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
[1] |
Zhi-Wei Sun .
New series for powers of |
[2] | Harman Kaur, Meenakshi Rana . Congruences for sixth order mock theta functions λ(q) and ρ(q). Electronic Research Archive, 2021, 29(6): 4257-4268. doi: 10.3934/era.2021084 |
[3] |
Jorge Garcia Villeda .
A computable formula for the class number of the imaginary quadratic field |
[4] | Fedor Petrov, Zhi-Wei Sun . Proof of some conjectures involving quadratic residues. Electronic Research Archive, 2020, 28(2): 589-597. doi: 10.3934/era.2020031 |
[5] |
Jin-Yun Guo, Cong Xiao, Xiaojian Lu .
On |
[6] | Dušan D. Repovš, Mikhail V. Zaicev . On existence of PI-exponents of unital algebras. Electronic Research Archive, 2020, 28(2): 853-859. doi: 10.3934/era.2020044 |
[7] |
Victor J. W. Guo .
A family of |
[8] | Chen Wang . Two congruences concerning Apéry numbers conjectured by Z.-W. Sun. Electronic Research Archive, 2020, 28(2): 1063-1075. doi: 10.3934/era.2020058 |
[9] | Dmitry Krachun, Zhi-Wei Sun . On sums of four pentagonal numbers with coefficients. Electronic Research Archive, 2020, 28(1): 559-566. doi: 10.3934/era.2020029 |
[10] | Hai-Liang Wu, Zhi-Wei Sun . Some universal quadratic sums over the integers. Electronic Research Archive, 2019, 27(0): 69-87. doi: 10.3934/era.2019010 |
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
∞∑k=0bk+cmka(k)=λ√dπ,(∗) |
where
(2kk)3, (2kk)2(3kk), (2kk)2(4k2k), (2kk)(3kk)(6k3k). |
In 1997 Van Hamme [47] conjectured that such a series
p−1∑k=0bk+cmka(k)≡cp(εddp) (mod p3), |
where
∑pn−1k=0(21k+8)(2kk)3−p∑n−1k=0(21k+8)(2kk)3(pn)3(2nn)3∈Zp, |
where
During the period 2002–2010, some new Ramanujan-type series of the form
∞∑n=05n+164nDn=8√3π, |
where
p−1∑k=05k+164kDk≡p(p3) (mod p3)for any prime p>3. |
The author [45,Conjecture 77] conjectured further that
1(pn)3(pn−1∑k=05k+164kDk−(p3)pn−1∑k=05k+164rDk)∈Zp |
for each odd prime
Let
Tn(b,c)=⌊n/2⌋∑k=0(n2k)(2kk)bn−2kck=⌊n/2⌋∑k=0(nk)(n−kk)bn−2kck. |
Note also that
T0(b,c)=1, T1(b,c)=b, |
and
(n+1)Tn+1(b,c)=(2n+1)bTn(b,c)−n(b2−4c)Tn−1(b,c) |
for all
For
Pn(x):=n∑k=0(nk)(n+kk)(x−12)k. |
It is well-known that if
Tn(b,c)=(√b2−4c)nPn(b√b2−4c)for all n∈N. |
Via the Laplace-Heine asymptotic formula for Legendre polynomials, for any positive real numbers
Tn(b,c)∼(b+2√c)n+1/224√c√nπas n→+∞ |
(cf. [40]). For any real numbers
limn→∞n√|Tn(b,c)|=√b2−4c. |
In 2011, the author posed over 60 conjectural series for
Type Ⅰ.
Type Ⅱ.
Type Ⅲ.
Type Ⅳ.
Type Ⅴ.
Type Ⅵ.
Type Ⅶ.
