American Institute of Mathematical Sciences

Advanced
June  2018, 12(3): 677-696. doi: 10.3934/ipi.2018029

On a gesture-computing technique using electromagnetic waves

Jingzhi Li 1, , Hongyu Liu 2,3, and  Hongpeng Sun 4,,

1. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
2. 

Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
3. 

HKBU Institute of Research and Continuing Education, Virtual University Park, Shenzhen, China
4. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author: hpsun@amss.ac.cn

Received  August 2017 Revised  October 2017 Published  March 2018

This paper is concerned with a conceptual gesture-based instruction/input technique using electromagnetic wave detection. The gestures are modeled as the shapes of some impenetrable or penetrable scatterers from a certain admissible class, called a dictionary. The gesture-computing device generates time-harmonic electromagnetic point signals for the gesture recognition and detection. It then collects the scattered wave in a relatively small backscattering aperture on a bounded surface containing the point sources. The recognition algorithm consists of two stages and requires only two incident waves of different wavenumbers. The location of the scatterer is first determined approximately by using the measured data at a small wavenumber and the shape of the scatterer is then identified using the computed location of the scatterer and the measured data at a regular wavenumber. We provide the corresponding mathematical principle with rigorous analysis. Numerical experiments show that the proposed device works effectively and efficiently.

Keywords: Gesture recognition, instruction/input device, electromagnetic wave propagation, inverse scattering.
Mathematics Subject Classification: Primary: 35R30, 35P25; Secondary: 78A46.
Citation: Jingzhi Li, Hongyu Liu, Hongpeng Sun. On a gesture-computing technique using electromagnetic waves. Inverse Problems & Imaging, 2018, 12 (3) : 677-696. doi: 10.3934/ipi.2018029
References:
[1]

H. Ammari, T. Boulier and J. Garnier, Modeling active electrolocation in weakly electric fish, SIAM J. Imaging Sci., 6 (2013), 285-321.  doi: 10.1137/12086858X.  Google Scholar

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H. Ammari, T. Boulier, J. Garnier, W. Jing, H. Kang and H. Wang, Target identification using dictionary matching of generalized polarization tensors, Found. Comput. Math., 14 (2014), 27-62.  doi: 10.1007/s10208-013-9168-6.  Google Scholar

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O. Bondarenko, A. Kirsch and X. Liu, The factorization method for inverse acoustic scattering in a layered medium, Inverse Problems, 29(2013), 045010, 19pp.  Google Scholar

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M. Born and E. Wolf, Princinples of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, $ 7^{nd}$ Edition, Cambridge University Press, 1999. Google Scholar

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D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $ 3^{nd}$ EditionSpringer, Applied Mathematical Sciences, 2013.  Google Scholar

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G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000.  Google Scholar

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A. Erol, G. Bebis, M. Nicolescu, R. D. Boyle and X. Twombly, Vision-based hand pose estimation: A review, Computer Vision and Image Understanding, 108 (2007), 52-73.  doi: 10.1016/j.cviu.2006.10.012.  Google Scholar

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A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations Expansion-, Integral-, and Variational Methods, Applied Mathematical Sciences, Springer, 2015.  Google Scholar

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J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

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H. Liu, Y. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.  Google Scholar

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A. Makris, N. Kyriazis and A. A. Argyros, Hierarchical particle filtering for 3D hand tracking, IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), 2015. doi: 10.1109/CVPRW.2015.7301343.  Google Scholar

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[17]

J. C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. Google Scholar

[18]

V. I. Pavlovic, R. Sharma and T. S. Huang, Visual interpretation of hand gestures for human-computer interaction: A review, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19 (1997), 677-695.  doi: 10.1109/34.598226.  Google Scholar

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R. Pike and P. Sabatier eds., Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002. Google Scholar

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show all references

References:
[1]

H. Ammari, T. Boulier and J. Garnier, Modeling active electrolocation in weakly electric fish, SIAM J. Imaging Sci., 6 (2013), 285-321.  doi: 10.1137/12086858X.  Google Scholar

[2]

H. Ammari, T. Boulier, J. Garnier, W. Jing, H. Kang and H. Wang, Target identification using dictionary matching of generalized polarization tensors, Found. Comput. Math., 14 (2014), 27-62.  doi: 10.1007/s10208-013-9168-6.  Google Scholar

[3]

H. Ammari, M. Tran and H. Wang, Shape identification and classification in echolocation, SIAM J. Imaging Sci., 7 (2014), 1883-1905.  doi: 10.1137/14096164X.  Google Scholar

