American Institute of Mathematical Sciences

Advanced
doi: 10.3934/mfc.2021003

Semi-Supervised classification of hyperspectral images using discrete nonlocal variation Potts Model

Linyao Ge , Baoxiang Huang , Weibo Wei and  Zhenkuan Pan ,

College of Computer Science and Technology, Qingdao University, Qingdao, 266071, China

* Corresponding author: Zhenkuan Pan

Received  October 2020 Revised  February 2021 Published  March 2021

The classification of Hyperspectral Image (HSI) plays an important role in various fields. To achieve more precise multi-target classification in a short time, a method for combining discrete non-local theory with traditional variable fraction Potts models is presented in this paper. The nonlocal operator makes better use of the information in a certain region centered on that pixel. Meanwhile, adding the constraint in the model can ensure that every pixel in HSI has only one class. The proposed model has the characteristics of non-convex, nonlinear, and non-smooth so that it is difficult to achieve global optimization results. By introducing a series of auxiliary variables and using the alternating direction method of multipliers, the proposed classification model is transformed into a series of convex subproblems. Finally, we conducted comparison experiments with support vector machine (SVM), K-nearest neighbor (KNN), and convolutional neural network (CNN) on five different dimensional HSI data sets. The numerical results further illustrate that the proposed method is stable and efficient and our algorithm can get more accurate predictions in a shorter time, especially when classifying data sets with more spectral layers.

Keywords: Hyperspectral images, Alternating Direction Method of Multipliers (ADMM), discrete nonlocal variation Potts model, semi-supervised classification.
Mathematics Subject Classification: Primary: 58F15, 58F17;Secondary: 53C35.
Citation: Linyao Ge, Baoxiang Huang, Weibo Wei, Zhenkuan Pan. Semi-Supervised classification of hyperspectral images using discrete nonlocal variation Potts Model. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021003
References:
[1]

A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Model. Simul., 10 (2012), 1090-1118.  doi: 10.1137/11083109X.  Google Scholar

[2]

C. Bo, H. Lu and D. Wang, Spectral-spatial K-Nearest Neighbor approach for hyperspectral image classification, Multimedia Tools Appl., 77 (2018), 10419-10436.  doi: 10.1007/s11042-017-4403-9.  Google Scholar

[3]

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A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2015), 490-530.  doi: 10.1137/040616024.  Google Scholar

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Y. Cai, X. F. Zhu and Z. Sun, Semi-supervised and ensemble learning: A review, Comput. Sci., 44 (2017), 7-13.   Google Scholar

[6]

G. Camps-Valls and L. Bruzzone, Kernel-based methods for hyperspectral image classification, IEEE Transactions on Geoscience and Remote Sensing, 43 (2005), 1351-1362.  doi: 10.1109/TGRS.2005.846154.  Google Scholar

[7]

G. Camps-Valls, L. Gomez-Chova, J. Munoz-Mari and et al., Composite kernels for hyperspectral image classification, IEEE Geoscience and Remote Sensing Letters, 3 (2006), 93-97.  doi: 10.1109/LGRS.2005.857031.  Google Scholar

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Z. Dou, B. Zhang and X. Yu, A new alternating minimization algorithm for image segmentation, 6th International Conference on Wireless, Mobile and Multi-Media (ICWMMN), 2015. doi: 10.1049/cp. 2015.0936.  Google Scholar

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M. D. Farrell and R. M. Mersereau, On the impact of PCA dimension reduction for hyperspectral detection of difficult targets, Geoscience and Remote Sensing Letters, 2 (2005), 192-195.  doi: 10.1109/LGRS.2005.846011.  Google Scholar

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T. Goldstein, B. O'Donoghue, S. Setzer and R. Baraniuk, Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2014), 1588-1623.  doi: 10.1137/120896219.  Google Scholar

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[20]

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[21]

S. Kaewpijit, J. Le Moigne and T. El-Ghazawi, Automatic reduction of hyperspectral imagery using wavelet spectral analysis, IEEE Transactions on Geoscience and Remote Sensing, 41 (2003), 863-871.  doi: 10.1109/TGRS.2003.810712.  Google Scholar