In general, the corresponding
∞∑k=03990k+1147(−288)3kTk(62,952)3=43295π(94√2+195√14) |
as well as its
p−1∑k=03990k+1147(−288)3kTk(62,952)3≡p19(4230(−2p)+17563(−14p)) (mod p2), |
where
In 1905, J. W. L. Glaisher [15] proved that
∞∑k=0(4k−1)(2kk)4(2k−1)4256k=−8π2. |
This actually follows from the following finite identity observed by the author [38]:
n∑k=0(4k−1)(2kk)4(2k−1)4256k=−(8n2+4n+1)(2nn)4256n for all n∈N. |
Motivated by Glaisher's identity and Ramanujan-type series for
Theorem 1.1. We have the following identities:
∞∑k=0k(4k−1)(2kk)3(2k−1)2(−64)k=−1π, | (1.1) |
∞∑k=0(4k−1)(2kk)3(2k−1)3(−64)k=2π, | (1.2) |
∞∑k=0(12k2−1)(2kk)3(2k−1)2256k=−2π, | (1.3) |
∞∑k=0k(6k−1)(2kk)3(2k−1)3256k=12π, | (1.4) |
∞∑k=0(28k2−4k−1)(2kk)3(2k−1)2(−512)k=−3√2π, | (1.5) |
∞∑k=0(30k2+3k−2)(2kk)3(2k−1)3(−512)k=27√28π, | (1.6) |
∞∑k=0(28k2−4k−1)(2kk)3(2k−1)24096k=−3π, | (1.7) |
∞∑k=0(42k2−3k−1)(2kk)3(2k−1)34096k=278π, | (1.8) |
∞∑k=0(34k2−3k−1)(2kk)2(3kk)(2k−1)(3k−1)(−192)k=−10√3π, | (1.9) |
∞∑k=0(64k2−11k−7)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−192)k=−125√39π, | (1.10) |
∞∑k=0(14k2+k−1)(2kk)2(3kk)(2k−1)(3k−1)216k=−√3π, | (1.11) |
∞∑k=0(90k2+7k+1)(2kk)2(3kk)(k+1)(2k−1)(3k−1)216k=9√32π, | (1.12) |
∞∑k=0(34k2−3k−1)(2kk)2(3kk)(2k−1)(3k−1)(−12)3k=−2√3π, | (1.13) |
∞∑k=0(17k+5)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−12)3k=9√3π, | (1.14) |
∞∑k=0(111k2−7k−4)(2kk)2(3kk)(2k−1)(3k−1)1458k=−454π, | (1.15) |
∞∑k=0(1524k2+899k+263)(2kk)2(3kk)(k+1)(2k−1)(3k−1)1458k=33754π, | (1.16) |
∞∑k=0(522k2−55k−13)(2kk)2(3kk)(2k−1)(3k−1)(−8640)k=−54√155π, | (1.17) |
∞∑k=0(1836k2+2725k+541)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−8640)k=2187√155π, | (1.18) |
∞∑k=0(529k2−45k−16)(2kk)2(3kk)(2k−1)(3k−1)153k=−55√32π, | (1.19) |
∞∑k=0(77571k2+68545k+16366)(2kk)2(3kk)(k+1)(2k−1)(3k−1)153k=59895√32π, | (1.20) |
∞∑k=0(574k2−73k−11)(2kk)2(3kk)(2k−1)(3k−1)(−48)3k=−20√3π, | (1.21) |
∞∑k=0(8118k2+9443k+1241)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−48)3k=2250√3π, | (1.22) |
∞∑k=0(978k2−131k−17)(2kk)2(3kk)(2k−1)(3k−1)(−326592)k=−990√749π, | (1.23) |
∞∑k=0(592212k2+671387k2+77219)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−326592)k=4492125√749π, | (1.24) |
∞∑k=0(116234k2−17695k−1461)(2kk)2(3kk)(2k−1)(3k−1)(−300)3k=−2650√3π, | (1.25) |
∞∑k=0(223664832k2+242140765k+18468097)(2kk)2(3kk)(k+1)(2k−1)(3k−1)(−300)3k=33497325√3π, | (1.26) |
∞∑k=0(122k2+3k−5)(2kk)2(4k2k)(2k−1)(4k−1)648k=−212π, | (1.27) |
∞∑k=0(1903k2+114k+41)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)648k=3432π, | (1.28) |
∞∑k=0(40k2−2k−1)(2kk)2(4k2k)(2k−1)(4k−1)(−1024)k=−4π, | (1.29) |
∞∑k=0(8k2−2k−1)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−1024)k=−165π, | (1.30) |
∞∑k=0(176k2−6k−5)(2kk)2(4k2k)(2k−1)(4k−1)482k=−8√3π, | (1.31) |
∞∑k=0(208k2+66k+23)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)482k=128√3π, | (1.32) |
∞∑k=0(6722k2−411k−152)(2kk)2(4k2k)(2k−1)(4k−1)(−632)k=−195√7π, | (1.33) |
∞∑k=0(281591k2−757041k−231992)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−632)k=−274625√7π, | (1.34) |
