# American Institute of Mathematical Sciences

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

## On a gesture-computing technique using electromagnetic waves

 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.

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. [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. [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. [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. [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. [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ EditionSpringer, Applied Mathematical Sciences, 2013. [7] G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [16] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003. [17] J. C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. [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. [19] R. Pike and P. Sabatier eds., Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002. [20] R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC Research Notes in Mathematics, 2001. [21] Project Soli, Google ATAP, https://www.google.com/atap/project-soli/.

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. [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. [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. [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. [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. [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, $3^{nd}$ EditionSpringer, Applied Mathematical Sciences, 2013. [7] G. Dassios and R. Kleinman, Low Frequency Scattering, Clarendon Press, Oxford, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [16] P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003. [17] J. C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. [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. [19] R. Pike and P. Sabatier eds., Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Academic Press, 2002. [20] R. Potthast, Point Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall/CRC Research Notes in Mathematics, 2001. [21] Project Soli, Google ATAP, https://www.google.com/atap/project-soli/.
Dictionary
PEC location test.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9838$ $149.9611$ $149.9742$ $149.9632$ $150.0075$ $149.9853$ $\mathring{z}_{0}^{2}$ $0.0400$ $0.0280$ $-0.0096$ $0.0442$ $-0.0440$ $0.0321$ $\mathring{z}_{0}^{3}$ $-0.0131$ $-0.0110$ $-0.0404$ $-0.0456$ $-0.0265$ $-0.0485$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0451$ $0.0492$ $0.0489$ $0.0734$ $0.0519$ $0.0600$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9838$ $149.9611$ $149.9742$ $149.9632$ $150.0075$ $149.9853$ $\mathring{z}_{0}^{2}$ $0.0400$ $0.0280$ $-0.0096$ $0.0442$ $-0.0440$ $0.0321$ $\mathring{z}_{0}^{3}$ $-0.0131$ $-0.0110$ $-0.0404$ $-0.0456$ $-0.0265$ $-0.0485$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0451$ $0.0492$ $0.0489$ $0.0734$ $0.0519$ $0.0600$
PEC gesture test.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$ $D_{2}$ $0.9233$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$ $D_{3}$ $0.9290$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$ $D_{4}$ $0.8660$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$ $D_{5}$ $0.8249$ $0.9107$ $0.9492$ $0.9300$ $\boldsymbol{1.0000}$ $0.9159$ $D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9149$ $\boldsymbol{1.0000}$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$ $D_{2}$ $0.9233$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$ $D_{3}$ $0.9290$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$ $D_{4}$ $0.8660$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$ $D_{5}$ $0.8249$ $0.9107$ $0.9492$ $0.9300$ $\boldsymbol{1.0000}$ $0.9159$ $D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9149$ $\boldsymbol{1.0000}$
PEC location test with $5\%$ noise.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0278$ $150.0547$ $150.0965$ $150.0158$ $150.0971$ $150.0958$ $\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$ $\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0278$ $150.0547$ $150.0965$ $150.0158$ $150.0971$ $150.0958$ $\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$ $\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$
PEC gesture test with $5\%$ noise.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$ $D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$ $D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$ $D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$ $D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$ $D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$ $D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$ $D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$ $D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$ $D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$ $D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$
Medium location test.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0417$ $150.0254$ $149.9576$ $150.0279$ $150.0069$ $149.9837$ $\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$ $\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0417$ $150.0254$ $149.9576$ $150.0279$ $150.0069$ $149.9837$ $\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$ $\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$
Medium gesture test.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9311$ $0.8848$ $0.9393$ $0.8716$ $0.9418$ $D_{2}$ $0.9312$ $\boldsymbol{1.0000}$ $0.9174$ $0.8838$ $0.9100$ $0.9086$ $D_{3}$ $0.8834$ $0.9179$ $\boldsymbol{1.0000}$ $0.9295$ $0.9740$ $0.8854$ $D_{4}$ $0.9398$ $0.8899$ $0.9004$ $\boldsymbol{1.0000}$ $0.9200$ $0.9864$ $D_{5}$ $0.8737$ $0.9171$ $0.9422$ $0.9183$ $\boldsymbol{1.0000}$ $0.9131$ $D_{6}$ $0.9346$ $0.9087$ $0.8557$ $0.9131$ $0.9089$ $\boldsymbol{1.0000}$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9311$ $0.8848$ $0.9393$ $0.8716$ $0.9418$ $D_{2}$ $0.9312$ $\boldsymbol{1.0000}$ $0.9174$ $0.8838$ $0.9100$ $0.9086$ $D_{3}$ $0.8834$ $0.9179$ $\boldsymbol{1.0000}$ $0.9295$ $0.9740$ $0.8854$ $D_{4}$ $0.9398$ $0.8899$ $0.9004$ $\boldsymbol{1.0000}$ $0.9200$ $0.9864$ $D_{5}$ $0.8737$ $0.9171$ $0.9422$ $0.9183$ $\boldsymbol{1.0000}$ $0.9131$ $D_{6}$ $0.9346$ $0.9087$ $0.8557$ $0.9131$ $0.9089$ $\boldsymbol{1.0000}$
Medium location test with $5\%$ noise.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9758$ $149.9762$ $149.9722$ $149.9819$ $149.9586$ $149.9529$ $\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$ $\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9758$ $149.9762$ $149.9722$ $149.9819$ $149.9586$ $149.9529$ $\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$ $\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$
Medium gesture test with $5\%$ noise.
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.8800$ $0.9109$ $0.9321$ $0.8895$ $0.9280$ $D_{2}$ $0.8738$ $\boldsymbol{1.0000}$ $0.8666$ $0.9520$ $0.9303$ $0.9032$ $D_{3}$ $0.9105$ $0.8656$ $\boldsymbol{1.0000}$ $0.9095$ $0.8817$ $0.9021$ $D_{4}$ $0.9320$ $0.9221$ $0.9802$ $\boldsymbol{1.0000}$ $0.9160$ $0.9287$ $D_{5}$ $0.8863$ $0.9285$ $0.8819$ $0.9162$ $\boldsymbol{1.0000}$ $0.9169$ $D_{6}$ $0.9279$ $0.9030$ $0.9256$ $0.9141$ $0.9171$ $\boldsymbol{1.0000}$
 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.8800$ $0.9109$ $0.9321$ $0.8895$ $0.9280$ $D_{2}$ $0.8738$ $\boldsymbol{1.0000}$ $0.8666$ $0.9520$ $0.9303$ $0.9032$ $D_{3}$ $0.9105$ $0.8656$ $\boldsymbol{1.0000}$ $0.9095$ $0.8817$ $0.9021$ $D_{4}$ $0.9320$ $0.9221$ $0.9802$ $\boldsymbol{1.0000}$ $0.9160$ $0.9287$ $D_{5}$ $0.8863$ $0.9285$ $0.8819$ $0.9162$ $\boldsymbol{1.0000}$ $0.9169$ $D_{6}$ $0.9279$ $0.9030$ $0.9256$ $0.9141$ $0.9171$ $\boldsymbol{1.0000}$
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