[22]

Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444.  doi: 10.1038/nature14539.  Google Scholar

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F. Li, M. K. Ng, T. Y. Zeng and C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM J. Imaging Sci., 3 (2010), 277-299.  doi: 10.1137/080736752.  Google Scholar

[24]

G. Li, C. Zhang, F. Gao and X. Zhang, Doubleconvpool-structured 3D-CNN for hyperspectral remote sensing image classification, J. Image and Graphics, 24 (2019), 639-654.  doi: 10.11834/jig.180422.  Google Scholar

[25]

F. Melgani and L. Bruzzone, Classification of hyperspectral remote sensing images with support vector machines, IEEE Transactions on Geoscience and Remote Sensing, 42 (2004), 1778-1790.  doi: 10.1109/TGRS.2004.831865.  Google Scholar

[26]

E. Merkurjev, J. Sunu and A. L. Bertozzi, Graph MBO method for multiclass segmentation of hyperspectral stand-off detection video, 2014 IEEE International Conference on Image Processing (ICIP), (2014), 689–693. doi: 10.1109/ICIP. 2014.7025138.  Google Scholar

[27]

B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple junctions: A level set approach, J. Comput. Phys., 112 (1994), 334-363.  doi: 10.1006/jcph.1994.1105.  Google Scholar

[28]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth function and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[29]

M. Myllykoski R. Glowinski, T. Kärkkäinen and T. Rossi, A new augmented Lagrangian approach for $L^1$-mean curvature image denoising, SIAM J. Imaging Sci., 8 (2015), 95-125.  doi: 10.1137/140962164.  Google Scholar

[30]

R. B. Potts, Some generalized order-disorder transformations, in Mathematical Proceedings of the Cambridge Philosophical Society, 48, Cambridge Philosophical Society, 1952, 106–109. doi: 10.1017/S0305004100027419.  Google Scholar

[31]

L. Shen and S. Jia, Three-dimensional Gabor wavelets for pixel-based hyperspectral imagery classification, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 5039-5046.  doi: 10.1109/TGRS.2011.2157166.  Google Scholar

[32]

J. Wang and C.-I. Chang, Independent component analysis-based dimensionality reduction with applications in hyperspectral image analysis, IEEE Transactions on Geoscience and Remote Sensing, 44 (2006), 1586-1600.  doi: 10.1109/TGRS.2005.863297.  Google Scholar

[33]

P. Wang and X. Y. Zhu, Model selection of SVM with RBF kernel and its application, Comput Engrg. Appl., (2003), 72–73. Google Scholar

[34]

Y. Wang and L. Wang, Local Gabor convolutional neural network for hyperspectral image classification, Comput. Sci., 47 (2020), 151-156.  doi: 10.11896/jsjkx.190500147.  Google Scholar

[35]

C. Yi, L. F. Zhang and X. Zhang, et al., Aerial hyperspectral remote sensing classification dataset of Xiongan New Area (Matiwan Village), J. Remote Sensing, (2019). doi: 10.11834/jrs. 20209065.  Google Scholar

[36]

L. Zhang, H. Sun, Z. Rao and H. Ji, Hyperspectral imaging technology combined with deep forest model to identify frost-damaged rice seeds, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 229 (2020). doi: 10.1016/j. saa. 2019.117973.  Google Scholar

[37]

X. Zhang, B. Zhang, L. Zhang and Y. Sun, Hyperspectral remote sensing dataset for tea farm, Global Change Data Repository, (2017). doi: 10.3974/geodb. 2017.03.04. V1.  Google Scholar

[38]

W. Zhu, V. Chayes, A. Tiard and et al., Unsupervised classification in hyperspectral imagery with nonlocal total variation and primal-dual hybrid gradient algorithm, IEEE Transactions on Geoscience and Remote Sensing, 55 (2017), 2786-2798.  doi: 10.1109/TGRS.2017.2654486.  Google Scholar

[39]