∞∑k=0(560k2−42k−11)(2kk)2(4k2k)(2k−1)(4k−1)124k=−24√2π, | (1.35) |
∞∑k=0(112k2+114k+23)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)124k=256√25π, | (1.36) |
∞∑k=0(248k2−18k−5)(2kk)2(4k2k)(2k−1)(4k−1)(−3×212)k=−28√3π, | (1.37) |
∞∑k=0(680k2+1482k+337)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−3×212)k=5488√39π, | (1.38) |
∞∑k=0(1144k2−102k−19)(2kk)2(4k2k)(2k−1)(4k−1)(−21034)k=−60π, | (1.39) |
∞∑k=0(3224k2+4026k+637)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−21034)k=2000π, | (1.40) |
∞∑k=0(7408k2−754k−103)(2kk)2(4k2k)(2k−1)(4k−1)284k=−560√33π, | (1.41) |
∞∑k=0(3641424k2+4114526k+493937)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)284k=896000√3π, | (1.42) |
∞∑k=0(4744k2−534k−55)(2kk)2(4k2k)(2k−1)(4k−1)(−214345)k=−1932√525π, | (1.43) |
∞∑k=0(18446264k2+20356230k+1901071)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−214345)k=66772496√525π, | (1.44) |
∞∑k=0(413512k2−50826k−3877)(2kk)2(4k2k)(2k−1)(4k−1)(−210214)k=−12180π, | (1.45) |
∞∑k=0(1424799848k2+1533506502k+108685699)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)(−210214)k=341446000π, | (1.46) |
∞∑k=0(71312k2−7746k−887)(2kk)2(4k2k)(2k−1)(4k−1)15842k=−840√11π, | (1.47) |
∞∑k=0(50678512k2+56405238k+5793581)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)15842k=5488000√11π, | (1.48) |
∞∑k=0(7329808k2−969294k−54073)(2kk)2(4k2k)(2k−1)(4k−1)3964k=−120120√2π, | (1.49) |
∞∑k=0(2140459883152k2+2259867244398k+119407598201)(2kk)2(4k2k)(k+1)(2k−1)(4k−1)3964k=44×18203√2π, | (1.50) |
∞∑k=0(164k2−k−3)(2kk)(3kk)(6k3k)(2k−1)(6k−1)203k=−7√52π, | (1.51) |
∞∑k=0(2696k2+206k+93)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)203k=686√5π, | (1.52) |
∞∑k=0(220k2−8k−3)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−215)k=−7√2π, | (1.53) |
∞∑k=0(836k2−1048k−309)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−215)k=−686√2π, | (1.54) |
∞∑k=0(504k2−11k−8)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−15)3k=−9√15π, | (1.55) |
∞∑k=0(189k2−11k−8)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−15)3k=−243√1535π, | (1.56) |
∞∑k=0(516k2−19k−7)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(2×303)k=−11√152π, | (1.57) |
∞∑k=0(3237k2+1922k+491)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(2×303)k=3993√1510π, | (1.58) |
∞∑k=0(684k2−40k−7)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−96)3k=−9√6π, | (1.59) |
∞∑k=0(2052k2+2536k+379)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−96)3k=486√6π, | (1.60) |
∞∑k=0(2556k2−131k−29)(2kk)(3kk)(6k3k)(2k−1)(6k−1)663k=−63√334π, | (1.61) |
∞∑k=0(203985k2+212248k+38083)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)663k=83349√334π, | (1.62) |
∞∑k=0(5812k2−408k−49)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−3×1603)k=−253√309π, | (1.63) |
∞∑k=0(3471628k2+3900088k+418289)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−3×1603)k=32388554√30135π, | (1.64) |
∞∑k=0(35604k2−2936k−233)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−960)3k=−189√15π, | (1.65) |
∞∑k=0(13983084k2+15093304k+1109737)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−960)3k=4500846√155π, | (1.66) |
