=152π, (1.88)
    k=012k+5(28)kSk(1,7)=67π, (1.89)
    k=084k+2980kSk(1,20)=2415π, (1.90)
    k=03k+1(100)kSk(1,25)=258π, (1.91)
    k=0228k+67224kSk(1,56)=807π, (1.92)
    k=0399k+101(676)kSk(1,169)=25358π, (1.93)
    k=02604k+5632600kSk(1,650)=850393π, (1.94)
    k=039468k+7817(6076)kSk(1,1519)=441031π, (1.95)
    k=041667k+78799800kSk(1,2450)=4042564π, (1.96)
    k=074613k+10711(5302)kSk(1,2652)=161517548π. (1.97)

    Remark 1.1. The author found the 10 series in Theorem 1.3 in Nov. 2019.

    We shall prove Theorems 1.1-1.3 in the next section. In Sections 3-10, we propose 117 new conjectural series for powers of π involving generalized central trinomial coefficients. In particular, we will present in Section 3 four conjectural series for 1/π of the following new type:

    Type Ⅷ. k=0a+dkmkTk(b,c)Tk(b,c)2,

    where a,b,b,c,c,d,m are integers with mbbccd(b24c)(b24c)(b2cb2c)0.

    Unlike Ramanujan-type series given by others, all our series for 1/π of types Ⅰ-Ⅷ have the general term involving a product of three generalized central trinomial coefficients.

    Motivated by the author's effective way to find new series for 1/π (cf. Sun [35]), we formulate the following general characterization of rational Ramanujan-type series for 1/π via congruences.

    Conjecture 1.1 (General Criterion for Rational Ramanujan-type Series for 1/π). Suppose that the series k=0bk+cmkak converges, where (ak)k0 is an integer sequence and b,c,m are integers with bcm0. Suppose also that there are no a,xZ such that an=a(2nn)nk=0(2kk)2(2(nk)nk)xnk for all nN. Let r{1,2,3} and let d1,,drZ+ with di/dj irrational for all distinct i,j{1,,r}. Then

    k=0bk+cmkak=ri=1λidiπ (1.98)

    for some nonzero rational numbers λ1,,λr if and only if there are positive integers dj (r<j3) and rational numbers c1,c2,c3 with ri=1ci0, such that for any prime p>3 with pmri=1di and c1,c2,c3Zp we have

    p1k=0bk+cmkakp(ri=1ci(εidip)+r<j3cj(djp)) (mod p2), (1.99)

    where εi{±1}, εi=1 if di is not an integer square, and c2=c3=0 if r=1 and ε1=1.

    For a Ramanujan-type series of the form (1.98), we call r its rank. We believe that there are some Ramanujan-type series of rank three but we have not yet found such a series.

    Conjecture 1.2. Let (an)n0 be an integer sequence with no a,xZ such that an=a(2nn)nk=0(2kk)2(2(nk)nk)xnk for all nN, and let b,c,m,d1,d2,d3Z with bcm0. Assume that limn+n|an|=r<|m|, and πk=0bk+cmkak is an algebraic number. Suppose that c1,c2,c3Q with c1+c2+c3=a0c , and

    p1k=0bk+cmkakp(c1(d1p)+c2(d2p)+c3(d3p)) (mod p2) (1.100)

    for all primes p>3 with pd1d2d3m and c1,c2,c3Zp. Then, for any prime p>3 with pm, c1,c2,c3Zp and (d1p)=(d2p)=(d3p)=δ{±1}, we have

    1(pn)2(pn1k=0bk+cmkakpδn1k=0bk+cmkak)Zp  for all nZ+.

    Joint with the author's PhD student Chen Wang, we pose the following conjecture.

    Conjecture 1.3 (Chen Wang and Z.-W. Sun). Let (ak)k0 be an integer sequence with a0=1. Let b,c,m,d1,d2,d3Z with bm0, and let c1,c2,c3 be rational numbers. If πk=0bk+cmkak is an algebraic number, and the congruence (1.100) holds for all primes p>3 with pd1d2d3m and c1,c2,c3Zp, then we must have c1+c2+c3=c.

    Remark 1.2. The author [39,Conjecture 1.1(i)] conjectured that

    p1k=0(8k+5)T2k3p(3p) (mod p2)

    for any prime p>3, which was confirmed by Y.-P. Mu and Z.-W. Sun [26]. This is not a counterexample to Conjecture 1.3 since k=0(8k+5)T2k diverges.

    All the new series and related congruences in Sections 3-9 support Conjectures 1.1-1.3. We discover the conjectural series for 1/π in Sections 3-9 based on the author's previous PhilosophyaboutSeriesfor 1/π} stated in [35], the PSLQ algorithm to discover integer relations (cf. [13]), and the following DualityPrinciple based on the author's experience and intuition.

    Conjecture 1.4 (Duality Principle). Let (ak)k0 be an integer sequence such that

    ak(dp)Dkap1k (mod p) (1.101)

    for any prime p6dD and k{0,,p1}, where d and D are fixed nonzero integers. If a0,a1, are not all zero and m is a nonzero integer such that

    k=0bk+cmkak=λ1d1+λ2d2+λ3d3π

    for some b,d1,d2,d3Z+, cZ and λ1,λ2,λ3Q, then m divides D, and

    p1k=0akmk(dp)p1k=0ak(D/m)k (mod p2) (1.102)

    for any prime p>3 with pdD.

    Remark 1.3 (ⅰ) For any prime p>3 with pdDm, the congruence (1.102) holds modulo p by (1.101) and Fermat's little theorem. We call p1k=0ak/(Dm)k the dual of the sum p1k=0ak/mk.

    (ⅱ) For any b,cZ and odd prime pb24c, it is known (see, e.g., [39,Lemma 2.2]) that

    Tk(b,c)(b24cp)(b24c)kTp1k(b,c) (mod p) (1.103)

    for all k=0,1,,p1.

    For a series k=0ak with a0,a1, real numbers, if limk+ak+1/ak=r(1,1) then we say that the series converge at a geometric rate with ratio r. Except for (7.1), all other conjectural series in Sections 3-9 converge at geometric rates and thus one can easily check them numerically via a computer.

    In Section 10, we pose two curious conjectural series for π involving the central trinomial coefficients.

