December  2015, 8(6): 1385-1400. doi: 10.3934/dcdss.2015.8.1385

Weighted two-phase supervised sparse representation based on Gaussian for face recognition

1. 

College of Computer Science & Technology, Zhejiang University of Technology, Hang Zhou, China, China

Received  April 2015 Revised  September 2015 Published  December 2015

As recently newly techniques, two-phase sparse representation algorithms have been presented, which achieve an excellent performance in face recognition via different phase sparse representation, capturing more local structural information of samples. However, there are some defects in these algorithms:1) The Euclidean distance metric applied in these algorithms fails to capture nonlinear structural information, leading to that the performance of these algorithms is sensitive to the geometric structure of facial images. 2) To select the m nearest neighbors of the test sample is achieved directly by applying sparse representation in training samples, which ignores prior information to construct the sparse representation model. In order to solve these problems, a Weighted Two-Phase Supervised Sparse Representation based on Gaussian (GWTPSSR) algorithm is proposed on basic of existing two-phase sparse representation algorithm, in which the nonlinear local information of samples is captured by exploiting effectively the Gaussian distance metric instead of the Euclidean distance metric. Besides, GWTPSSR recreates reconstruction set from training samples in the sparse representation model for each test sample, making full use of prior information to eliminate some training samples far from the test sample. Compared with existing two-phase sparse representation algorithms, experimental results on standard face datasets show that GWTPSSR has better robustness and classification performance.
Citation: Shuhua Xu, Fei Gao. Weighted two-phase supervised sparse representation based on Gaussian for face recognition. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1385-1400. doi: 10.3934/dcdss.2015.8.1385
References:
[1]

, Available, from: , ().

[2]

A. Back and E. Frenod, Geometric two-scale convergence on manifold and applications to the vlasov equation,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 223. doi: 10.3934/dcdss.2015.8.223.

[3]

H. Beirao da Veiga and F. Crispo, On the global regularity for nonlinear systems of the p-laplacian type,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173. doi: 10.3934/dcdss.2013.6.1173.

[4]

M. Debruyne and T. Verdonck, Robust kernel principal component analysis and classification,, Adv. Data Anal. Classification, 4 (2010), 151. doi: 10.1007/s11634-010-0068-1.

[5]

X. Fan, Learning a Hierarchy of Classifiers for Multi-Class Shape Detection,, Ph.D. dissertation, (2006).

[6]

Z. Fan, M. Ni, Q. Zhu and E. Liu, Weighted sparse representation for face recognition,, Neurocomputing, 151 (2015), 304. doi: 10.1016/j.neucom.2014.09.035.

[7]

E. Frenod, An attempt at classifying homogenization-based numerical methods,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015).

[8]

S. Gangaputra and D. Geman, A design principle for coarse-to-fine classification,, in Proc. IEEE Comput. Soc. Conf. CVPR, (2006), 1877. doi: 10.1109/CVPR.2006.21.

[9]

S. Gao, I. Tsang and L.-T. Chia, Kernel sparse representation for image classification and face recognition,, in Computer Vision ECCV (eds. K. Daniilidis, (2010), 1.

[10]

J. Huang, K. Su, J. El-Den, T. Hu and J. Li, An MPCA/LDA based dimensionality reduction algorithm for face recognition,, Mathematical Problems in Engineering, 2014 (2014), 1. doi: 10.1155/2014/393265.

[11]

S. Huang, J. Ye, T. Wang, L. Jiang, X. Wu and Y. Li, Extracting refined low-rank features of robust pca for human action recognition,, Computer Engineering and Computer Science, 40 (2015), 1427. doi: 10.1007/s13369-015-1635-8.

[12]

M. Kirby and L. Sirovich, Application of the KL phase for the characterization of human faces,, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 103.

[13]

Z. Lai, Z. Jin, J. Yang and W. K. Wong, Sparse local discriminant projections for feature extraction,, in Proceedings of ICPR, (2010), 926.