∞∑k=0(157752k2−11243k−1304)(2kk)(3kk)(6k3k)(2k−1)(6k−1)2553k=−513√2552π, | (1.67) |
∞∑k=0(28240947k2+31448587k+3267736)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)2553k=45001899√25570π, | (1.68) |
∞∑k=0(2187684k2−200056k−11293)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−5280)3k=−1953√330π, | (1.69) |
∞∑k=0(101740699836k2+107483900696k+5743181813)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−5280)3k=4966100118√3305π, | (1.70) |
∞∑k=0(16444841148k2−1709536232k−53241371)(2kk)(3kk)(6k3k)(2k−1)(6k−1)(−640320)3k=−1672209√10005π, | (1.71) |
and
∞∑k=0P(k)(2kk)(3kk)(6k3k)(k+1)(2k−1)(6k−1)(−640320)3k=18×5574033√100055π, | (1.72) |
where
P(k):=637379600041024803108k2+657229991696087780968k+19850391655004126179. |
Recall that the Catalan numbers are given by
Cn:=(2nn)n+1=(2nn)−(2nn+1) (n∈N). |
For
(2kk)2k−1={−1if k=0,2Ck−1if k>0. |
Thus, for any
∞∑k=0(ak2+bk+c)(2kk)3(2k−1)3mk=−c+∞∑k=1(ak2+bk+c)(2Ck−1)3mk=−c+8m∞∑k=0a(k+1)2+b(k+1)+cmkC3k. |
For example, (1.2) has the equivalent form
∞∑k=04k+3(−64)kC3k=8−16π.(1.2′) |
For any odd prime
(p+1)/2∑k=0(4k−1)(2kk)3(2k−1)3(−64)k≡p(−1p)+p3(Ep−3−2) (mod p4) |
(where
(p−1)/2∑k=04k+3(−64)kC3k≡8(1−p(−1p)−p3(Ep−3−2)) (mod p4). |
Recently, C. Wang [50] proved that for any prime
(p+1)/2∑k=0(3k−1)(2kk)3(2k−1)216k≡p+2p3(−1p)(Ep−3−3) (mod p4) |
and
p−1∑k=0(3k−1)(2kk)3(2k−1)216k≡p−2p3 (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(2k−1)4096k=−1728π. |
Now we state our second theorem.
Theorem 1.2. We have the identities
∞∑k=128k2+31k+8(2k+1)2k3(2kk)3=π2−82, | (1.73) |
∞∑k=142k2+39k+8(2k+1)3k3(2kk)3=9π2−882, | (1.74) |
∞∑k=1(8k2+5k+1)(−8)k(2k+1)2k3(2kk)3=4−6G, | (1.75) |
∞∑k=1(30k2+33k+7)(−8)k(2k+1)3k3(2kk)3=54G−52, | (1.76) |
∞∑k=1(3k+1)16k(2k+1)2k3(2kk)3=π2−82, | (1.77) |
∞∑k=1(4k+1)(−64)k(2k+1)2k2(2kk)3=4−8G, | (1.78) |
∞∑k=1(4k+1)(−64)k(2k+1)3k3(2kk)3=16G−16, | (1.79) |
∞∑k=1(2k2−11k−3)8k(2k+1)(3k+1)k3(2kk)2(3kk)=48−5π22, | (1.80) |
∞∑k=2(178k2−103k−39)8k(k−1)(2k+1)(3k+1)k3(2kk)2(3kk)=1125π2−1109636, | (1.81) |
∞∑k=1(5k+1)(−27)k(2k+1)(3k+1)k2(2kk)2(3kk)=6−9K, | (1.82) |
∞∑k=2(45k2+5k−2)(−27)k−1(k−1)(2k+1)(3k+1)k3(2kk)2(3kk)=37−48K16, | (1.83) |
∞∑k=1(98k2−21k−8)81k(2k+1)(4k+1)k3(2kk)2(4k2k)=216−20π2, | (1.84) |
∞∑k=2(1967k2−183k−104)81k(k−1)(2k+1)(4k+1)k3(2kk)2(4k2k)=20000π2−190269120, | (1.85) |
∞∑k=1(46k2+3k−1)(−144)k(2k+1)(4k+1)k3(2kk)2(4k2k)=72−2252K, | (1.86) |
∞∑k=2(343k2+18k−16)(−144)k(k−1)(2k+1)(4k+1)k3(2kk)2(4k2k)=9375K−704810, | (1.87) |
where
G:=∞∑k=0(−1)k(2k+1)2 and K:=∞∑k=0(k3)k2. |
For
(k−1)k(2kk)=2(2j+1)j(2jj). |
Thus, for any
∞∑j=1(aj2+bj+c)mj(2j+1)3j3(2jj)3=8m∞∑k=2(a(k−1)2+b(k−1)+c)mk(k−1)3k3(2kk)3. |
For example, (1.77) has the following equivalent form
∞∑k=2(2k−1)(3k−2)16k(k−1)3k3(2kk)3=π2−8.(1.77′) |
In contrast with the Domb numbers, we introduce a new kind of numbers
Sn:=n∑k=0(nk)2TkTn−k (n=0,1,2,…). |
The values of
1,2,10,68,586,5252,49204,475400,4723786,47937812,494786260 |
respectively. We may extend the numbers
Sn(b,c):=n∑k=0(nk)2Tk(b,c)Tn−k(b,c) (n=0,1,2,…). |
Note that
Now we state our third theorem.