    Lemma 2.1. Let m0 and n0 be integers. Then

    nk=0((64m)k332k216k+8)(2kk)3(2k1)2mk=8(2n+1)mn(2nn)3, (2.1)
    nk=0((64m)k396k2+48k8)(2kk)3(2k1)3mk=8mn(2nn)3, (2.2)
    nk=0((108m)k354k212k+6)(2kk)2(3kk)(2k1)(3k1)mk=6(3n+1)mn(2nn)2(3nn), (2.3)
    nk=0((108m)k3(54+m)k212k+6)(2kk)2(3kk)(k+1)(2k1)(3k1)mk=6(3n+1)(n+1)mn(2nn)2(3nn), (2.4)
    (2.5)
    (2.6)
    (2.7)
    (2.8)

    Remark 2.1. The eight identities in Lemma 2.1 can be easily proved by induction on . In light of Stirling's formula, as , we have

    (2.9)
    (2.10)

    Proof of Theorem 1.1. Just apply Lemma 2.1 and the 36 known rational Ramanujan-type series listed in [16]. Let us illustrate the proofs by showing (1.1), (1.2), (1.71) and (1.72) in details.

    By (2.1) with , we have

    Note that

    and recall Bauer's series

    So, we get

    This proves (1.1). By (2.2) with , we have

    and hence

    Combining this with we immediately get .

    In view of (2.7) with , we have

    and hence

    In 1987, D. V. Chudnovsky and G. V. Chudnovsky [8] got the formula

    which enabled them to hold the world record for the calculation of during 1989–1994. Note that

    and hence

    This proves .

    By (2.8) with , we have

    and hence

    Note that

    Therefore, with the help of we get

    This proves .

    The identities (1.3)–(1.70) can be proved similarly.

    Lemma 2.2. Let and be integers. Then

    (2.11)
    (2.12)
    (2.13)
    (2.14)
    (2.15)
    (2.16)

    Remark 2.2. This can be easily proved by induction on .

    Proof of Theorem 1.2. We just apply Lemma 2.2 and use the known identities:

    Here, the first identity was found and proved by D. Zeilberger [52] in 1993. The second, third and fourth identities were obtained by J. Guillera [17] in 2008. The fifth identity on was conjectured by Sun [33] and later confirmed by K. Hessami Pilehrood and T. Hessami Pilehrood [22] in 2012. The last four identities were also conjectured by Sun [33], and they were later proved in the paper [18,Theorem 3] by Guillera and M. Rogers.

    Let us illustrate our proofs by proving (1.77)-(1.79) and (1.82)-(1.83) in details.

    In view of (2.11) with , we have

    for all , and hence

    Notice that

    So we have

    and hence (1.77) holds.

    By (2.11) with , we have

    for all , and hence

    Since and

    we see that

    and hence (1.78) holds. In light of (2.12) with , we have

    for all , and hence

    Since , with the aid of (1.78) we obtain

    This proves (1.79).

    By (2.13) with , we have

    As

    and

    we see that (1.82) follows. By (2.14) with , we have

    and hence

    As

    with the aid of (1.82) we get

    and hence follows.

    Other identities in Theorem 1.2 can be proved similarly.

    For integers , we define

    (2.17)

    For we set

    (2.18)

    Lemma 2.3. For any , we have

    (2.19)

    and

    (2.20)

    where denotes the Franel number .

    Proof. For with , we set

    By the telescoping method for double summation [7], for

    with , we find that

    where

    and

    with regarded as , and and given by

    and

    respectively. Therefore

    and hence

    satisfies the recurrence relation

    As pointed out by J. Franel [14], the Franel numbers satisfy the same recurrence. Note also that and . So we always have . This proves (2.19).

    The identity (2.20) can be proved similarly. In fact, if we use denote the left-hand side or the right-hand side of (2.20), then we have the recurrence

    In view of the above, we have completed the proof of Lemma 2.3.

    Lemma 2.4. For any and , we have

    (2.21)

    Proof. For each , we have

    If and , then

    with the aid of the Chu-Vandermonde identity. Therefore

    This proves (2.21).

    Lemma 2.5. For and , we have

    (2.22)

    Proof. Let . Then

    and

    Hence

    This proves (2.22).

    To prove Theorem 1.3, we need an auxiliary theorem.

    Theorem 2.6. Let and be real numbers. For any integer with , we have

    (2.23)

    Proof. Let . In view of (2.21),

    and similarly

    where we consider as .

    If is an integer in the interval , then by Lemma 2.5 we have

    where is the Legendre polynomial of degree . Thus

    and

    Recall that

    As , we have and hence

    converges. Thus

    and hence by the above we have

    and

    Therefore, with the aid of (2.19), we obtain

    (2.24)

    and

    (2.25)

    In view of (2.25) and (2.20),

    Combining this with (2.24), we immediately obtain the desired (2.23).

    Proof of Theorem 1.3. Let with . Since

    for any , we have for all . Thus, in light of Theorem 2.6,

    Therefore

    It is known (cf. [5,4]) that

    So we get the identities (1.88)-(1.97) finally.

    Now we pose a conjecture related to the series (Ⅰ1)-(Ⅰ4) of Sun [34,40].

    Conjecture 3.1. We have the following identities:

    Remark 3.1. For each , we have

    since and . Thus, for example, [40,(I1)] and (I1) together imply that

    and (I5) and (I5) imply that

    For the conjectural identities in Conjecture 3.1, we have conjectures for the corresponding -adic congruences. For example, in contrast with (I2), we conjecture that for any prime we have the congruences

    and

    Concerning (I5) and (I5), we conjecture that

    and

    for each , and that for any prime with we have

    and

    By [40,Theorem 5.1], we have

    for any prime . The identities (I5), (I5) and (I5) were formulated by the author on Dec. 9, 2019.

    Next we pose a conjecture related to the series (Ⅱ1)-(Ⅱ7) and (Ⅱ10)-(Ⅱ12) of Sun [34,40].

    Conjecture 3.2. We have the following identities:

    Remark 3.2. We also have conjectures on related congruences. For example, concerning (Ⅱ), for any prime we conjecture that

    and that

    where and are integers. The identities (Ⅱ13), (Ⅱ13), (Ⅱ14) and (Ⅱ14) were found by the author on Dec. 11, 2019.

    The following conjecture is related to the series (Ⅲ1)-(Ⅲ10) and (Ⅲ12) of Sun [34,40].

    Conjecture 3.3. We have the following identities:

    and

    The following conjecture is related to the series (Ⅳ1)-(Ⅳ21) of Sun [34,40].

    Conjecture 3.4. We have the following identities:

    For the five open conjectural series (Ⅵ1), (Ⅵ2), (Ⅵ3), (ⅥI2) and (ⅥI7) of Sun [34,40], we make the following conjecture on related supercongruences.

    Conjecture 3.5. Let be an odd prime and let . If , then

    divided by is a -adic integer. If , then

    divided by is a -adic integer. If but , then

    divided by is a -adic integer. If but , then

    divided by is a -adic integer. If but , then

    divided by is a -adic integer.

    Now we pose four conjectural series for of type Ⅷ.

