Algorithm 2 Procedures of Foot Axis Detection

Input data: contact region Output data: foot axis connecting the centers Variables -$cn$: centroid of the contact region -$i$: index of the boudnary point in the heel side part -$b_i$: i-th boudnary point in the heel side part -$l_i$: distance between $cn$ and $b_i$ -${\widehat{l}}_i$: smoothed distance between $cn$ and $b_i$ -${l^{\prime }}_i$: first derivvatnormal vector of ${\widehat{l}}_i$ -$j$: index of the salient point -$s_j$: j-th salient point Procedures Aligning the contact region: calculate the centroid of contact region $cn$ and rotate the contact region by using PCA Defining the heel side part: 2.1. devide the aligned contact region into two parts. 2.2. for each part, find maximum width perpendicular to the first principal axis. 2.3. define a part with a larger maximum width as a heel side part. Calculating the distances from $cn$ to the boundary points in the heel side part: for all $b_i$, calculate distance $l_i$ from $cn$. Smoothing the distances: for all $l_i$, compute the smoothed distance ${\widehat{l}}_i$ using a 1D Gaussian smoothing filter. Deriving the derivative of the smoothed distances: for all ${\widehat{l}}_i$, compute ${l^{\prime }}_i$ the first derivative of the smoothed distance ${\widehat{l}}_i$ using the central difference method. Detecting the salient points: for all ${\widehat{l}}_i$, find the local maximum of which ${l^{\prime }}_i$ is a zero-crossing point, and define the local maximum as the salient point $s_j$. Detecting the center of heel: for all $s_j$, find the farthest point from $cn$ and define it as the center of heel. Defining the foot axis: detect the line connecting the detected center of heel and the center of second toe selected by a user.