[14]

Z. Y. Liu, K. C. Chiu and L. Xu, Improved system for object detection and star/galaxy classification via local subspace analysis,, Neural Netw., 16 (2003), 437. doi: 10.1016/S0893-6080(03)00015-7.

[15]

Z. Liu, J. Pu, M. Xu and Y. Qiu, Face recognition via weighted two phase test sample sparse representation,, Neural Process. Lett., 41 (2015), 43. doi: 10.1007/s11063-013-9333-6.

[16]

C.-Y. Lu, H. Min, J. Gui, L. Zhu and Y.-K. Lei, Face recognition via weighted sparse representation,, J. Vis. Commun. Image Represent., 24 (2013), 111.

[17]

X. Luan, B. Fang, L. Liu, W. Yang and J. Qian, Extracting sparse error of robust PCA for face recognition in the presence of varying illumination and occlusion,, Pattern Recognition, 47 (2014), 495. doi: 10.1016/j.patcog.2013.06.031.

[18]

K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, An introduction to kernel-based learning algorithms,, IEEE Trans. Neural Netw., 12 (2001), 181. doi: 10.1109/72.914517.

[19]

M. Murtaza, M. Sharif, M. Raza and J. Hussain Shah, Face Recognition Using Adaptive Margin Fisher's Criterion and Linear Discriminant Analysis (AMFC-LDA),, The International Arab Journal of Information Technology, 11 (2014), 149.

[20]

S. W. Park and M. Savvides, A multifactor extension of linear discriminant analysis for face recognition under varying pose and illumination,, EURASIP J. Adv. Signal Process., 2010 (2010). doi: 10.1155/2010/158395.

[21]

P. J. Phillips, The Facial Recognition Technology (FERET) Database [Online]., Available from: , ().

[22]

P. J. Phillips, H. Moon, S. A. Rizvi and P. J. Rauss, The FERET evaluation methodology for face-recognition algorithms,, in 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (1997), 137. doi: 10.1109/CVPR.1997.609311.

[23]

H. Ryu, J.-C. Yoon, S. S. Chun and S. Sull, Coarse-to-fine classification for image-based face detection,, in Image and Video Retrieval, (2006), 291. doi: 10.1007/11788034_30.

[24]

F. Samaria and A. Harter, Parameterisation of a stochastic model for human face identification,, in Second IEEE workshop on Applications of Computer Vision, (1994), 138. doi: 10.1109/ACV.1994.341300.

[25]

B. Scholkopf and A. Smola, Learning with Kernels,, MIT Press, (2002).

[26]

Q. Shi, A. Eriksson, A. Hengel and C. Shen, Is face recognition really a compressive sensing problem?,, in 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2011), 553. doi: 10.1109/CVPR.2011.5995556.

[27]

M. Sugiyama, Dimensionality reduction of multimodal labeled data by local Fisher discriminant analysis,, J. Mach. Learn. Res., 8 (2007), 1027.

[28]

D. Tao , X. Li , X. Wu and S. J. Maybank, Geometric mean for subspace selection,, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 260.

[29]

D. Tao and X. Tang, Kernel full-space biased discriminant analysis,, in 2004 IEEE International Conference on Multimedia and Expo, (2004), 1287. doi: 10.1109/ICME.2004.1394460.

[30]

M. E. Tipping, Sparse kernel principal component analysis,, in in Neural Information Processing Systems (eds. T. K. Leen, (2000), 633.

[31]

V. Vural, G. Fung, B. Krishnapuram, J. G. Dy and B. Rao, Using local dependencies within batches to improve large margin classifiers,, J. Mach. Learn. Res., 10 (2009), 183.

[32]

L. Wang, H. Wu and C. Pan, Manifold regularized local sparse representation for face recognition,, Ieee Transactions on Circuits and Systems for Video Technology, 25 (2015), 651. doi: 10.1109/TCSVT.2014.2335851.