S. Zhuo, X. S. Guo and J. Wan, et al., Fast classification algorithm for polynomial kernel support vector machines, Comput. Engrg., 33 (2007). Google Scholar

show all references

References:
[1]

A. L. Bertozzi and A. Flenner, Diffuse interface models on graphs for classification of high dimensional data, Multiscale Model. Simul., 10 (2012), 1090-1118.  doi: 10.1137/11083109X.  Google Scholar

[2]

C. Bo, H. Lu and D. Wang, Spectral-spatial K-Nearest Neighbor approach for hyperspectral image classification, Multimedia Tools Appl., 77 (2018), 10419-10436.  doi: 10.1007/s11042-017-4403-9.  Google Scholar

[3]

B. E. Boser, I. M. Guyon and V. N. Vapnik, A training algorithm for optimal margin classifier, Proceedings of the Fifth Annual Workshop on Computational Learning Theory, 5 (1992), 144-152.  doi: 10.1145/130385.130401.  Google Scholar

[4]

A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4 (2015), 490-530.  doi: 10.1137/040616024.  Google Scholar

[5]

Y. Cai, X. F. Zhu and Z. Sun, Semi-supervised and ensemble learning: A review, Comput. Sci., 44 (2017), 7-13.   Google Scholar

[6]

G. Camps-Valls and L. Bruzzone, Kernel-based methods for hyperspectral image classification, IEEE Transactions on Geoscience and Remote Sensing, 43 (2005), 1351-1362.  doi: 10.1109/TGRS.2005.846154.  Google Scholar

[7]

G. Camps-Valls, L. Gomez-Chova, J. Munoz-Mari and et al., Composite kernels for hyperspectral image classification, IEEE Geoscience and Remote Sensing Letters, 3 (2006), 93-97.  doi: 10.1109/LGRS.2005.857031.  Google Scholar

[8]

C. -I. Chang, Hyperspectral Imaging: Techniques for Spectral Detection and Classification, Springer, 2003. doi: 10.1007/978-1-4419-9170-6.  Google Scholar

[9]

Z. Dou, B. Zhang and X. Yu, A new alternating minimization algorithm for image segmentation, 6th International Conference on Wireless, Mobile and Multi-Media (ICWMMN), 2015. doi: 10.1049/cp. 2015.0936.  Google Scholar

[10]

M. D. Farrell and R. M. Mersereau, On the impact of PCA dimension reduction for hyperspectral detection of difficult targets, Geoscience and Remote Sensing Letters, 2 (2005), 192-195.  doi: 10.1109/LGRS.2005.846011.  Google Scholar

[11]

C. Garcia-Cardona, E. Merkurjev, A. L. Bertozzi and et al., Multiclass data segmentation using diffuse interface methods on graphs, IEEE Transactions on Pattern Analysis and Machine Intelligence, 36 (2014), 1600-1613.  doi: 10.1109/TPAMI.2014.2300478.  Google Scholar

[12]

G. Gilboa and S. Osher, Nonlocal Operators with Applications to Image Processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[13]

T. Goldstein, B. O'Donoghue, S. Setzer and R. Baraniuk, Fast alternating direction optimization methods, SIAM J. Imaging Sci., 7 (2014), 1588-1623.  doi: 10.1137/120896219.  Google Scholar

[14]

J. A. Gualtieri and R. F. Cromp, Support vector machines for hyperspectral remote sensing classification, Proc. SPIE, 3584 (1999). doi: 10.1117/12.339824.  Google Scholar

[15]

X. Hao, G. Zhang and S. Ma, Deep learning, International J. Semantic Computing, 10 (2016), 417-439.  doi: 10.1142/S1793351X16500045.  Google Scholar

[16]

M. He, B. Li and H. Chen, Deep multi-scale 3D deep convolutional neural network for hyperspectral image classification, 2017 IEEE International Conference on Image Processing (ICIP), (2017), 3904–3908. doi: 10.1109/ICIP. 2017.8297014.  Google Scholar

[17]

