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### Najeeb Alam Khan and Amber Shaikh

### Md Wasiur Rahman, Fatema Tuz Zohra and Marina L. Gavrilova

## Abstract

Computational intelligence firmly made its way into the areas of consumer applications, banking, education, social networks, and security. Among all the applications, biometric systems play a significant role in ensuring an uncompromised and secure access to resources and facilities. This article presents a first multimodal biometric system that combines KINECT gait modality with KINECT face modality utilizing the rank level and the score level fusion. For the KINECT gait modality, a new approach is proposed based on the skeletal information processing. The gait cycle is calculated using three consecutive local minima computed for the distance between left and right ankles. The feature distance vectors are calculated for each person’s gait cycle, which allows extracting the biometric features such as the mean and the variance of the feature distance vector. For Kinect face recognition, a novel method based on HOG features has been developed. Then, K-nearest neighbors feature matching algorithm is applied as feature classification for both gait and face biometrics. Two fusion algorithms are implemented. The combination of Borda count and logistic regression approaches are used in the rank level fusion. The weighted sum method is used for score level fusion. The recognition accuracy obtained for multi-modal biometric recognition system tested on KINECT Gait and KINECT Eurocom Face datasets is 93.33% for Borda count rank level fusion, 96.67% for logistic regression rank-level fusion and 96.6% for score level fusion.

### Kourosh Parand, Mehdi Delkhosh and Mehran Nikarya

, Commun. Nonlinear Sci. Numer. Simulat., 15(2) (2010) 360-367. [42] K. Parand, A. Taghavi, M. Shahini, Comparison between rational Chebyshev and modified generalized Laguerre functions pseudospectral methods for solving Lane-Emden and unsteady gas equations, Acta Phys. Pol. B, 40(12) (2009) 1749-1763. [43] K. Parand, A.R. Rezaei, A. Taghavi, Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison, Math. Method. Appl. Sci., 33(17) (2010) 2076

### Hossein Aminikhah

differential transform method. Journal of Computational and Applied Mathematics 215, 142-151. AYAZ, F. 2004. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation 147, 547-567. GHORBANI, A. 2009. Beyond Adomian polynomials: He polynomials. Chaos, Solitons & Fractals 39, 14861492. SHAWAGFEH, N.T. 1993. Nonperturbative approximate solution for Lane-Emden equation. J. Math. Phys. 34, 4364-4369.

### Asmat Ara, Oyoon Abdul Razzaq and Najeeb Alam Khan

problems, Neural Processing Letters , 4 , (2017), 59–74 [20] H. Zhao and J. Zhang , Functional link neural network cascaded with Chebyshev orthogonal polynomial for nonlinear channel equalization, Neural Processing Letters , 88 , (2008), 1946–1957 [21] S. Mall and S. Chakraverty , Chebyshev neural network based model for solving Lane-Emden type equations, Applied Mathematics and Computation , 247 , (2014), 100–114 [22] Y. Shirvany M. Hayati and R. Moradian , Multilayer perceptron neural networks with novel unsupervised training method for

### Kourosh Parand, Kobra Rabiei and Mehdi Delkhosh

stochastic factor models with jumps using a meshless local Petrov-Galerkin method, Appl. Numer. Math ., 115 (2017), 252–274. [8] K. Parand, P. Mazaheri, M. Delkhosh, A. Ghaderi, New numerical solutions for solving Kidder equation by using the rational Jacobi functions, SeMA J ., (2017) doi:10.1007/s40324-016-0103-z. [9] K. Parand, M. Nikarya, J. A. Rad, Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocation method, Celest. Mech. Dyn. Astr ., 116 (2013), 97–107. [10] D. Funaro and O. Kavian, approximation of some

### Mona D. Aljoufi and Abdelhalim Ebaid

Solving the Volterra Integral Form of the Lane-Emden Equations with Initial Values and Boundary Conditions. Appl. Math. Comput., 219 (2013), No. 10, 5004-5019. [7] EBAID, A. Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method. Computational and Mathematical Methods in Medicine, Volume 2013, Article ID 547954, 8 pages. [8] EBAID, A. Approximate Periodic Solutions for the Non-linear Relativistic Harmonic Oscillator via Differential Transformation Method. Commun. Nonlin. Sci. Numer

### R. A. Mundewadi and S. Kumbinarasaiah

. Vessella S. Abel integral equations, analysis and applications Lecture notes in mathematics Heidelberg Springer 1991 [15] C.T.H. Baker, The numerical treatment of integral equations, Clarendon Press, Oxford, 1977. Baker C.T.H. The numerical treatment of integral equations Clarendon Press Oxford 1977 [16] S.C. Shiralashetti, S. Kumbinarasaiah, Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations, Appl. Math. Comput., 315 (2017), 591–602. Shiralashetti S

### Kuppalapalle Vajravelu, Ronald Li, Mangalagama Dewasurendra, Joseph Benarroch, Nicholas Ossi, Ying Zhang, Michael Sammarco and K.V. Prasad

papers, including: Lane-Emden equation [ 33 ], time-dependent Michaelis-Menton equation [ 34 ], non-local Whitham equation [ 35 ], and Zakharov system [ 36 ] to name a few. Here we outline the solution method and later discuss the convergence and accuracy of the HAM solution. For the present problem, we choose the auxiliary linear operator 𝓛 as L = ∂ 3 ∂ η 3 + β ∂ 2 ∂ η 2 , $$\begin{array}{} \displaystyle \mathscr{L}=\frac{\partial^3}{\partial \eta^3}+\beta\frac{\partial^2}{\partial\eta^2}, \end{array}$$ (13) with an initial approximation to f ( η ) as