[33]

J. Wright, Y. Ma and J. Mairal, et al., Sparse representation for computer vision and pattern recognition,, in Proceedings of IEEE, (2009), 1.

[34]

J. Wright, A. Y. Yang and A. Ganesh, et al., Robust face recognition via sparse representation,, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 210.

[35]

X. Xiang, J. Yang and Q. Chen, Color face recognition by PCA-like approach,, Neurocomputing, 152 (2015), 231. doi: 10.1016/j.neucom.2014.10.074.

[36]

Y. Xu and D. Zhang, Represent and fuse bimodal biometric images at the feature level: Complex-matrix-based fusion scheme,, Opt. Eng., 49 (2010).

[37]

Y. Xu, D. Zhang, F. Song, J.-Y. Yang, Z. Jing and M. Li, A method for speeding up feature extraction based on KPCA,, Neurocomputing, 70 (2007), 1056. doi: 10.1016/j.neucom.2006.09.005.

[38]

Y. Xu, D. Zhang, J. Yang and J.-Y. Yang, An approach for directly extracting features from matrix data and its application in face recognition,, Neurocomputing, 71 (2008), 1857. doi: 10.1016/j.neucom.2007.09.021.

[39]

Y. Xu, D. Zhang, J. Yang and J.-Y.Yang, A two-phase test sample sparse representation method for use with face recognition,, IEEE Trans. Circuits Syst. Video Technol., 21 (2011), 1255.

[40]

Y. Xu and Q. Zhu, A simple and fast representation-based face recognition method,, Neural Comput Appl., 22 (2013), 1543.

[41]

Y. Xu, W. Zuo and Z. Fan, Supervised sparse presentation method with a heuristic strategy and face recognition experiments,, Neurocomputing, 79 (2011), 125.

[42]

H. Yan, J. Lu, X. Zhou and Y. Shang, Multi-feature multi-manifold learning for single-sample face recognition,, Neurocomputing, 143 (2014), 134. doi: 10.1016/j.neucom.2014.06.012.

[43]

J. Yang, D. Zhang, A. F. Frangi and J.-Y. Yang, Two-dimensional PCA: A new approach to appearance-based face representation and recognition,, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), 131.

[44]

M. Yang, L. Zhang, J. Yang and D. Zhang, Robust sparse coding for face recognition,, in IEEE International Conference Computer Vision and Pattern Recognition, (2011), 625.

[45]

K. Yu and T. Zhang, Improved local coordinate coding using local tangents,, in International Conference on Machine Learning, (2010), 1215.

[46]

K. Yu, T. Zhang and Y. Gong, Nonlinear learning using local coordinate coding,, Adv. Neural Inf. Process. Syst., 22 (2009), 2223.

[47]

Y. Zeng, Y. Yang and L. Zhao, Nonparametric classification based on local mean and class statistics,, Expert Syst. Appl., 36 (2009), 8443. doi: 10.1016/j.eswa.2008.10.041.

[48]

L. Zhang, et al., Sparse representation or collaborative representation: Which helps face recognition?,, in 2011 IEEE International Conference on Computer Vision (ICCV), (2011), 471. doi: 10.1109/ICCV.2011.6126277.

[49]

C. Zhou, L. Wang, Q. Zhang and X. Wei, Face recognition based on PCA and logistic regression analysis,, Optik., 125 (2014), 5916. doi: 10.1016/j.ijleo.2014.07.080.

show all references

References:
[1]

, Available, from: , ().

[2]

A. Back and E. Frenod, Geometric two-scale convergence on manifold and applications to the vlasov equation,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 223. doi: 10.3934/dcdss.2015.8.223.

[3]

H. Beirao da Veiga and F. Crispo, On the global regularity for nonlinear systems of the p-laplacian type,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173. doi: 10.3934/dcdss.2013.6.1173.

[4]

M. Debruyne and T. Verdonck, Robust kernel principal component analysis and classification,, Adv. Data Anal. Classification, 4 (2010), 151. doi: 10.1007/s11634-010-0068-1.