Theorem 1.3. We have
∞∑k=07k+324kSk(1,−6)=15√2π, | (1.88) |
∞∑k=012k+5(−28)kSk(1,7)=6√7π, | (1.89) |
∞∑k=084k+2980kSk(1,−20)=24√15π, | (1.90) |
∞∑k=03k+1(−100)kSk(1,25)=258π, | (1.91) |
∞∑k=0228k+67224kSk(1,−56)=80√7π, | (1.92) |
∞∑k=0399k+101(−676)kSk(1,169)=25358π, | (1.93) |
∞∑k=02604k+5632600kSk(1,−650)=850√393π, | (1.94) |
∞∑k=039468k+7817(−6076)kSk(1,1519)=4410√31π, | (1.95) |
∞∑k=041667k+78799800kSk(1,−2450)=40425√64π, | (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
Type Ⅷ.
where
Unlike Ramanujan-type series given by others, all our series for
Motivated by the author's effective way to find new series for
Conjecture 1.1 (General Criterion for Rational Ramanujan-type Series for
∞∑k=0bk+cmkak=r∑i=1λi√diπ | (1.98) |
for some nonzero rational numbers
p−1∑k=0bk+cmkak≡p(r∑i=1ci(εidip)+∑r<j≤3cj(djp)) (mod p2), | (1.99) |
where
For a Ramanujan-type series of the form (1.98), we call
Conjecture 1.2. Let
p−1∑k=0bk+cmkak≡p(c1(d1p)+c2(d2p)+c3(d3p)) (mod p2) | (1.100) |
for all primes
1(pn)2(pn−1∑k=0bk+cmkak−pδn−1∑k=0bk+cmkak)∈Zp for all n∈Z+. |
Joint with the author's PhD student Chen Wang, we pose the following conjecture.
Conjecture 1.3 (Chen Wang and Z.-W. Sun). Let
Remark 1.2. The author [39,Conjecture 1.1(i)] conjectured that
p−1∑k=0(8k+5)T2k≡3p(−3p) (mod p2) |
for any prime
All the new series and related congruences in Sections 3-9 support Conjectures 1.1-1.3. We discover the conjectural series for
Conjecture 1.4 (Duality Principle). Let
ak≡(dp)Dkap−1−k (mod p) | (1.101) |
for any prime
∞∑k=0bk+cmkak=λ1√d1+λ2√d2+λ3√d3π |
for some
p−1∑k=0akmk≡(dp)p−1∑k=0ak(D/m)k (mod p2) | (1.102) |
for any prime
Remark 1.3 (ⅰ) For any prime
(ⅱ) For any
Tk(b,c)≡(b2−4cp)(b2−4c)kTp−1−k(b,c) (mod p) | (1.103) |
for all
For a series
In Section 10, we pose two curious conjectural series for
Lemma 2.1. Let
n∑k=0((64−m)k3−32k2−16k+8)(2kk)3(2k−1)2mk=8(2n+1)mn(2nn)3, | (2.1) |
n∑k=0((64−m)k3−96k2+48k−8)(2kk)3(2k−1)3mk=8mn(2nn)3, | (2.2) |
n∑k=0((108−m)k3−54k2−12k+6)(2kk)2(3kk)(2k−1)(3k−1)mk=6(3n+1)mn(2nn)2(3nn), | (2.3) |
n∑k=0((108−m)k3−(54+m)k2−12k+6)(2kk)2(3kk)(k+1)(2k−1)(3k−1)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
(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
Note that
and recall Bauer's series
So, we get
This proves (1.1). By (2.2) with
and hence
Combining this with
In view of (2.7) with
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
and hence
This proves
By (2.8) with
and hence
Note that
Therefore, with the help of
This proves
The identities (1.3)–(1.70) can be proved similarly.
Lemma 2.2. Let
(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
Let us illustrate our proofs by proving (1.77)-(1.79) and (1.82)-(1.83) in details.
In view of (2.11) with
for all
Notice that
So we have
and hence (1.77) holds.
By (2.11) with
for all
Since
we see that
and hence (1.78) holds. In light of (2.12) with
for all
Since
This proves (1.79).