W. Hu, Y. Huang, L. Wei, F. Zhang and H. Li, Deep convolutional neural networks for hyperspectral image classification, J. Sensors, 2015 (2015), 1-12.  doi: 10.1155/2015/258619.  Google Scholar

[18]

K. Huang, S. Li, X. Kang and L. Fang, Spectral-spatial hyperspectral image classification based on KNN, Sensing and Imaging, 17 (2016). doi: 10.1007/s11220-015-0126-z.  Google Scholar

[19]

G. Huo, S. X. Yang, Q. Li and Y. Zhou, A robust and fast method for sidescan sonar image segmentation using nonlocal despeckling and active contour model, IEEE Transactions on Cybernetics, 47 (2017), 855-872.  doi: 10.1109/TCYB.2016.2530786.  Google Scholar

[20]

S. Jia, L. Shen and Q. Li, Gabor feature-based collaborative representation for hyperspectral imagery classification, IEEE Transactions on Geoscience and Remote Sensing, 53 (2015), 1118-1129.  doi: 10.1109/TGRS.2014.2334608.  Google Scholar

[21]

S. Kaewpijit, J. Le Moigne and T. El-Ghazawi, Automatic reduction of hyperspectral imagery using wavelet spectral analysis, IEEE Transactions on Geoscience and Remote Sensing, 41 (2003), 863-871.  doi: 10.1109/TGRS.2003.810712.  Google Scholar

[22]

Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444.  doi: 10.1038/nature14539.  Google Scholar

[23]

F. Li, M. K. Ng, T. Y. Zeng and C. Shen, A multiphase image segmentation method based on fuzzy region competition, SIAM J. Imaging Sci., 3 (2010), 277-299.  doi: 10.1137/080736752.  Google Scholar

[24]

G. Li, C. Zhang, F. Gao and X. Zhang, Doubleconvpool-structured 3D-CNN for hyperspectral remote sensing image classification, J. Image and Graphics, 24 (2019), 639-654.  doi: 10.11834/jig.180422.  Google Scholar

[25]

F. Melgani and L. Bruzzone, Classification of hyperspectral remote sensing images with support vector machines, IEEE Transactions on Geoscience and Remote Sensing, 42 (2004), 1778-1790.  doi: 10.1109/TGRS.2004.831865.  Google Scholar

[26]

E. Merkurjev, J. Sunu and A. L. Bertozzi, Graph MBO method for multiclass segmentation of hyperspectral stand-off detection video, 2014 IEEE International Conference on Image Processing (ICIP), (2014), 689–693. doi: 10.1109/ICIP. 2014.7025138.  Google Scholar

[27]

B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple junctions: A level set approach, J. Comput. Phys., 112 (1994), 334-363.  doi: 10.1006/jcph.1994.1105.  Google Scholar

[28]

D. Mumford and J. Shah, Optimal approximation by piecewise smooth function and associated variational problems, Comm. Pure Appl. Math., 42 (1989), 577-685.  doi: 10.1002/cpa.3160420503.  Google Scholar

[29]

M. Myllykoski R. Glowinski, T. Kärkkäinen and T. Rossi, A new augmented Lagrangian approach for $L^1$-mean curvature image denoising, SIAM J. Imaging Sci., 8 (2015), 95-125.  doi: 10.1137/140962164.  Google Scholar

[30]

R. B. Potts, Some generalized order-disorder transformations, in Mathematical Proceedings of the Cambridge Philosophical Society, 48, Cambridge Philosophical Society, 1952, 106–109. doi: 10.1017/S0305004100027419.  Google Scholar

[31]

L. Shen and S. Jia, Three-dimensional Gabor wavelets for pixel-based hyperspectral imagery classification, IEEE Transactions on Geoscience and Remote Sensing, 49 (2011), 5039-5046.  doi: 10.1109/TGRS.2011.2157166.  Google Scholar

[32]

J. Wang and C.-I. Chang, Independent component analysis-based dimensionality reduction with applications in hyperspectral image analysis, IEEE Transactions on Geoscience and Remote Sensing, 44 (2006), 1586-1600.  doi: 10.1109/TGRS.2005.863297.  Google Scholar