    Conjecture 3.6. We have

    Remark 3.3. The author found the identity (Ⅷ1) on Nov. 3, 2019. The identities (Ⅷ2), (Ⅷ3) and (Ⅷ4) were formulated on Nov. 4, 2019.

    Below we present some conjectures on congruences related to Conjecture 3.6.

    Conjecture 3.7. (ⅰ) For each , we have

    (3.1)

    and this number is odd if and only if is a power of two i.e., .

    (ⅱ) Let be a prime. Then

    (3.2)

    If , then

    (3.3)

    for all .

    (ⅲ) Let be a prime. Then

    (3.4)

    Remark 3.4. The imaginary quadratic field has class number two.

    Conjecture 3.8. (ⅰ) For any , we have

    (3.5)

    and the number is odd if and only if is a power of two.

    (ⅱ) Let be a prime. Then

    (3.6)

    If , then

    (3.7)

    for all .

    (ⅲ) Let be a prime. Then

    (3.8)

    Remark 3.5. This conjecture can be viewed as the dual of Conjecture 3.7. Note that the series diverges.

    Conjecture 3.9. (ⅰ) For each , we have

    (3.9)

    (ⅱ) Let be a prime. Then

    (3.10)

    If , then

    (3.11)

    divided by is a -adic integer for each .

    (ⅲ) Let be a prime. Then

    (3.12)

    Remark 3.6. The imaginary quadratic field has class number two.

    Conjecture 3.10. (ⅰ) For each , we have

    (3.13)

    (ⅱ) Let be a prime. Then

    (3.14)

    If , then

    (3.15)

    divided by is a -adic integer for each .

    (ⅲ) Let be a prime. Then

    (3.16)

    Remark 3.7. This conjecture can be viewed as the dual of Conjecture 3.9. Note that the series

    diverges.

    Conjecture 3.11. (ⅰ) For each , we have

    (3.17)

    (ⅱ) Let be a prime. Then

    (3.18)

    If , then

    (3.19)

    divided by is an -adic integer for any .

    (ⅲ Let be a prime. Then

    (3.20)

    where and are integers.

    Remark 3.8. Note that the imaginary quadratic field has class number .

    Conjecture 3.12. (ⅰ) For each , we have

    (3.21)

    and this number is odd if and only if is a power of two.

    (ⅱ) Let be a prime. Then

    (3.22)

    If , then

    (3.23)

    divided by is a -adic integer for each .

    (ⅲ) Let be a prime with . Then

    (3.24)

    where and are integers.

    Remark 3.9. Note that the imaginary quadratic field has class number .

    Conjectures 4.1–4.14 below provide congruences related to (1.88)–(1.97).

    Conjecture 4.1. (ⅰ) For any , we have

    (4.1)

    (ⅱ) Let be a prime. Then

    (4.2)

    If , then

    (4.3)

    for all .

    (ⅲ) For any prime , we have

    (4.4)

    Conjecture 4.2. (ⅰ) For any , we have

    (4.5)

    and this number is odd if and only if is a power of two.

    (ⅱ) Let be an odd prime. Then

    (4.6)

    and moreover

    (4.7)

    for all .

    (ⅲ) For any prime , we have

    (4.8)

    where and are integers.

    Conjecture 4.3. (ⅰ) For any , we have

    (4.9)

    and this number is odd if and only if is a power of two.

    (ⅱ) Let be an odd prime with . Then

    (4.10)

    If , then

    (4.11)

    for all .

    (ⅲ) For any prime , we have

    (4.12)

    where and are integers.

    Conjecture 4.4. (ⅰ) For any , we have

    (4.13)

    (ⅱ) Let be an odd prime. Then

    (4.14)

    for all .

    (ⅲ) For any prime with , we have

    (4.15)

    where and are integers.

    Conjecture 4.5. (ⅰ) For any , we have

    (4.16)

    and this number is odd if and only if is a power of two.

    (ⅱ) Let be an odd prime with . Then

    (4.17)

    If , then

    (4.18)

    for all .

    (ⅲ) For any prime , we have

    (4.19)

    where and are integers.

    Conjecture 4.6. (ⅰ) For any , we have

    (4.20)

    (ⅱ) Let be an odd prime. Then

    (4.21)

    divided by is a -adic integer for any .

    (ⅲ) For any prime with , we have

    (4.22)

    where and are integers.

    Conjecture 4.7. (ⅰ) For any , we have

    (4.23)

    and this number is odd if and only if .

    (ⅱ) Let be an odd prime with . Then

    (4.24)

    If , then

    (4.25)

    divided by is a -adic integer for any .

    (ⅲ)For any odd prime , we have

    (4.26)

    where and are integers.

    Conjecture 4.8. (ⅰ) For any , we have

    (4.27)

    and this number is odd if and only if .

    (ⅱ) Let be an odd prime. Then

    (4.28)

    divided by is a -adic integer for any .

    (ⅲ) For any prime with , we have

    (4.29)

    where and are integers.

    Conjecture 4.9. (ⅰ) For any , we have

    (4.30)

    (ⅱ) Let be an odd prime. Then

    (4.31)

    If , then

    (4.32)

    divided by is a -adic integer for any .

    (ⅲ) For any prime with , we have

    (4.33)

    where and are integers.

    Conjecture 4.10. (ⅰ) For any , we have

    (4.34)

    Let be an odd prime. Then

    (4.35)

    divided by is a -adic integer for any .

    For any prime with , we have

    (4.36)

    where and are integers.

    Conjecture 4.11. For any odd prime ,

    (4.37)

    Also, for any prime we have

    (4.38)

    Conjecture 4.12. For any , we have

    (4.39)

    and this number is odd if and only if is a power of two.

    For any odd prime and positive integer , we have

    (4.40)

    Let be an odd prime. Then

    (4.41)

    Conjecture 4.13. For any , we have

    (4.42)

    and this number is odd if and only if is a power of two.

    Let be a prime. Then

    (4.43)

    If , then

    (4.44)

    for all .

    For any prime , we have

    (4.45)

    Conjecture 4.14. For any , we have

    (4.46)

    Let be an odd prime. Then

    (4.47)

    If , then

    (4.48)

    for all .

    For any prime , we have

    (4.49)

    where and are integers.

    Conjecture 4.15. Let be an odd prime with . Then

    (4.50)

    where and are integers. If , then

    Remark 4.1. We also have some similar conjectures involving

    modulo , where is a prime greater than .

    Motivated by Theorem 2.6, we pose the following general conjecture.

    Conjecture 4.16. For any odd prime and integer , we have

    (4.51)

    and

    (4.52)

    Remark 4.2 We have checked this conjecture via . In view of the proof of Theorem 2.6, both (4.51) and (4.52) hold modulo .