By (2.13) with
As
and
we see that (1.82) follows. By (2.14) with
and hence
As
with the aid of (1.82) we get
and hence
Other identities in Theorem 1.2 can be proved similarly.
For integers
(2.17) |
For
(2.18) |
Lemma 2.3. For any
(2.19) |
and
(2.20) |
where
Proof. For
By the telescoping method for double summation [7], for
with
where
and
with
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
The identity (2.20) can be proved similarly. In fact, if we use
In view of the above, we have completed the proof of Lemma 2.3.
Lemma 2.4. For any
(2.21) |
Proof. For each
If
with the aid of the Chu-Vandermonde identity. Therefore
This proves (2.21).
Lemma 2.5. For
(2.22) |
Proof. Let
and
Hence
This proves (2.22).
To prove Theorem 1.3, we need an auxiliary theorem.
Theorem 2.6. Let
(2.23) |
Proof. Let
and similarly
where we consider
If
where
and
Recall that
As
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
for any
Therefore
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
since
and (I5) and (I5
For the conjectural identities in Conjecture 3.1, we have conjectures for the corresponding
and
Concerning (I5) and (I5
and
for each
and
By [40,Theorem 5.1], we have
for any prime
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
and that
where
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
divided by
divided by
divided by
divided by
divided by
Now we pose four conjectural series for
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
(3.1) |
and this number is odd if and only if
(ⅱ) Let
(3.2) |
If
(3.3) |
for all
(ⅲ) Let
(3.4) |
Remark 3.4. The imaginary quadratic field
Conjecture 3.8. (ⅰ) For any
(3.5) |
and the number is odd if and only if
(ⅱ) Let
(3.6) |
If
(3.7) |
for all
(ⅲ) Let
(3.8) |
Remark 3.5. This conjecture can be viewed as the dual of Conjecture 3.7. Note that the series
Conjecture 3.9. (ⅰ) For each
(3.9) |
(ⅱ) Let
(3.10) |
If
(3.11) |
divided by
(ⅲ) Let
(3.12) |
Remark 3.6. The imaginary quadratic field
Conjecture 3.10. (ⅰ) For each
(3.13) |
(ⅱ) Let
(3.14) |
If
(3.15) |
divided by
(ⅲ) Let
(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
(3.17) |
(ⅱ) Let
(3.18) |
If
(3.19) |
divided by
(ⅲ Let
(3.20) |
where
Remark 3.8. Note that the imaginary quadratic field
Conjecture 3.12. (ⅰ) For each
(3.21) |
and this number is odd if and only if
(ⅱ) Let
(3.22) |
If
(3.23) |
divided by
(ⅲ) Let
(3.24) |
where
Remark 3.9. Note that the imaginary quadratic field
Conjectures 4.1–4.14 below provide congruences related to (1.88)–(1.97).
Conjecture 4.1. (ⅰ) For any
(4.1) |
(ⅱ) Let
(4.2) |
If
(4.3) |
for all
(ⅲ) For any prime
(4.4) |
Conjecture 4.2. (ⅰ) For any
(4.5) |
and this number is odd if and only if
(ⅱ) Let
(4.6) |
and moreover
(4.7) |
for all
(ⅲ) For any prime
(4.8) |
where
Conjecture 4.3. (ⅰ) For any
(4.9) |
and this number is odd if and only if
(ⅱ) Let
(4.10) |
If
(4.11) |
for all
(ⅲ) For any prime
(4.12) |
where
Conjecture 4.4. (ⅰ) For any
(4.13) |
(ⅱ) Let
(4.14) |
for all
(ⅲ) For any prime
(4.15) |
where
Conjecture 4.5. (ⅰ) For any
(4.16) |
and this number is odd if and only if
(ⅱ) Let
(4.17) |
If
(4.18) |
for all
(ⅲ) For any prime
(4.19) |
where
Conjecture 4.6. (ⅰ) For any
(4.20) |
(ⅱ) Let
(4.21) |
divided by
(ⅲ) For any prime
(4.22) |
where
Conjecture 4.7. (ⅰ) For any
(4.23) |
and this number is odd if and only if
(ⅱ) Let
(4.24) |
If
(4.25) |
divided by
(ⅲ)For any odd prime
(4.26) |
where
Conjecture 4.8. (ⅰ) For any
(4.27) |
and this number is odd if and only if
(ⅱ) Let
(4.28) |
divided by
(ⅲ) For any prime
(4.29) |
where
Conjecture 4.9. (ⅰ) For any
(4.30) |
(ⅱ) Let
(4.31) |
If
(4.32) |
divided by
(ⅲ) For any prime
(4.33) |
where
Conjecture 4.10. (ⅰ) For any
(4.34) |
(4.35) |
divided by
(4.36) |
where
Conjecture 4.11. For any odd prime
(4.37) |
Also, for any prime
(4.38) |
Conjecture 4.12.