[5]

X. Fan, Learning a Hierarchy of Classifiers for Multi-Class Shape Detection,, Ph.D. dissertation, (2006).

[6]

Z. Fan, M. Ni, Q. Zhu and E. Liu, Weighted sparse representation for face recognition,, Neurocomputing, 151 (2015), 304. doi: 10.1016/j.neucom.2014.09.035.

[7]

E. Frenod, An attempt at classifying homogenization-based numerical methods,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015).

[8]

S. Gangaputra and D. Geman, A design principle for coarse-to-fine classification,, in Proc. IEEE Comput. Soc. Conf. CVPR, (2006), 1877. doi: 10.1109/CVPR.2006.21.

[9]

S. Gao, I. Tsang and L.-T. Chia, Kernel sparse representation for image classification and face recognition,, in Computer Vision ECCV (eds. K. Daniilidis, (2010), 1.

[10]

J. Huang, K. Su, J. El-Den, T. Hu and J. Li, An MPCA/LDA based dimensionality reduction algorithm for face recognition,, Mathematical Problems in Engineering, 2014 (2014), 1. doi: 10.1155/2014/393265.

[11]

S. Huang, J. Ye, T. Wang, L. Jiang, X. Wu and Y. Li, Extracting refined low-rank features of robust pca for human action recognition,, Computer Engineering and Computer Science, 40 (2015), 1427. doi: 10.1007/s13369-015-1635-8.

[12]

M. Kirby and L. Sirovich, Application of the KL phase for the characterization of human faces,, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 103.

[13]

Z. Lai, Z. Jin, J. Yang and W. K. Wong, Sparse local discriminant projections for feature extraction,, in Proceedings of ICPR, (2010), 926.

[14]

Z. Y. Liu, K. C. Chiu and L. Xu, Improved system for object detection and star/galaxy classification via local subspace analysis,, Neural Netw., 16 (2003), 437. doi: 10.1016/S0893-6080(03)00015-7.

[15]

Z. Liu, J. Pu, M. Xu and Y. Qiu, Face recognition via weighted two phase test sample sparse representation,, Neural Process. Lett., 41 (2015), 43. doi: 10.1007/s11063-013-9333-6.

[16]

C.-Y. Lu, H. Min, J. Gui, L. Zhu and Y.-K. Lei, Face recognition via weighted sparse representation,, J. Vis. Commun. Image Represent., 24 (2013), 111.

[17]

X. Luan, B. Fang, L. Liu, W. Yang and J. Qian, Extracting sparse error of robust PCA for face recognition in the presence of varying illumination and occlusion,, Pattern Recognition, 47 (2014), 495. doi: 10.1016/j.patcog.2013.06.031.

[18]

K.-R. Muller, S. Mika, G. Ratsch, K. Tsuda and B. Scholkopf, An introduction to kernel-based learning algorithms,, IEEE Trans. Neural Netw., 12 (2001), 181. doi: 10.1109/72.914517.

[19]

M. Murtaza, M. Sharif, M. Raza and J. Hussain Shah, Face Recognition Using Adaptive Margin Fisher's Criterion and Linear Discriminant Analysis (AMFC-LDA),, The International Arab Journal of Information Technology, 11 (2014), 149.

[20]

S. W. Park and M. Savvides, A multifactor extension of linear discriminant analysis for face recognition under varying pose and illumination,, EURASIP J. Adv. Signal Process., 2010 (2010). doi: 10.1155/2010/158395.

[21]

P. J. Phillips, The Facial Recognition Technology (FERET) Database [Online]., Available from: , ().

[22]

P. J. Phillips, H. Moon, S. A. Rizvi and P. J. Rauss, The FERET evaluation methodology for face-recognition algorithms,, in 1997 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (1997), 137. doi: 10.1109/CVPR.1997.609311.