    The numbers

    were first introduced by D. Zagier in his paper [51] the preprint of which was released in 2002. Thus we name such numbers as Zagier numbers. As pointed out by the author [41,Remark 4.3], for any the number coincides with the so-called CLF (Catalan-Larcombe-French) number

    Let be an odd prime. For any , we have

    by F. Jarvis and H.A. Verrill [24,Corollary 2.2], and hence

    Combining this with Remark 1.3(ⅱ), we see that

    for any with .

    J. Wan and Zudilin [49] obtained the following irrational series for involving the Legendre polynomials and the Zagier numbers:

    Via our congruence approach (including Conjecture 1.4), we find 24 rational series for involving and the Zagier numbers. Theorem 1 of [49] might be helpful to solve some of them.

    Conjecture 5.1. We have the following identities for .

    (5.1)
    (5.2)
    (5.3)
    (5.4)
    (5.5)
    (5.6)
    (5.7)
    (5.8)
    (5.9)
    (5.10)
    (5.11)
    (5.12)
    (5.13)
    (5.14)
    (5.15)
    (5.16)
    (5.17)
    (5.18)
    (5.19)
    (5.20)
    (5.21)
    (5.22)
    (5.23)
    (5.24)

    Below we present some conjectures on congruences related to , , and .

    Conjecture 5.2. (ⅰ) For any , we have

    (5.25)

    Let be an odd prime with . Then

    (5.26)

    If , then

    (5.27)

    for all .

    For any prime , we have

    (5.28)

    Conjecture 5.3. (ⅰ) For any , we have

    (5.29)

    Let be a prime with . Then

    (5.30)

    If , then

    (5.31)

    for all .

    For any prime with , we have

    (5.32)

    Conjecture 5.4. For any , we have

    (5.33)

    Let be a prime. Then

    (5.34)

    If , then

    (5.35)

    for all .

    For any prime , we have

    (5.36)

    Conjecture 5.5. For any , we have

    (5.37)

    Let be an odd prime with . Then

    (5.38)

    If , then

    (5.39)

    for all .

    For any prime , we have

    (5.40)

    where and are integers.

    Sun [36,37] obtained some supercongruences involving the Franel numbers . M. Rogers and A. Straub [30] confirmed the -series for involving Franel polynomials conjectured by Sun [34].

    Let be an odd prime. By [24,Lemma 2.6], we have for each . Combining this with Remark 1.3(ⅱ), we see that

    for any with .

    Wan and Zudilin [49] deduced the following irrational series for involving the Legendre polynomials and the Franel numbers:

    Via our congruence approach (including Conjecture 1.4), we find rational series for involving and the Franel numbers; Theorem 1 of [49] might be helpful to solve some of them.

    Conjecture 6.1. We have

    (6.1)
    (6.2)
    (6.3)
    (6.4)
    (6.5)
    (6.6)
    (6.7)
    (6.8)
    (6.9)
    (6.10)
    (6.11)
    (6.12)

    We now present a conjecture on congruence related to .

    Conjecture 6.2. For any , we have

    (6.13)

    Let be a prime. Then

    (6.14)

    If , then

    (6.15)

    for all .

    For any prime , we have

    (6.16)

    Remark 6.1 This conjecture was formulated by the author on Oct. 25, 2019.

    Conjecture 6.3. For any , we have

    (6.17)

    Let be an odd prime. Then

    (6.18)

    If , then

    (6.19)

    divided by is a -adic integer for any .

    Let be a prime. Then

    (6.20)

    Remark 6.2. This conjecture is the dual of Conjecture 6.2.

    The following conjecture is related to the identity .

    Conjecture 6.4. For any , we have

    (6.21)

    Let be a prime with . Then

    (6.22)

    If , then

    (6.23)

    divided by is a -adic integer for any .

    Let be a prime with . Then

    (6.24)

    where and are integers.

    Remark 6.3. Note that the imaginary quadratic field has class number .

    The following conjecture is related to the identity .

    Conjecture 6.5. For any , we have

    (6.25)

    Let be a prime with . Then

    (6.26)

    If , then

    (6.27)

    divided by is a -adic integer for any .

    Let be a prime with . Then

    (6.28)

    where and are integers.

    Remark 6.4. Note that the imaginary quadratic field has class number .

    The following conjecture is related to the identity .

    Conjecture 6.6. For any , we have

    (6.29)

    Let be an odd prime with . Then

    (6.30)

    If , then

    (6.31)

    divided by is a -adic integer for any .

    Let be a prime with . Then

    (6.32)

    where and are integers.

    Remark 6.5. Note that the imaginary quadratic field has class number .

    The identities are related to the quadratic fields

    (with class number ) respectively. We also have conjectures on related congruences similar to Conjectures 6.4, 6.5 and 6.6.

    For let

    It is known that for all . See [43,20,26] for some congruences on polynomials related to these numbers.

    Let be a prime. For any , we have

    by [24,Lemma 2.7(ⅱ)]. Combining this with Remark 1.3(ⅱ), we see that

    for any with .

    Wan and Zudilin [49] obtained the following irrational series for involving the Legendre polynomials and the sequence :

    Using our congruence approach (including Conjecture 1.4), we find 12 rational series for involving and ; Theorem 1 of [49] might be helpful to solve some of them.

    Conjecture 7.1. We have the following identities.

    (7.1)
    (7.2)
    (7.3)
    (7.4)
    (7.5)
    (7.6)
    (7.7)
    (7.8)
    (7.9)
    (7.10)
    (7.11)
    (7.12)

    Now we present a conjecture on congruences related to .

    Conjecture 7.2. For any , we have

    (7.13)

    and this number is odd if and only if .

    Let be a prime. Then

    (7.14)

    If , then

    (7.15)

    divided by is a -adic integer for any .

    Let be a prime. Then

    (7.16)

    where and are integers.

    Remark 7.1. Note that the imaginary quadratic field has class number .

    The following conjecture is related to the identity .

    Conjecture 7.3. For any , we have

    (7.17)

    and this number is odd if and only if .

    Let be a prime. Then

    (7.18)

    If , then

    (7.19)

    divided by is a -adic integer for any .

    Let be a prime. Then

    (7.20)

    where and are integers.

    Remark 7.2. Note that the imaginary quadratic field has class number .

    Now we pose a conjecture related to the identity .

    Conjecture 7.4. For any , we have

    (7.21)

    Let be a prime with . Then

    (7.22)

    If , then

    (7.23)

    is a -adic integer for any .

    Let be a prime with . Then

    (7.24)

    where and are integers.

    Remark 7.3. Note that the imaginary quadratic field has class number .

    Now we pose a conjecture related to the identity .

    Conjecture 7.5. For any , we have

    (7.25)

    and this number is odd if and only if .

