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# On a gesture-computing technique using electromagnetic waves

• 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.

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

 Citation: • • Figure 1.  Dictionary

Table 1.  PEC location test.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9838$ $149.9611$ $149.9742$ $149.9632$ $150.0075$ $149.9853$ $\mathring{z}_{0}^{2}$ $0.0400$ $0.0280$ $-0.0096$ $0.0442$ $-0.0440$ $0.0321$ $\mathring{z}_{0}^{3}$ $-0.0131$ $-0.0110$ $-0.0404$ $-0.0456$ $-0.0265$ $-0.0485$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0451$ $0.0492$ $0.0489$ $0.0734$ $0.0519$ $0.0600$

Table 2.  PEC gesture test.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9234$ $0.9291$ $0.8660$ $0.8249$ $0.9453$ $D_{2}$ $0.9233$ $\boldsymbol{1.0000}$ $0.9245$ $0.9502$ $0.9109$ $0.9146$ $D_{3}$ $0.9290$ $0.9242$ $\boldsymbol{1.0000}$ $0.9040$ $0.9494$ $0.9851$ $D_{4}$ $0.8660$ $0.9510$ $0.9040$ $\boldsymbol{1.0000}$ $0.9301$ $0.9742$ $D_{5}$ $0.8249$ $0.9107$ $0.9492$ $0.9300$ $\boldsymbol{1.0000}$ $0.9159$ $D_{6}$ $0.9451$ $0.9147$ $0.9849$ $0.9732$ $0.9149$ $\boldsymbol{1.0000}$

Table 3.  PEC location test with $5\%$ noise.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0278$ $150.0547$ $150.0965$ $150.0158$ $150.0971$ $150.0958$ $\mathring{z}_{0}^{2}$ $0.0679$ $0.0743$ $0.0655$ $0.0706$ $0.0277$ $0.0097$ $\mathring{z}_{0}^{3}$ $0.0758$ $0.0392$ $0.0171$ $0.0032$ $0.0046$ $0.0823$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.1055$ $0.1003$ $0.1173$ $0.1196$ $0.0322$ $0.1277$

Table 4.  PEC gesture test with $5\%$ noise.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $D_{1}$ $\boldsymbol{1.0000}$ $0.9453$ $0.9632$ $0.8213$ $0.9182$ $0.9649$ $D_{2}$ $0.9431$ $\boldsymbol{1.0000}$ $0.9374$ $0.8864$ $0.9205$ $0.9061$ $D_{3}$ $0.9651$ $0.9255$ $\boldsymbol{1.0000}$ $0.9213$ $0.9070$ $0.9124$ $D_{4}$ $0.8268$ $0.8811$ $0.9219$ $\boldsymbol{1.0000}$ $0.9621$ $0.9450$ $D_{5}$ $0.9152$ $0.9213$ $0.9071$ $0.9491$ $\boldsymbol{1.0000}$ $0.9378$ $D_{6}$ $0.9649$ $0.9066$ $0.9123$ $0.9459$ $0.9367$ $\boldsymbol{1.0000}$

Table 5.  Medium location test.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $150.0417$ $150.0254$ $149.9576$ $150.0279$ $150.0069$ $149.9837$ $\mathring{z}_{0}^{2}$ $-0.0214$ $-0.0120$ $-0.0446$ $0.0434$ $-0.0031$ $-0.0338$ $\mathring{z}_{0}^{3}$ $0.0257$ $0.0068$ $0.0031$ $-0.0370$ $-0.0488$ $0.0294$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0535$ $0.0289$ $0.0616$ $0.0635$ $0.0494$ $0.0477$

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 7.  Medium location test with $5\%$ noise.

 $D_{1}$ $D_{2}$ $D_{3}$ $D_{4}$ $D_{5}$ $D_{6}$ $\mathring{z}_{0}^{1}$ $149.9758$ $149.9762$ $149.9722$ $149.9819$ $149.9586$ $149.9529$ $\mathring{z}_{0}^{2}$ $-0.0091$ $0.0103$ $-0.0383$ $-0.0076$ $-0.0238$ $0.0429$ $\mathring{z}_{0}^{3}$ $0.0095$ $0.0211$ $-0.0203$ $0.0008$ $0.0301$ $0.0230$ $\left|\mathring{z}_{0}-z_{0}\right|$ $0.0275$ $0.0334$ $0.0515$ $0.0197$ $0.0565$ $0.0677$

Table 8.  Medium gesture test with $5\%$ noise.

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