[33]

P. Wang and X. Y. Zhu, Model selection of SVM with RBF kernel and its application, Comput Engrg. Appl., (2003), 72–73. Google Scholar

[34]

Y. Wang and L. Wang, Local Gabor convolutional neural network for hyperspectral image classification, Comput. Sci., 47 (2020), 151-156.  doi: 10.11896/jsjkx.190500147.  Google Scholar

[35]

C. Yi, L. F. Zhang and X. Zhang, et al., Aerial hyperspectral remote sensing classification dataset of Xiongan New Area (Matiwan Village), J. Remote Sensing, (2019). doi: 10.11834/jrs. 20209065.  Google Scholar

[36]

L. Zhang, H. Sun, Z. Rao and H. Ji, Hyperspectral imaging technology combined with deep forest model to identify frost-damaged rice seeds, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 229 (2020). doi: 10.1016/j. saa. 2019.117973.  Google Scholar

[37]

X. Zhang, B. Zhang, L. Zhang and Y. Sun, Hyperspectral remote sensing dataset for tea farm, Global Change Data Repository, (2017). doi: 10.3974/geodb. 2017.03.04. V1.  Google Scholar

[38]

W. Zhu, V. Chayes, A. Tiard and et al., Unsupervised classification in hyperspectral imagery with nonlocal total variation and primal-dual hybrid gradient algorithm, IEEE Transactions on Geoscience and Remote Sensing, 55 (2017), 2786-2798.  doi: 10.1109/TGRS.2017.2654486.  Google Scholar

[39]

S. Zhuo, X. S. Guo and J. Wan, et al., Fast classification algorithm for polynomial kernel support vector machines, Comput. Engrg., 33 (2007). Google Scholar