[4]

O. Bondarenko, A. Kirsch and X. Liu, The factorization method for inverse acoustic scattering in a layered medium, Inverse Problems, 29(2013), 045010, 19pp.  Google Scholar

[5]

M. Born and E. Wolf, Princinples of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, $ 7^{nd}$ Edition, Cambridge University Press, 1999. Google Scholar

[6]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $ 3^{nd}$ EditionSpringer, Applied Mathematical Sciences, 2013.  Google Scholar

[7]

G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000.  Google Scholar

[8]

A. Erol, G. Bebis, M. Nicolescu, R. D. Boyle and X. Twombly, Vision-based hand pose estimation: A review, Computer Vision and Image Understanding, 108 (2007), 52-73.  doi: 10.1016/j.cviu.2006.10.012.  Google Scholar

[9]

A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations Expansion-, Integral-, and Variational Methods, Applied Mathematical Sciences, Springer, 2015.  Google Scholar

[10]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford, Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008.  Google Scholar

[11]

M. Kolsch and M. Turk, Fast 2D hand tracking with flocks of features and multi-cue integration, IEEE Computer Vision and Pattern Recognition Workshop, 2004. CVPRW 04, 2004. doi: 10.1109/CVPR.2004.345.  Google Scholar

[12]

V. I. Lebedev and D. N. Laikov, A quadrature formula for the sphere of the 131st algebraic order of accuracy, Doklady Mathematics, 59 (1999), 477-481.   Google Scholar

[13]

J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, SIAM Multiscale Model. Simul., 12 (2014), 927-952.  doi: 10.1137/13093409X.  Google Scholar

[14]

H. Liu, Y. Wang and C. Yang, Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9 (2016), 822-841.  doi: 10.1137/16M1063551.  Google Scholar

[15]

A. Makris, N. Kyriazis and A. A. Argyros, Hierarchical particle filtering for 3D hand tracking, IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), 2015. doi: 10.1109/CVPRW.2015.7301343.  Google Scholar

[16]

P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003.  Google Scholar

[17]

J. C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. Google Scholar

[18]

V. I. Pavlovic, R. Sharma and T. S. Huang, Visual interpretation of hand gestures for human-computer interaction: A review, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19 (1997), 677-695.  doi: 10.1109/34.598226.  Google Scholar

[19]

R. Pike and P. Sabatier eds., Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002. Google Scholar

[20]

R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC Research Notes in Mathematics, 2001.  Google Scholar

[21]

Project Soli, Google ATAP, https://www.google.com/atap/project-soli/. Google Scholar

Figure 1.  Dictionary
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Table 1.  PEC location test.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $149.9838$ $149.9611$ $149.9742$ $149.9632$ $150.0075$ $149.9853$
$\mathring{z}_{0}^{2}$ $0.0400$ $0.0280$ $-0.0096$ $0.0442$ $-0.0440$ $0.0321$
$\mathring{z}_{0}^{3}$ $-0.0131$ $-0.0110$ $-0.0404$ $-0.0456$ $-0.0265$ $-0.0485$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0451$ $0.0492$ $0.0489$ $0.0734$ $0.0519$ $0.0600$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $149.9838$ $149.9611$ $149.9742$ $149.9632$ $150.0075$ $149.9853$
$\mathring{z}_{0}^{2}$ $0.0400$ $0.0280$ $-0.0096$ $0.0442$ $-0.0440$ $0.0321$
$\mathring{z}_{0}^{3}$ $-0.0131$ $-0.0110$ $-0.0404$ $-0.0456$ $-0.0265$ $-0.0485$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0451$ $0.0492$ $0.0489$ $0.0734$ $0.0519$ $0.0600$
Table 2.  PEC gesture test.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$
$D_{2}$ $0.9233$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$
$D_{3}$ $0.9290$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$
$D_{4}$ $0.8660$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$
$D_{5}$ $0.8249$ $0.9107$ $0.9492$ $0.9300$ $\boldsymbol{1.0000}$ $0.9159$
$D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9149$ $\boldsymbol{1.0000}$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$
$D_{2}$ $0.9233$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$
$D_{3}$ $0.9290$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$
$D_{4}$ $0.8660$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$
$D_{5}$ $0.8249$ $0.9107$ $0.9492$ $0.9300$ $\boldsymbol{1.0000}$ $0.9159$
$D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9149$ $\boldsymbol{1.0000}$
Table 3.  PEC location test with $5\%$ noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $150.0278$ $150.0547$ $150.0965$ $150.0158$ $150.0971$ $150.0958$
$\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$
$\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $150.0278$ $150.0547$ $150.0965$ $150.0158$ $150.0971$ $150.0958$
$\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$
$\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$
Table 4.  PEC gesture test with $5\%$ noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$
$D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$
$D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$
$D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$
$D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$
$D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$
$D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$
$D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$
$D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$
$D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$
$D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$
Table 5.  Medium location test.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $150.0417$ $150.0254$ $149.9576$ $150.0279$ $150.0069$ $149.9837$
$\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$
$\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $150.0417$ $150.0254$ $149.9576$ $150.0279$ $150.0069$ $149.9837$
$\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$
$\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$
Table 6.  Medium gesture test.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9311$ $0.8848$ $0.9393$ $0.8716$ $0.9418$
$D_{2}$ $0.9312$ $\boldsymbol{1.0000}$ $0.9174$ $0.8838$ $0.9100$ $0.9086$
$D_{3}$ $0.8834$ $0.9179$ $\boldsymbol{1.0000}$ $0.9295$ $0.9740$ $0.8854$
$D_{4}$ $0.9398$ $0.8899$ $0.9004$ $\boldsymbol{1.0000}$ $0.9200$ $0.9864$
$D_{5}$ $0.8737$ $0.9171$ $0.9422$ $0.9183$ $\boldsymbol{1.0000}$ $0.9131$
$D_{6}$ $0.9346$ $0.9087$ $0.8557$ $0.9131$ $0.9089$ $\boldsymbol{1.0000}$
Table Options

