Two gesture-computing approaches by using electromagnetic waves

Xianchao Wang; Yukun Guo; Jingzhi Li; Hongyu Liu

doi:10.3934/ipi.2019040

Article Contents

2019, Volume 13, Issue 4: 879-901. Doi: 10.3934/ipi.2019040

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Two gesture-computing approaches by using electromagnetic waves

1.
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
2.
Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China
3.
Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

^* Corresponding authors: Jingzhi Li and Hongyu Liu
^* Corresponding authors: Jingzhi Li and Hongyu Liu

Received: February 2019

Published: May 2019

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Abstract

We are concerned with a novel sensor-based gesture input/ instruction technology which enables human beings to interact with computers conveniently. The human being wears an emitter on the finger or holds a digital pen that generates a time harmonic point charge. The inputs/instructions are performed through moving the finger or the digital pen. The computer recognizes the instruction by determining the motion trajectory of the dynamic point charge from the collected electromagnetic field measurement data. The identification process is mathematically modelled as a dynamic inverse source problem for time-dependent Maxwell's equations. From a practical point of view, the point source should be assumed to move in an unknown inhomogeneous background medium, which models the human body and the surroundings. Moreover, a salient feature is that the electromagnetic radiated data are only collected in a limited aperture. For the inverse problem, we develop, from the respectively deterministic and stochastic viewpoints, a dynamic direct sampling method and a modified particle filter method. Both approaches can effectively recover the motion trajectory. Rigorous theoretical justifications are presented for the mathematical modelling and the proposed recovery methods. Extensive numerical experiments are conducted to illustrate the promising features of the two proposed recognition approaches.

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Mathematics Subject Classification: Primary: 35R30, 35P25; Secondary: 78A46.

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Figure 1. Schematic illustration of the proposed input/ instruction technology using a moving emitter

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Figure 2. The measurement points and human being's motion domain

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Figure 3. Reconstruction of the trajectory "C". (a) exact moving trajectory in the homogeneous medium, (b) exact moving trajectory in the inhomogeneous medium, (c), (d), (e), the electric field in the three components with the receiver at $ (2,2,1) $, where the red solid line denotes the homogeneous medium and the blue dotted line denotes the inhomogeneous medium, (f) the relationship between the $ \alpha $ and the error, (g) reconstruction by the particle filter method in the homogeneous medium, (h) reconstruction by the particle filter method in the inhomogeneous medium

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Figure 4. Reconstruction of the trajectory "3". (a) exact moving trajectory, (b) reconstruction by the direct sampling method, (c) plots of Indicator function $ I(\mathit{\boldsymbol{y}},t) $ in the instant $ t = 4.5 $ s (slices at $ x = 1 $ and $ y = 1.2 $), (d) plots of Indicator function $ I(\mathit{\boldsymbol{y}},t) $ in the instant $ t = 7.5 $ s (slices at $ x = 1 $ and $ y = 1.2 $), (e) reconstruction by the particle filter method with $ N_s = 100 $, (f) point-wise error between the exact and reconstructed trajectory

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Figure 5. Reconstruction of a conical spiral shaped trajectory. (a) exact moving trajectory, (b) reconstruction result by the direct sampling method, (c) reconstruction by the particle filter method with $ N_s = 200 $, (d) point-wise error between the exact and reconstructed trajectory, (e) relationship between the number of particles and relative $ L^2 $ error

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Figure 6. Reconstruct the trajectory of a text. (a) exact moving trajectory, (b) reconstruction by the direct sampling method, (c) reconstruction by the particle filter method with $ N_s = 500 $, (d) point-wise error between the exact and reconstructed trajectory

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Figure 7. Reconstruct the moving trajectory by the particle filter method and snapshots at different instants, where the black points denotes the particles. (a) $ t = 3.0 $ s, (b) $ t = 6.6 $ s, (c) $ t = 9.0 $ s, (d) $ t = 9.1 $ s, (e) $ t = 9.2 $ s, (f) $ t = 12.0 $ s

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Table 1. The relative $ L^2 $ error and the CPU time for reconstructing the trajectory "3" ($ N_h $ and $ N_s $ denote, respectively, the number of sampling points and the number of particle samples)

	Direct sampling method			Particle filter technique
	$ N_h=25^3 $	$ N_h=50^3 $	$ N_h=100^3 $	$ N_s=50 $	$ N_s=100 $	$ N_s=500 $
Error	2.09%	0.79%	0.68%	6.76%	2.62%	1.59%
Time	16 s	115 s	897 s	0.23 s	0.48 s	10.5 s

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