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Table 3 Mathematical notation

From: A low-cost virtual coach for 2D video-based compensation assessment of upper extremity rehabilitation exercises

Equation Description
\(p_j^t = [ x_j^t y_j^t ]'\) \([ x_j^t y_j^t ]'\) denotes the transposed vector of 2D coordinates in the image of a body joint j from a set of joints J (Fig. 1); t denotes the frame number
\(P^t(j_1,j_2) = p^t_{j_2} - p^t_{j_1} = [ x^t_{j_2} - x^t_{j_1} y^t_{j_2} - y^t_{j_1} ]'\) Vector directed from joint \(j_1\) to joint \(j_2\)
\(\Vert P^t(j_1,j_2)\Vert\) \(d^t(j_1,j_2) = \Vert p_{j_1}^t - p_{j_2}^t \Vert\) \(\Vert P^t(j_1,j_2)\Vert\) is the euclidean norm of vector \(P^t(j_1,j_2)\) and, alternatively, \(d^t(j_1,j_2)\) is the euclidean distance between two selected joints, \(j_1\) and \(j_2\)
\(\Delta x^t(j_1,j_2) = x^t_{j_1} - x^t_{j_2}\) \(\Delta y^t(j_1,j_2) = y^t_{j_1} - y^t_{j_2}\) Displacement between two selected joints, \(j_1\) and \(j_2\), in the X (\(\Delta x\)) and Y (\(\Delta y\)) axis
\(a^t(j_1,j_2,j_3) = \arccos \bigg ( \frac{P^t(j_2,j_1) \cdot P^t(j_2,j_3)}{\Vert P^t(j_2,j_1) \Vert \cdot \Vert P^t(j_2,j_3) \Vert } \bigg )\) Angle between two vectors, \(P^t(j_2,j_1)\) and \(P^t(j_2,j_3)\), defined by two points, \(j_2\) to \(j_1\) and \(j_2\) to \(j_3\)