    Let be a prime. Then

    (7.26)

    If , then

    (7.27)

    divided by is a -adic integer for each .

    Let be a prime. Then

    (7.28)

    where and are integers.

    Remark 7.4. Note that the imaginary quadratic field has class number . We believe that is the largest positive squarefree number for which the imaginary quadratic field can be used to construct a Ramanujan-type series for .

    The identities are related to the imaginary quadratic fields , , , (with class number ) respectively. We also have conjectures on related congruences similar to Conjectures 7.2, 7.3, 7.4 and 7.5.

    To conclude this section, we confirm an open series for conjectured by the author (cf. [34,(3.28)] and [35,Conjecture 7.9]) in 2011.

    Theorem 7.1. We have

    (7.29)

    where

    Proof. The Franel numbers of order are given by . Note that

    By [11,(8.1)], for and , we have

    (7.30)

    Since

    putting , and in (7.30) we obtain

    As

    by Cooper [9], we finally get

    This concludes the proof of (7.29).

    Recall that the numbers

    are a kind of Apéry numbers. Let be an odd prime. For any , we have

    by [24,Lemma 2.7(ⅰ)]. Combining this with Remark 1.3(ⅱ), we see that

    for any with .

    Wan and Zudilin [49] obtained the following irrational series for involving the Legendre polynomials and the numbers :

    Using our congruence approach (including Conjecture 1.4), we find one rational series for involving and the Apéry numbers (see (8.1) below); Theorem 1 of [49] might be helpful to solve it.

    Conjecture 8.1. (ⅰ) We have

    (8.1)

    Also, for any we have

    (8.2)

    Let be a prime. Then

    (8.3)

    If , then

    (8.4)

    for all .

    Let be a prime. Then

    (8.5)

    Remark 8.1. This conjecture was formulated by the author on Oct. 27, 2019.

    Conjecture 8.2. For any , we have

    (8.6)

    and this number is odd if and only if .

    Let be a prime. Then

    (8.7)

    If i.e., , then

    (8.8)

    divided by is a -adic integer for any .

    For any prime , we have

    (8.9)

    Remark 8.2. This conjecture was formulated by the author on Nov. 13, 2019.

    Conjecture 8.3. For any , we have

    (8.10)

    and this number is odd if and only if is a power of two.

    Let be a prime. Then

    (8.11)

    If , then

    (8.12)

    for all .

    For any odd prime , we have

    (8.13)

    Remark 8.3. This conjecture was formulated by the author on Nov. 13, 2019.

    Conjecture 8.4. For any , we have

    (8.14)

    and this number is odd if and only if .

    Let be any odd prime. Then

    (8.15)

    If , then

    (8.16)

    for all .

    For any odd prime , we have

    (8.17)

    Conjecture 8.5. For any , we have

    (8.18)

    Let be a prime. Then

    (8.19)

    If , then

    (8.20)

    for all .

    Let be a prime. Then

    (8.21)

    Conjecture 8.6. For any , we have

    (8.22)

    and this number is odd if and only if is a power of two.

    Let be an odd prime. Then

    (8.23)

    If , then

    (8.24)

    for all .

    Let be a prime. Then

    (8.25)

    Conjecture 8.7. For any , we have

    (8.26)

    and this number is odd if and only if is a power of two.

    Let be a prime. Then

    (8.27)

    If , then

    (8.28)

    for all .

    For any prime , we have

    (8.29)

    The numbers

    were first introduced by Zagier [51] during his study of Apéry-like integer sequences, who noted the recurrence

    Lemma 9.1. Let be a prime. Then

    Proof. Note that

    with the help of the known congruence conjectured by F. Rodriguez-Villegas [28] and proved by E. Mortenson [25]. Similarly,

    By induction,

    for all . In particular,

    So we have for . (Note that and .)

    Now let and assume that

    Then

    and hence

    In view of the above, we have proved the desired result by induction.

    For Lemma 9.1 one may also consult [31,Corollary 3.1]. Let be a prime. In view of Lemma 9.1 and Remark 1.3(ⅱ), we have

    for any with .

    Wan and Zudilin [49] obtained the following irrational series for involving the Legendre polynomials and the numbers :

    Using our congruence approach (including Conjecture 1.4), we find five rational series for involving and the numbers ; Theorem 1 of [49] might be helpful to solve them.

    Conjecture 9.1. We have

    (9.1)
    (9.2)
    (9.3)
    (9.4)
    (9.5)

    Below we present our conjectures on congruences related to the identities (9.2) and (9.5).

    Conjecture 9.2. For any , we have

    (9.6)

    and this number is odd if and only if .

    Let be a prime. Then

    (9.7)

    If i.e., , then

    (9.8)

    for all .

    For any prime , we have

    (9.9)

    Conjecture 9.3. For any , we have

    (9.10)

    and this number is odd if and only if .

    Let be a prime with . Then

    (9.11)

    If i.e., , then

    (9.12)

    for all .

    For any prime with , we have

    (9.13)

    Now we give one more conjecture in this section.

    Conjecture 9.4. For any integer , we have

    (9.14)

    Let be a prime. Then

    (9.15)

    If , then

    (9.16)

    for all .

    For any prime , we have

    (9.17)

    Remark 9.1. For primes with , in general the congruence (9.16) is not always valid for all . This does not violate Conjecture 1.2 since . If the series converges, its value times should be a rational number.

    Let be an odd prime and let with . Then

    with the aid of [33,Lemma 2.1]. Thus

    in view of Remark 1.3(ⅱ).

    Let be a prime. By the above, the author's conjectural congruence (cf. [35,Conjecture 1.3])

    implies that

    Motivated by this, we pose the following curious conjecture.

    Conjecture 10.1. We have the following identities:

    (10.1)
    (10.2)

    Remark 10.1. The two identities were conjectured by the author on Dec. 7, 2019. One can easily check them numerically via as the two series converge fast.

    Now we state our related conjectures on congruences.

    Conjecture 10.2. For any prime , we have

    (10.3)

    and

    (10.4)

    Conjecture 10.3. (ⅰ) We have

    for all , and also

    for each prime .

    For any prime and , we have

    (10.5)

    Remark 10.2. See also [45,Conjecture 67] for a similar conjecture.

    Let be an odd prime. We conjecture that

    (10.6)

    and

    (10.7)

    Though (10.6) implies the congruence

    and (10.7) with implies the congruence

    we are unable to find the exact values of the two converging series

    The author would like to thank Prof. Qing-Hu Hou at Tianjin Univ. for his helpful comments on the proof of Lemma 2.3.