Figure 1.  Patch and Search window
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Figure 2.  Classification results of Fanglu data set. The picture of the left is the ground truth, and the remaining four images are the classification result of the SVM, KNN, CNN, and NLVP
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Figure 3.  Classification results of Indian data set. The picture of the left is the ground truth, and the remaining four images are the classification result of the SVM, KNN, CNN, and NLVP
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Figure 4.  Classification results of Salinas Scene data set. The picture of the left is the ground truth, and the remaining four images are the classification result of the SVM, KNN, CNN, and NLVP
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Figure 5.  Classification results of Pavia University data set. The picture of the left is the ground truth, and the remaining four images are the classification result of the SVM, KNN, CNN, and NLVP
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Figure 6.  Classification results of XiongAn data set. The picture of the left is the ground truth, and the remaining four images are the classification result of the SVM, KNN, CNN, and NLVP
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Table 1.  Comparison of numerical results in the Fanglu and Indian data sets
Algorithm Fanglu Indian
Run-Time(s) Accuracy($ \% $) Run-Time(s) Accuracy($ \% $)
SVM 5.625 93.626 2.318s 79.002
KNN 121.741 93.015 11.001s 66.387
CNN 11.002(GPU) 99.196 4.756s(GPU) 84.590
NLVP 16.809 99.961 37.449 99.618
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Algorithm Fanglu Indian
Run-Time(s) Accuracy($ \% $) Run-Time(s) Accuracy($ \% $)
SVM 5.625 93.626 2.318s 79.002
KNN 121.741 93.015 11.001s 66.387
CNN 11.002(GPU) 99.196 4.756s(GPU) 84.590
NLVP 16.809 99.961 37.449 99.618
Table 2.  Comparison of numerical results in the Salinas and Pavia University data sets
Algorithm Salinas Pavia University
Run-Time(s) Accuracy($ \% $) Run-Time(s) Accuracy($ \% $)
SVM 17.938 93.003 6.275 93.098
KNN 283.259 89.227 90.298 86.076
CNN 25.804(GPU) 98.029 12.984(GPU) 99.578
NLVP 31.175 98.889 20.835 99.783
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Algorithm Salinas Pavia University
Run-Time(s) Accuracy($ \% $) Run-Time(s) Accuracy($ \% $)
SVM 17.938 93.003 6.275 93.098
KNN 283.259 89.227 90.298 86.076
CNN 25.804(GPU) 98.029 12.984(GPU) 99.578
NLVP 31.175 98.889 20.835 99.783
Table 3.  The accuracy of all algorithms in each category of the Salinas Scene data set
Label Samples SVM($ \% $) KNN($ \% $) CNN($ \% $) NLVP($ \% $)
1 2009 97.412 94.873 99.104 99.801
2 3726 98.631 98.148 99.973 99.866
3 1976 98.330 97.217 99.241 99.798
4 1394 98.350 98.494 100 99.785
5 2678 97.498 95.183 99.664 98.394
6 3959 98.232 98.055 99.949 99.571
7 3579 98.044 97.346 99.721 99.749
8 11271 89.726 84.624 95.333 97.746
9 6203 98.436 97.469 99.919 99.532
10 3278 94.356 87.340 99.847 96.827
11 1068 96.536 88.390 98.315 97.659
12 1927 97.146 96.886 100 100
13 916 97.489 96.397 100 98.690
14 1070 95.234 86.729 98.972 93.738
15 7268 64.364 54.100 96.863 98.005
16 1807 97.620 95.849 99.779 97.731
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Label Samples SVM($ \% $) KNN($ \% $) CNN($ \% $) NLVP($ \% $)
1 2009 97.412 94.873 99.104 99.801
2 3726 98.631 98.148 99.973 99.866
3 1976 98.330 97.217 99.241 99.798
4 1394 98.350 98.494 100 99.785
5 2678 97.498 95.183 99.664 98.394
6 3959 98.232 98.055 99.949 99.571
7 3579 98.044 97.346 99.721 99.749
8 11271 89.726 84.624 95.333 97.746
9 6203 98.436 97.469 99.919 99.532
10 3278 94.356 87.340 99.847 96.827
11 1068 96.536 88.390 98.315 97.659
12 1927 97.146 96.886 100 100
13 916 97.489 96.397 100 98.690
14 1070 95.234 86.729 98.972 93.738
15 7268 64.364 54.100 96.863 98.005
16 1807 97.620 95.849 99.779 97.731
Table 4.  The accuracy of all algorithms in each category of the Pavia University Scene data set
Label Samples SVM($ \% $) KNN($ \% $) CNN($ \% $) NLVP($ \% $)
1 6631 96.139 87.830 97.994 99.774
2 18649 96.960 98.665 99.882 100
3 2099 74.845 64.078 94.140 99.809
4 3064 93.603 77.742 99.641 98.172
5 1345 99.628 99.257 100 99.777
6 5029 85.942 47.345 99.920 99.980
7 1330 80.000 82.556 91.429 99.925
8 3682 88.403 84.465 98.588 99.891
9 947 100 99.789 99.578 100
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Label Samples SVM($ \% $) KNN($ \% $) CNN($ \% $) NLVP($ \% $)
1 6631 96.139 87.830 97.994 99.774
2 18649 96.960 98.665 99.882 100
3 2099 74.845 64.078 94.140 99.809
4 3064 93.603 77.742 99.641 98.172
5 1345 99.628 99.257 100 99.777
6 5029 85.942 47.345 99.920 99.980
7 1330 80.000 82.556 91.429 99.925
8 3682 88.403 84.465 98.588 99.891
9 947 100 99.789 99.578 100
Table 5.  Comparison of numerical results on the XiongAn data set
Algorithm Run-Time(s) Accuracy($ \% $)
SVM 1571.159 86.361
PCA-SVM 75.926 80.188
PCA-KNN 515.325 80.114
CNN 135.598(GPU) 93.037
NLVP 135.789 99.507
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Algorithm Run-Time(s) Accuracy($ \% $)
SVM 1571.159 86.361
PCA-SVM 75.926 80.188
PCA-KNN 515.325 80.114
CNN 135.598(GPU) 93.037
NLVP 135.789 99.507
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