44:10 Y.Yin et al. problems such as reversal and skew of the projected contour,which needs further calibration.It is worth mentioning that lowercase letters are different from capital letters;the shapes of different lowercase letters observed from different viewing angles can be similar,e.g."b"and"q.""d"and "p,"thus the orientation and writing order of a character are both important for lowercase letter recognition.It is essential to calibrate the projected 2D contour in right orientation and normalized size under a uniform view for character recognition. 4.2 Reference Coordinate Plane Detection According to Section 4.1 and Figure 9,the projected 2D contour in the principal plane has a high probability of keeping the shape feature of the in-air contour,while still having the problems such as reversal and skew,i.e.,the orientation of the contour is changed.Thus,we need to calibrate the 2D contour in the principal plane.To achieve this goal,we detect the reference coordinate plane and determine the viewing angle first.Here,the reference coordinate plane is used to indicate the viewing angle and possible orientation of the projected contour in the principal plane.In regard to the user,she/he can perform the gesture towards arbitrary directions;the writing plane may be not parallel to any coordinate plane. 4.2.1 Axis Projection Calculation.We project the xh-axis,yh-axis,zh-axis of human-frame into the principal plane and then compare the length of the projected axis to determine the reference coordinate plane.According to Section 4.1,in the principal plane,the orthonormal basis vectors @@z represent the xp-axis and yp-axis,respectively.With xp-axis and yp-axis,we further cal- culate the zp-axis as @3=@ix @2 to establish the principal-plane coordinate system(principal- frame for short).Here,@1,@2,@3 are described in human-frame.While in the principal-frame,we can represent xp-axis,yp-axis,zp-axis as the unit vector(1,0,0),(0,1,0),(0,0,1),respectively. By comparing @1,@2,@3 in human-frame and xp-axis,yp-axis,zp-axis in principal-frame,we can get the rotation matrix Rhp.which transforms coordinates from human-frame to principal-frame, as shown in Equation(5)and Equation(6): 10 0 0 1 0 =Rhp[@1 @2 03], (5) 00 1 Rhp=[01 @2 @3]-1. (6) With the rotation matrix Rhp,we then calculate the projection of each axis of human-frame in principal plane.For convenience,we use ui,ie[1,3]to represent xhaxis,yh-axis,zh-axis, respectively.For ui,its coordinates in the principal-frame is gi,where qi =Rpui.Then,we get the projection vi of qi in the principal plane with M,i.e.,setting the coordinate value in zp-axis to zero,as shown in Equation(7): [1001 =Mq:=0109i (7) 000 As shown in Figure 10(b),Figure 11(b),Figure 12(b),Figure 13(b),and Figure 14(b),we represent the projected axis of xhaxis,yh-axis,zh-axis (of human-frame)in the principal plane with black, green,and fuchsia dashed line,respectively. 4.2.2 Reference Plane Detection.In Figure 15,we show how to utilize the length of the pro- jected axis to detect the reference coordinate plane.Intuitively,if the projection of axis @has the shortest length in the principal plane,it indicates that the coordinate plane perpendicular to r has the highest probability of being parallel to the principal plane and should be selected as the ACM Transactions on Sensor Networks,Vol 15.No.4.Article 44.Publication date:October 2019.44:10 Y. Yin et al. problems such as reversal and skew of the projected contour, which needs further calibration. It is worth mentioning that lowercase letters are different from capital letters; the shapes of different lowercase letters observed from different viewing angles can be similar, e.g., “b” and “q,” “d” and “p,” thus the orientation and writing order of a character are both important for lowercase letter recognition. It is essential to calibrate the projected 2D contour in right orientation and normalized size under a uniform view for character recognition. 4.2 Reference Coordinate Plane Detection According to Section 4.1 and Figure 9, the projected 2D contour in the principal plane has a high probability of keeping the shape feature of the in-air contour, while still having the problems such as reversal and skew, i.e., the orientation of the contour is changed. Thus, we need to calibrate the 2D contour in the principal plane. To achieve this goal, we detect the reference coordinate plane and determine the viewing angle first. Here, the reference coordinate plane is used to indicate the viewing angle and possible orientation of the projected contour in the principal plane. In regard to the user, she/he can perform the gesture towards arbitrary directions; the writing plane may be not parallel to any coordinate plane. 4.2.1 Axis Projection Calculation. We project the xh-axis, yh-axis, zh-axis of human-frame into the principal plane and then compare the length of the projected axis to determine the reference coordinate plane. According to Section 4.1, in the principal plane, the orthonormal basis vectors ω1,ω2 represent the xp -axis and yp -axis, respectively. With xp -axis and yp -axis, we further cal￾culate the zp -axis as ω3 = ω1 × ω2 to establish the principal-plane coordinate system (principal￾frame for short). Here, ω1, ω2, ω3 are described in human-frame. While in the principal-frame, we can represent xp -axis, yp -axis, zp -axis as the unit vector (1, 0, 0) T , (0, 1, 0) T , (0, 0, 1) T , respectively. By comparing ω1, ω2, ω3 in human-frame and xp -axis, yp -axis, zp -axis in principal-frame, we can get the rotation matrix Rhp , which transforms coordinates from human-frame to principal-frame, as shown in Equation (5) and Equation (6): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 100 010 001 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = Rhp [ω1 ω2 ω3 ], (5) Rhp = [ω1 ω2 ω3 ] −1 . (6) With the rotation matrix Rhp , we then calculate the projection of each axis of human-frame in principal plane. For convenience, we use ui , i ∈ [1, 3] to represent xh-axis, yh-axis, zh-axis, respectively. For ui , its coordinates in the principal-frame is qi , where qi = Rhpui . Then, we get the projection vi of qi in the principal plane with M, i.e., setting the coordinate value in zp -axis to zero, as shown in Equation (7): vi = Mqi = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 100 010 000 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ qi . (7) As shown in Figure 10(b), Figure 11(b), Figure 12(b), Figure 13(b), and Figure 14(b), we represent the projected axis of xh-axis, yh-axis, zh-axis (of human-frame) in the principal plane with black, green, and fuchsia dashed line, respectively. 4.2.2 Reference Plane Detection. In Figure 15, we show how to utilize the length of the pro￾jected axis to detect the reference coordinate plane. Intuitively, if the projection of axis ωi has the shortest length in the principal plane, it indicates that the coordinate plane perpendicular to ωi has the highest probability of being parallel to the principal plane and should be selected as the ACM Transactions on Sensor Networks, Vol. 15, No. 4, Article 44. Publication date: October 2019