[23]

H. Ryu, J.-C. Yoon, S. S. Chun and S. Sull, Coarse-to-fine classification for image-based face detection,, in Image and Video Retrieval, (2006), 291. doi: 10.1007/11788034_30.

[24]

F. Samaria and A. Harter, Parameterisation of a stochastic model for human face identification,, in Second IEEE workshop on Applications of Computer Vision, (1994), 138. doi: 10.1109/ACV.1994.341300.

[25]

B. Scholkopf and A. Smola, Learning with Kernels,, MIT Press, (2002).

[26]

Q. Shi, A. Eriksson, A. Hengel and C. Shen, Is face recognition really a compressive sensing problem?,, in 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2011), 553. doi: 10.1109/CVPR.2011.5995556.

[27]

M. Sugiyama, Dimensionality reduction of multimodal labeled data by local Fisher discriminant analysis,, J. Mach. Learn. Res., 8 (2007), 1027.

[28]

D. Tao , X. Li , X. Wu and S. J. Maybank, Geometric mean for subspace selection,, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 260.

[29]

D. Tao and X. Tang, Kernel full-space biased discriminant analysis,, in 2004 IEEE International Conference on Multimedia and Expo, (2004), 1287. doi: 10.1109/ICME.2004.1394460.

[30]

M. E. Tipping, Sparse kernel principal component analysis,, in in Neural Information Processing Systems (eds. T. K. Leen, (2000), 633.

[31]

V. Vural, G. Fung, B. Krishnapuram, J. G. Dy and B. Rao, Using local dependencies within batches to improve large margin classifiers,, J. Mach. Learn. Res., 10 (2009), 183.

[32]

L. Wang, H. Wu and C. Pan, Manifold regularized local sparse representation for face recognition,, Ieee Transactions on Circuits and Systems for Video Technology, 25 (2015), 651. doi: 10.1109/TCSVT.2014.2335851.

[33]

J. Wright, Y. Ma and J. Mairal, et al., Sparse representation for computer vision and pattern recognition,, in Proceedings of IEEE, (2009), 1.

[34]

J. Wright, A. Y. Yang and A. Ganesh, et al., Robust face recognition via sparse representation,, IEEE Trans. Pattern Anal. Mach. Intell., 31 (2009), 210.

[35]

X. Xiang, J. Yang and Q. Chen, Color face recognition by PCA-like approach,, Neurocomputing, 152 (2015), 231. doi: 10.1016/j.neucom.2014.10.074.

[36]

Y. Xu and D. Zhang, Represent and fuse bimodal biometric images at the feature level: Complex-matrix-based fusion scheme,, Opt. Eng., 49 (2010).

[37]

Y. Xu, D. Zhang, F. Song, J.-Y. Yang, Z. Jing and M. Li, A method for speeding up feature extraction based on KPCA,, Neurocomputing, 70 (2007), 1056. doi: 10.1016/j.neucom.2006.09.005.

[38]

Y. Xu, D. Zhang, J. Yang and J.-Y. Yang, An approach for directly extracting features from matrix data and its application in face recognition,, Neurocomputing, 71 (2008), 1857. doi: 10.1016/j.neucom.2007.09.021.

[39]

Y. Xu, D. Zhang, J. Yang and J.-Y.Yang, A two-phase test sample sparse representation method for use with face recognition,, IEEE Trans. Circuits Syst. Video Technol., 21 (2011), 1255.

[40]

Y. Xu and Q. Zhu, A simple and fast representation-based face recognition method,, Neural Comput Appl., 22 (2013), 1543.

[41]

Y. Xu, W. Zuo and Z. Fan, Supervised sparse presentation method with a heuristic strategy and face recognition experiments,, Neurocomputing, 79 (2011), 125.

[42]

H. Yan, J. Lu, X. Zhou and Y. Shang, Multi-feature multi-manifold learning for single-sample face recognition,, Neurocomputing, 143 (2014), 134. doi: 10.1016/j.neucom.2014.06.012.