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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.9311$ $0.8848$ $0.9393$ $0.8716$ $0.9418$
$D_{2}$ $0.9312$ $\boldsymbol{1.0000}$ $0.9174$ $0.8838$ $0.9100$ $0.9086$
$D_{3}$ $0.8834$ $0.9179$ $\boldsymbol{1.0000}$ $0.9295$ $0.9740$ $0.8854$
$D_{4}$ $0.9398$ $0.8899$ $0.9004$ $\boldsymbol{1.0000}$ $0.9200$ $0.9864$
$D_{5}$ $0.8737$ $0.9171$ $0.9422$ $0.9183$ $\boldsymbol{1.0000}$ $0.9131$
$D_{6}$ $0.9346$ $0.9087$ $0.8557$ $0.9131$ $0.9089$ $\boldsymbol{1.0000}$
Table 7.  Medium location test with $5\%$ noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $149.9758$ $149.9762$ $149.9722$ $149.9819$ $149.9586$ $149.9529$
$\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$
$\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$
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$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$\mathring{z}_{0}^{1}$ $149.9758$ $149.9762$ $149.9722$ $149.9819$ $149.9586$ $149.9529$
$\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$
$\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$
$\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$
Table 8.  Medium gesture test with $5\%$ noise.
$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.8800$ $0.9109$ $0.9321$ $0.8895$ $0.9280$
$D_{2}$ $0.8738$ $\boldsymbol{1.0000}$ $0.8666$ $0.9520$ $0.9303$ $0.9032$
$D_{3}$ $0.9105$ $0.8656$ $\boldsymbol{1.0000}$ $0.9095$ $0.8817$ $0.9021$
$D_{4}$ $0.9320$ $0.9221$ $0.9802$ $\boldsymbol{1.0000}$ $0.9160$ $0.9287$
$D_{5}$ $0.8863$ $0.9285$ $0.8819$ $0.9162$ $\boldsymbol{1.0000}$ $0.9169$
$D_{6}$ $0.9279$ $0.9030$ $0.9256$ $0.9141$ $0.9171$ $\boldsymbol{1.0000}$
Table Options

Download as excel

$D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$
$D_{1}$ $\boldsymbol{1.0000}$ $0.8800$ $0.9109$ $0.9321$ $0.8895$ $0.9280$
$D_{2}$ $0.8738$ $\boldsymbol{1.0000}$ $0.8666$ $0.9520$ $0.9303$ $0.9032$
$D_{3}$ $0.9105$ $0.8656$ $\boldsymbol{1.0000}$ $0.9095$ $0.8817$ $0.9021$
$D_{4}$ $0.9320$ $0.9221$ $0.9802$ $\boldsymbol{1.0000}$ $0.9160$ $0.9287$
$D_{5}$ $0.8863$ $0.9285$ $0.8819$ $0.9162$ $\boldsymbol{1.0000}$ $0.9169$
$D_{6}$ $0.9279$ $0.9030$ $0.9256$ $0.9141$ $0.9171$ $\boldsymbol{1.0000}$
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