    [1] K. Patel, H. Han, A. K. Jain, Secure face unlock: Spoof detection on smartphones, IEEE Trans. Inf. Forensics Secur., 11 (2016), 2268–2283. https://doi.org/10.1109/TIFS.2016.2578288 doi: 10.1109/TIFS.2016.2578288
    [2] A. Krizhevsky, I. Sutskever, G. E. Hinton, ImageNet classification with deep convolutional neural networks, Commun. ACM, 60 (2017), 84–90. https://doi.org/10.1145/3065386 doi: 10.1145/3065386
    [3] Y. Sun, Y. Chen, X. Wang, X. Tang, Deep learning face representation by joint identification-verification, in Proceedings of the 27th International Conference on Neural Information Processing Systems, 2 (2014), 1988–1996.
    [4] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, et al., An image is worth 1616 words: Transformers for image recognition at scale, preprint, arXiv: 2010.11929.
    [5] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, et al., Swin transformer: Hierarchical vision transformer using shifted windows, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), (2021), 9992–10002. https://doi.org/10.1109/ICCV48922.2021.00986
    [6] J. Yang, Z. Lei, S. Z. Li, Learn convolutional neural network for face anti-spoofing, preprint, arXiv: 1408.5601.
    [7] Z. Xu, S. Li, W. Deng, Learning temporal features using LSTM-CNN architecture for face anti-spoofing, in 2015 3rd IAPR Asian Conference on Pattern Recognition (ACPR), (2016), 141–145. https://doi.org/10.1109/ACPR.2015.7486482
    [8] Z. Yu, C. Zhao, Z. Wang, Y. Qin, Z. Su, X. Li, et al., Searching central difference convolutional networks for face anti-spoofing, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2020), 5294–5304. https://doi.org/10.1109/CVPR42600.2020.00534
    [9] R. Cai, Z. Li, R. Wan, H. Li, Y. Hu, A. C. Kot, Learning meta pattern for face anti-spoofing, IEEE Trans. Inf. Forensics Secur., 17 (2022), 1201–1213. https://doi.org/10.1109/TIFS.2022.3158551 doi: 10.1109/TIFS.2022.3158551
    [10] R. Cai, Z. Yu, C. Kong, H. Li, C. Chen, Y. Hu, et al., S-adapter: Generalizing vision transformer for face anti-spoofing with statistical tokens, IEEE Trans. Inf. Forensics Secur., 19 (2024), 8385–8397. https://doi.org/10.1109/TIFS.2024.3420699 doi: 10.1109/TIFS.2024.3420699
    [11] R. Cai, Y. Cui, Z. Li, Z. Yu, H. Li, Y. Hu, et al., Rehearsal-free domain continual face anti-spoofing: Generalize more and forget less, in 2023 IEEE/CVF International Conference on Computer Vision (ICCV), (2023), 8003–8014. https://doi.org/10.1109/ICCV51070.2023.00738
    [12] A. Liu, Z. Tan, Z. Yu, C. Zhao, J. Wan, Y. Liang, et al., FM-ViT: Flexible modal vision transformers for face anti-spoofing, IEEE Trans. Inf. Forensics Secur., 18 (2023), 4775–4786. https://doi.org/10.1109/TIFS.2023.3296330 doi: 10.1109/TIFS.2023.3296330
    [13] C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. J. Goodfellow, et al., Intriguing properties of neural networks, preprint, arXiv: 1312.6199.
    [14] A. Aldahdooh, W. Hamidouche, O. Deforges, Reveal of vision transformers robustness against adversarial attacks, preprint, arXiv: 2106.03734.
    [15] A. Kurakin, I. J. Goodfellow, S. Bengio, Adversarial examples in the physical world, preprint, arXiv: 1607.02533.
    [16] K. Eykholt, I. Evtimov, E. Fernandes, B. Li, A. Rahmati, C. Xiao, et al., Robust physical-world attacks on deep learning visual classification, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2018), 1625–1634. https://doi.org/10.1109/CVPR.2018.00175
    [17] M. Sharif, S. Bhagavatula, L. Bauer, M. K. Reiter, Accessorize to a crime: Real and stealthy attacks on state-of-the-art face recognition, in Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, (2016), 1528–1540. https://doi.org/10.1145/2976749.2978392
    [18] Y. Liu, X. Chen, C. Liu, D. Song, Delving into transferable adversarial examples and black-box attacks, preprint, arXiv: 1611.02770.
    [19] W. Ma, Y. Li, X. Jia, W. Xu, Transferable adversarial attack for both vision transformers and convolutional networks via momentum integrated gradients, in 2023 IEEE/CVF International Conference on Computer Vision (ICCV), (2023), 4607–4616. https://doi.org/10.1109/ICCV51070.2023.00427
    [20] P. Russu, A. Demontis, B. Biggio, G. Fumera, F. Roli, Secure kernel machines against evasion attacks, in Proceedings of the 2016 ACM workshop on artificial intelligence and security, (2016), 59–69. https://doi.org/10.1145/2996758.2996771
    [21] F. Tramèr, N. Papernot, I. Goodfellow, D. Boneh, P. McDaniel, The space of transferable adversarial examples, preprint, arXiv: 1704.03453.
    [22] F. Yang, Z. Chen, Using randomness to improve robustness of machine-learning models against evasion attacks, preprint, arXiv: 1808.03601.
    [23] Z. Zhou, J. Feng, Deep forest: Towards an alternative to deep neural networks, in Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, (2017), 3553–3559. https://doi.org/10.24963/ijcai.2017/497
    [24] J. Maatta, A. Hadid, M. Pietikäinen, Face spoofing detection from single images using micro-texture analysis, in 2011 International Joint Conference on Biometrics (IJCB), (2011), 1–7. https://doi.org/10.1109/IJCB.2011.6117510
    [25] J. Yang, Z. Lei, S. Liao, S. Z. Li, Face liveness detection with component dependent descriptor, in 2013 International Conference on Biometrics (ICB), (2013), 1–6. https://doi.org/10.1109/ICB.2013.6612955
    [26] R. Nosaka, Y. Ohkawa, K. Fukui, Feature extraction based on co-occurrence of adjacent local binary patterns, in Advances in Image and Video Technology, 7088 (2011), 82–91. https://doi.org/10.1007/978-3-642-25346-1_8