(4.39) |
and this number is odd if and only if
(4.40) |
(4.41) |
Conjecture 4.13.
(4.42) |
and this number is odd if and only if
(4.43) |
If
(4.44) |
for all
(4.45) |
Conjecture 4.14.
(4.46) |
(4.47) |
If
(4.48) |
for all
(4.49) |
where
Conjecture 4.15. Let
(4.50) |
where
Remark 4.1. We also have some similar conjectures involving
modulo
Motivated by Theorem 2.6, we pose the following general conjecture.
Conjecture 4.16. For any odd prime
(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
Let
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
J. Wan and Zudilin [49] obtained the following irrational series for
Via our congruence approach (including Conjecture 1.4), we find 24 rational series for
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
Conjecture 5.2. (ⅰ) For any
(5.25) |
(5.26) |
If
(5.27) |
for all
(5.28) |
Conjecture 5.3. (ⅰ) For any
(5.29) |
(5.30) |
If
(5.31) |
for all
(5.32) |
Conjecture 5.4.
(5.33) |
(5.34) |
If
(5.35) |
for all
(5.36) |
Conjecture 5.5.
(5.37) |
(5.38) |
If
(5.39) |
for all
(5.40) |
where
Sun [36,37] obtained some supercongruences involving the Franel numbers
Let
for any
Wan and Zudilin [49] deduced the following irrational series for
Via our congruence approach (including Conjecture 1.4), we find
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.
(6.13) |
(6.14) |
If
(6.15) |
for all
(6.16) |
Remark 6.1 This conjecture was formulated by the author on Oct. 25, 2019.
Conjecture 6.3. For any
(6.17) |
(6.18) |
If
(6.19) |
divided by
(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.
(6.21) |
(6.22) |
If
(6.23) |
divided by
(6.24) |
where
Remark 6.3. Note that the imaginary quadratic field
The following conjecture is related to the identity
Conjecture 6.5.
(6.25) |
(6.26) |
If
(6.27) |
divided by
(6.28) |
where
Remark 6.4. Note that the imaginary quadratic field
The following conjecture is related to the identity
Conjecture 6.6.
(6.29) |
(6.30) |
If
(6.31) |
divided by
(6.32) |
where
Remark 6.5. Note that the imaginary quadratic field
The identities
(with class number
For
It is known that
Let
by [24,Lemma 2.7(ⅱ)]. Combining this with Remark 1.3(ⅱ), we see that
for any
Wan and Zudilin [49] obtained the following irrational series for
Using our congruence approach (including Conjecture 1.4), we find 12 rational series for
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.
(7.13) |
and this number is odd if and only if
(7.14) |
If
(7.15) |
divided by
(7.16) |
where
Remark 7.1. Note that the imaginary quadratic field
The following conjecture is related to the identity
Conjecture 7.3.
(7.17) |
and this number is odd if and only if
(7.18) |
If
(7.19) |
divided by
(7.20) |
where
Remark 7.2. Note that the imaginary quadratic field
Now we pose a conjecture related to the identity
Conjecture 7.4.
(7.21) |
(7.22) |
If
(7.23) |
is a
(7.24) |
where
Remark 7.3. Note that the imaginary quadratic field
Now we pose a conjecture related to the identity
Conjecture 7.5.
(7.25) |
and this number is odd if and only if
(7.26) |
If
(7.27) |
divided by
(7.28) |
where
Remark 7.4. Note that the imaginary quadratic field
The identities
To conclude this section, we confirm an open series for
Theorem 7.1. We have
(7.29) |
where
Proof. The Franel numbers of order
By [11,(8.1)], for
(7.30) |
Since
putting
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
by [24,Lemma 2.7(ⅰ)]. Combining this with Remark 1.3(ⅱ), we see that
for any
Wan and Zudilin [49] obtained the following irrational series for
Using our congruence approach (including Conjecture 1.4), we find one rational series for
Conjecture 8.1. (ⅰ) We have
(8.1) |
Also, for any
(8.2) |
(8.3) |
If
(8.4) |
for all
(8.5) |
Remark 8.1. This conjecture was formulated by the author on Oct. 27, 2019.