[43]

J. Yang, D. Zhang, A. F. Frangi and J.-Y. Yang, Two-dimensional PCA: A new approach to appearance-based face representation and recognition,, IEEE Trans. Pattern Anal. Mach. Intell., 26 (2004), 131.

[44]

M. Yang, L. Zhang, J. Yang and D. Zhang, Robust sparse coding for face recognition,, in IEEE International Conference Computer Vision and Pattern Recognition, (2011), 625.

[45]

K. Yu and T. Zhang, Improved local coordinate coding using local tangents,, in International Conference on Machine Learning, (2010), 1215.

[46]

K. Yu, T. Zhang and Y. Gong, Nonlinear learning using local coordinate coding,, Adv. Neural Inf. Process. Syst., 22 (2009), 2223.

[47]

Y. Zeng, Y. Yang and L. Zhao, Nonparametric classification based on local mean and class statistics,, Expert Syst. Appl., 36 (2009), 8443. doi: 10.1016/j.eswa.2008.10.041.

[48]

L. Zhang, et al., Sparse representation or collaborative representation: Which helps face recognition?,, in 2011 IEEE International Conference on Computer Vision (ICCV), (2011), 471. doi: 10.1109/ICCV.2011.6126277.

[49]

C. Zhou, L. Wang, Q. Zhang and X. Wei, Face recognition based on PCA and logistic regression analysis,, Optik., 125 (2014), 5916. doi: 10.1016/j.ijleo.2014.07.080.

[1]

Huining Qiu, Xiaoming Chen, Wanquan Liu, Guanglu Zhou, Yiju Wang, Jianhuang Lai. A fast $\ell_1$-solver and its applications to robust face recognition. Journal of Industrial & Management Optimization, 2012, 8 (1) : 163-178. doi: 10.3934/jimo.2012.8.163

[2]

Honggang Yu. An efficient face recognition algorithm using the improved convolutional neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 901-914. doi: 10.3934/dcdss.2019060

[3]

Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135

[4]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[5]

Ville Kolehmainen, Matthias J. Ehrhardt, Simon R. Arridge. Incorporating structural prior information and sparsity into EIT using parallel level sets. Inverse Problems & Imaging, 2019, 13 (2) : 285-307. doi: 10.3934/ipi.2019015

[6]

Weiping Li, Haiyan Wu, Jie Yang. Intelligent recognition algorithm for social network sensitive information based on classification technology. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1385-1398. doi: 10.3934/dcdss.2019095

[7]

Alberto A. Pinto, João P. Almeida, Telmo Parreira. Local market structure in a Hotelling town. Journal of Dynamics & Games, 2016, 3 (1) : 75-100. doi: 10.3934/jdg.2016004

[8]

Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004

[9]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817

[10]

Jianping Zhang, Ke Chen, Bo Yu, Derek A. Gould. A local information based variational model for selective image segmentation. Inverse Problems & Imaging, 2014, 8 (1) : 293-320. doi: 10.3934/ipi.2014.8.293

[11]

Hideaki Takaichi, Izumi Takagi, Shoji Yotsutani. Global bifurcation structure on a shadow system with a source term - Representation of all solutions-. Conference Publications, 2011, 2011 (Special) : 1344-1350. doi: 10.3934/proc.2011.2011.1344

[12]

Thierry Goudon, Martin Parisot. Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime. Kinetic & Related Models, 2011, 4 (3) : 735-766. doi: 10.3934/krm.2011.4.735

[13]

Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations & Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61

[14]

Badr Saad T. Alkahtani, Ilknur Koca. A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 429-442. doi: 10.3934/dcdss.2020024

[15]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008

[16]

José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921

[17]

Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565

[18]

Augusto VisintiN. On the variational representation of monotone operators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046

[19]

Evans K. Afenya, Calixto P. Calderón. Growth kinetics of cancer cells prior to detection and treatment: An alternative view. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 25-28. doi: 10.3934/dcdsb.2004.4.25

[20]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]