    [27] I. Chingovska, A. Anjos, S. Marcel, On the effectiveness of local binary patterns in face anti-spoofing, in 2012 BIOSIG-Proceedings of the International Conference of Biometrics Special Interest Group (BIOSIG), (2012), 1–7.
    [28] Y. Atoum, Y. Liu, A. Jourabloo, X. Liu, Face anti-spoofing using patch and depth-based CNNs, in 2017 IEEE International Joint Conference on Biometrics (IJCB), (2017), 319–328. https://doi.org/10.1109/BTAS.2017.8272713
    [29] X. Tan, Y. Li, J. Liu, L. Jiang, Face liveness detection from a single image with sparse low rank bilinear discriminative model, in Computer Vision-ECCV 2010, 6316 (2010), 504–517. https://doi.org/10.1007/978-3-642-15567-3_37
    [30] D. Gragnaniello, G. Poggi, C. Sansone, L. Verdoliva, An investigation of local descriptors for biometric spoofing detection, IEEE Trans. Inf. Forensics Secur., 10 (2015), 849–863. https://doi.org/10.1109/TIFS.2015.2404294 doi: 10.1109/TIFS.2015.2404294
    [31] Z. Boulkenafet, J. Komulainen, A. Hadid, Face antispoofing using speeded-up robust features and fisher vector encoding, IEEE Signal Process. Lett., 24 (2017), 141–145. https://doi.org/10.1109/LSP.2016.2630740 doi: 10.1109/LSP.2016.2630740
    [32] T. D. F. Pereira, A. Anjos, J. M. D. Martino, S. Marcel, LBP-TOP based countermeasure against face spoofing attacks, in Computer Vision-ACCV 2012 Workshops, 7728 (2012), 121–132. https://doi.org/10.1007/978-3-642-37410-4_11
    [33] S. R. Arashloo, J. Kittler, W. Christmas, Face spoofing detection based on multiple descriptor fusion using multiscale dynamic binarized statistical image features, IEEE Trans. Inf. Forensics Secur., 10 (2015), 2396–2407. https://doi.org/10.1109/TIFS.2015.2458700 doi: 10.1109/TIFS.2015.2458700
    [34] Z. Boulkenafet, J. Komulainen, A. Hadid, Face spoofing detection using colour texture analysis, IEEE Trans. Inf. Forensics Secur., 11 (2016), 1818–1830. https://doi.org/10.1109/TIFS.2016.2555286 doi: 10.1109/TIFS.2016.2555286
    [35] D. Menotti, G. Chiachia, A. Pinto, W. R. Schwartz, H. Pedrini, A. X. Falcão, et al., Deep representations for iris, face, and fingerprint spoofing detection, IEEE Trans. Inf. Forensics Secur., 10 (2015), 864–879. https://doi.org/10.1109/TIFS.2015.2398817 doi: 10.1109/TIFS.2015.2398817
    [36] H. Li, P. He, S. Wang, A. Rocha, X. Jiang, A. C. Kot, Learning generalized deep feature representation for face anti-spoofing, IEEE Trans. Inf. Forensics Secur., 13 (2018), 2639–2652. https://doi.org/10.1109/TIFS.2018.2825949 doi: 10.1109/TIFS.2018.2825949
    [37] R. Cai, H. Li, S. Wang, C. Chen, A. C. Kot, DRL-FAS: A novel framework based on deep reinforcement learning for face anti-spoofing, IEEE Trans. Inf. Forensics Secur., 16 (2020), 937–951. https://doi.org/10.1109/TIFS.2020.3026553 doi: 10.1109/TIFS.2020.3026553
    [38] W. Sun, Y. Song, C. Chen, J. Huang, A. C. Kot, Face spoofing detection based on local ternary label supervision in fully convolutional networks, IEEE Trans. Inf. Forensics Secur., 15 (2020), 3181–3196. https://doi.org/10.1109/TIFS.2020.2985530 doi: 10.1109/TIFS.2020.2985530
    [39] A. George, S. Marcel, Deep pixel-wise binary supervision for face presentation attack detection, preprint, arXiv: 1907.04047.
    [40] Z. Yu, X. Li, J. Shi, Z. Xia, G. Zhao, Revisiting pixel-wise supervision for face anti-spoofing, IEEE Trans. Biom. Behav. Identity Sci., 3 (2021), 285–295. https://doi.org/10.1109/TBIOM.2021.3065526 doi: 10.1109/TBIOM.2021.3065526
    [41] R. Shao, X. Lan, J. Li, P. C. Yuen, Multi-adversarial discriminative deep domain generalization for face presentation attack detection, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2019), 10015–10023. https://doi.org/10.1109/CVPR.2019.01026
    [42] Y. Qin, Z. Yu, L. Yan, Z. Wang, C. Zhao, Z. Lei, Meta-teacher for face anti-spoofing, IEEE Trans. Pattern Anal. Mach. Intell., 44 (2022), 6311–6326. https://doi.org/10.1109/TPAMI.2021.3091167 doi: 10.1109/TPAMI.2021.3091167
    [43] Z. Yu, J. Wan, Y. Qin, X. Li, S. Z. Li, G. Zhao, NAS-FAS: Static-dynamic central difference network search for face anti-spoofing, IEEE Trans. Pattern Anal. Mach. Intell., 43 (2021), 3005–3023. https://doi.org/10.1109/tpami.2020.3036338 doi: 10.1109/tpami.2020.3036338
    [44] Y. Liu, J. Stehouwer, X. Liu, On disentangling spoof trace for generic face anti-spoofing, in Computer Vision-ECCV 2020, 12363 (2020), 406–422. https://doi.org/10.1007/978-3-030-58523-5_24
    [45] K. Zhang, T. Yao, J. Zhang, Y. Tai, S. Ding, J. Li, et al., Face anti-spoofing via disentangled representation learning, in Computer Vision-ECCV 2020, 12364 (2020), 641–657. https://doi.org/10.1007/978-3-030-58529-7_38
    [46] H. Wu, D. Zeng, Y. Hu, H. Shi, T. Mei, Dual spoof disentanglement generation for face anti-spoofing with depth uncertainty learning, IEEE Trans. Circuits Syst. Video Technol., 32 (2022), 4626–4638. https://doi.org/10.1109/TCSVT.2021.3133620 doi: 10.1109/TCSVT.2021.3133620
    [47] W. Yan, Y. Zeng, H. Hu, Domain adversarial disentanglement network with cross-domain synthesis for generalized face anti-spoofing, IEEE Trans. Circuits Syst. Video Technol., 32 (2022), 7033–7046. https://doi.org/10.1109/TCSVT.2022.3178723 doi: 10.1109/TCSVT.2022.3178723
    [48] Y. Jia, J. Zhang, S. Shan, X. Chen, Single-side domain generalization for face anti-spoofing, in 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2020), 8481–8490. https://doi.org/10.1109/CVPR42600.2020.00851
    [49] Z. Wang, Z. Wang, Z. Yu, W. Deng, J. Li, T. Gao, et al., Domain Generalization via Shuffled Style Assembly for Face Anti-Spoofing, in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), (2022), 4113–4123. https://doi.org/10.1109/CVPR52688.2022.00409