Conjecture 8.2.
(8.6) |
and this number is odd if and only if
(8.7) |
If
(8.8) |
divided by
(8.9) |
Remark 8.2. This conjecture was formulated by the author on Nov. 13, 2019.
Conjecture 8.3.
(8.10) |
and this number is odd if and only if
(8.11) |
If
(8.12) |
for all
(8.13) |
Remark 8.3. This conjecture was formulated by the author on Nov. 13, 2019.
Conjecture 8.4.
(8.14) |
and this number is odd if and only if
(8.15) |
If
(8.16) |
for all
(8.17) |
Conjecture 8.5.
(8.18) |
(8.19) |
If
(8.20) |
for all
(8.21) |
Conjecture 8.6.
(8.22) |
and this number is odd if and only if
(8.23) |
If
(8.24) |
for all
(8.25) |
Conjecture 8.7.
(8.26) |
and this number is odd if and only if
(8.27) |
If
(8.28) |
for all
(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
Proof. Note that
with the help of the known congruence
By induction,
for all
So we have
Now let
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
for any
Wan and Zudilin [49] obtained the following irrational series for
Using our congruence approach (including Conjecture 1.4), we find five rational series for
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.
(9.6) |
and this number is odd if and only if
(9.7) |
If
(9.8) |
for all
(9.9) |
Conjecture 9.3.
(9.10) |
and this number is odd if and only if
(9.11) |
If
(9.12) |
for all
(9.13) |
Now we give one more conjecture in this section.
Conjecture 9.4.
(9.14) |
(9.15) |
If
(9.16) |
for all
(9.17) |
Remark 9.1. For primes
Let
with the aid of [33,Lemma 2.1]. Thus
in view of Remark 1.3(ⅱ).
Let
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
(10.3) |
and
(10.4) |
Conjecture 10.3. (ⅰ) We have
for all
for each prime
(10.5) |
Remark 10.2. See also [45,Conjecture 67] for a similar conjecture.
Let
(10.6) |
and
(10.7) |
Though (10.6) implies the congruence
and (10.7) with
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
![]() |
1. | Ji-Cai Liu, On two supercongruences for sums of Apéry-like numbers, 2021, 115, 1578-7303, 10.1007/s13398-021-01092-6 | |
2. | Rong-Hua Wang, Michael X.X. Zhong, q-Rational reduction and q-analogues of series for π, 2023, 116, 07477171, 58, 10.1016/j.jsc.2022.08.020 | |
3. | Qing-hu Hou, Guo-jie Li, Gosper summability of rational multiples of hypergeometric terms, 2021, 27, 1023-6198, 1723, 10.1080/10236198.2021.2007903 | |
4. | Qing-Hu Hou, Ke Liu, Congruences and telescopings of P-recursive sequences, 2021, 27, 1023-6198, 686, 10.1080/10236198.2021.1934462 | |
5. | Ji-Cai Liu, On two congruences involving Franel numbers, 2020, 114, 1578-7303, 10.1007/s13398-020-00935-y | |
6. | Liuquan Wang, Yifan Yang, Ramanujan-type -series from bimodular forms, 2022, 59, 1382-4090, 831, 10.1007/s11139-021-00532-6 | |
7. | Zhi-Wei Sun, On Motzkin numbers and central trinomial coefficients, 2022, 136, 01968858, 102319, 10.1016/j.aam.2021.102319 | |
8. | Ji-Cai Liu, Ramanujan-Type Supercongruences Involving Almkvist–Zudilin Numbers, 2022, 77, 1422-6383, 10.1007/s00025-022-01607-6 | |
9. | Qing-Hu Hou, Zhi-Wei Sun, q-Analogues of Some Series for Powers of , 2021, 25, 0218-0006, 167, 10.1007/s00026-021-00522-x | |
10. | Rong-Hua Wang, Rational Reductions for Holonomic Sequences, 2024, 1009-6124, 10.1007/s11424-024-4034-y | |
11. | Chunli Li, Wenchang Chu, Infinite series about harmonic numbers inspired by Ramanujan–like formulae, 2023, 31, 2688-1594, 4611, 10.3934/era.2023236 | |
12. | Zhi-Wei Sun, 2025, Chapter 21, 978-3-031-65063-5, 413, 10.1007/978-3-031-65064-2_21 | |
13. | Sun Zhi-Wei, Infinite series involving binomial coefficients and harmonic numbers, 2024, 54, 1674-7216, 765, 10.1360/SSM-2024-0007 |