    [50] A. Liu, C. Zhao, Z. Yu, J. Wan, A. Su, X. Liu, et al., Contrastive context-aware learning for 3D high-fidelity mask face presentation attack detection, IEEE Trans. Inf. Forensics Secur., 17 (2022), 2497–2507. https://doi.org/10.1109/TIFS.2022.3188149 doi: 10.1109/TIFS.2022.3188149
    [51] A. George, S. Marcel, Learning one class representations for face presentation attack detection using multi-channel convolutional neural networks, IEEE Trans. Inf. Forensics Secur., 16 (2020), 361–375. https://doi.org/10.1109/TIFS.2020.3013214 doi: 10.1109/TIFS.2020.3013214
    [52] H. Li, W. Li, H. Cao, S. Wang, F. Huang, A. C. Kot, Unsupervised Domain Adaptation for Face Anti-Spoofing, IEEE Trans. Inf. Forensics Secur., 13 (2018), 1794–1809. https://doi.org/10.1109/TIFS.2018.2801312 doi: 10.1109/TIFS.2018.2801312
    [53] Y. Liu, Y. Chen, W. Dai, M. Gou, C. Huang, H. Xiong, Source-free domain adaptation with contrastive domain alignment and self-supervised exploration for face anti-spoofing, in Computer Vision-ECCV 2022, 13672 (2022), 511–528. https://doi.org/10.1007/978-3-031-19775-8_30
    [54] Y. Qin, C. Zhao, X. Zhu, Z. Wang, Z. Yu, T. Fu, et al., Learning meta model for zero-and few-shot face anti-spoofing, in Proceedings of the AAAI Conference on Artificial Intelligence, 34 (2020), 11916–11923. https://doi.org/10.1609/aaai.v34i07.6866
    [55] H. Huang, D. Sun, Y. Liu, W. Chu, T. Xiao, J. Yuan, et al., Adaptive transformers for robust few-shot cross-domain face anti-spoofing, in Computer Vision-ECCV 2022, 13673 (2022), 37–54. https://doi.org/10.1007/978-3-031-19778-9_3
    [56] L. Li, X. Feng, Z. Xia, X. Jiang, A. Hadid, Face spoofing detection with local binary pattern network, J. Visual Commun. Image Representation, 54 (2018), 182–192. https://doi.org/10.1016/j.jvcir.2018.05.009 doi: 10.1016/j.jvcir.2018.05.009
    [57] A. Roohi, S. Angizi, Efficient targeted bit-flip attack against the local binary pattern network, in 2022 IEEE International Symposium on Hardware Oriented Security and Trust (HOST), (2022), 89–92. https://doi.org/10.1109/HOST54066.2022.9839959
    [58] N. Bousnina, L. Zheng, M. Mikram, S. Ghouzali, K. Minaoui, Unraveling robustness of deep face anti-spoofing models against pixel attacks, Multimedia Tools Appl., 80 (2021), 7229–7246. https://doi.org/10.1007/s11042-020-10041-1 doi: 10.1007/s11042-020-10041-1
    [59] D. Deb, X. Liu, A. K. Jain, Unified detection of digital and physical face attacks, in 2023 IEEE 17th International Conference on Automatic Face and Gesture Recognition (FG), (2023) 1–8. https://doi.org/10.1109/FG57933.2023.10042500
    [60] T. K. Ho, The random subspace method for constructing decision forests, IEEE Trans. Pattern Anal. Mach. Intell., 20 (1998), 832–844. https://doi.org/10.1109/34.709601 doi: 10.1109/34.709601
    [61] L. Breiman, Random forest, Mach. Learn., 45 (2001), 5–32. https://doi.org/10.1023/A: 1010933404324
    [62] F. T. Liu, M. T. Kai, Y. Yu, Z. Zhou, Spectrum of variable-random trees, J. Artif. Intell. Res., 32 (2008), 355–384.
    [63] T. Chen, C. Guestrin, XGBoost: A scalable tree boosting system, preprint, arXiv: 1603.02754.
    [64] T. Ojala, M. Pietikainen, T. Maenpaa, Multiresolution gray-scale and rotation invariant texture classification with local binary patterns, IEEE Trans. Pattern Anal. Mach. Intell., 24 (2002), 971–987. https://doi.org/10.1109/TPAMI.2002.1017623 doi: 10.1109/TPAMI.2002.1017623
    [65] Z. Zhang, J. Yan, S. Liu, Z. Lei, D. Yi, S. Z. Li, A face antispoofing database with diverse attacks, in 2012 5th IAPR International Conference on Biometrics (ICB), (2012), 26–31. https://doi.org/10.1109/ICB.2012.6199754
    [66] P. Viola, M. Jones, Rapid object detection using a casacde of simple features, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2001). https://doi.org/10.1109/CVPR.2001.990517
    [67] K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, preprint, arXiv: 1409.1556.
    [68] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 770–778, . https://doi.org/10.1109/CVPR.2016.90
    [69] I. J. Goodfellow, J. Shlens, C. Szegedy, Explaining and harnessing adversarial examples, preprint, arXiv: 1412.6572.
    [70] Z. Sun, L. Sun, Q. Li, Investigation in spatial-temporal domain for face spoof detection, in 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2018), 1538–1542. https://doi.org/10.1109/ICASSP.2018.8461942
    [71] T. M. Oshiro, P. S. Perez, J. A. Baranauskas, How many trees in a random forest? in Machine Learning and Data Mining in Pattern Recognition, 7376 (2012), 154–168. https://doi.org/10.1007/978-3-642-31537-4_13
    [72] J. Lu, V. E. Liong, J. Zhou, Cost-sensitive local binary feature learning for facial age estimation, IEEE Trans. Image Process., 24 (2015), 5356–5368. https://doi.org/10.1109/TIP.2015.2481327 doi: 10.1109/TIP.2015.2481327
    [73] J. Lu, V. E. Liong, J. Zhou, Deep hashing for scalable image search, IEEE Trans. Image Process., 26 (2017), 2352–2367. https://doi.org/10.1109/TIP.2017.2678163 doi: 10.1109/TIP.2017.2678163
    [74] J. Lu, V. E. Liong, J. Zhou, Simultaneous local binary feature learning and encoding for homogeneous and heterogeneous face recognition, IEEE Trans. Pattern Anal. Mach. Intell., 40 (2018), 1979–1993. https://doi.org/10.1109/TPAMI.2017.2737538 doi: 10.1109/TPAMI.2017.2737538
    [75] Y. Duan, J. Lu, J. Feng, J. Zhou, Context-aware local binary feature learning for face recognition, IEEE Trans. Pattern Anal. Mach. Intell., 40 (2018), 1139–1153. https://doi.org/10.1109/TPAMI.2017.2710183 doi: 10.1109/TPAMI.2017.2710183
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