## 2 Methods
### Proposed Method  
In this study, we propose a method to automatically extract Doppler envelopes for different body parts from the Doppler map using contour analysis. Similar to how a contour connects points of equal elevation, the Doppler map is converted into a contour map, where points with the same signal intensity are connected. This allows the exploration of Doppler envelopes corresponding to various body parts. First, for each time frame in the Doppler map, the velocity value with the highest signal strength is extracted to construct a baseline envelope, as defined in (10). This baseline envelope serves as the most distinct and representative envelope in gait analysis. It is used as a reference when selecting envelopes corresponding to major body parts from the various envelopes extracted using the contour-based method. \[A(t)=\max P(t,\cdot) \tag{12}\]  
Figure 7 shows the baseline envelope \(A(t)\) extracted from the Doppler map. A total of \(n\) contour levels are extracted from the Doppler map. Higher levels correspond to stronger reflected signals, making it more likely that they correspond to specific moving body parts (Foot, Leg, Spine). Figure 8a shows the Doppler map for normal walking, and Figure 8b illustrates the 7 contours extracted from the Doppler map. Figure 6: Five Doppler envelopes extracted using the conventional method. Figure 7: Baseline envelope \(A(t)\). The various contours extracted from the Doppler map may result in multiple velocity values at the same time frame, so a process of connecting them into a single line is necessary. This can be achieved by calculating the average of the velocity values at each level to generate a representative velocity \(M_{i}(t)\), as shown in (13). \[\overline{M}(t)=\text{mean}_{\text{o}}P_{i}(t,\cdot) \tag{13}\]  
The index \(i\) refers to the level. For each \(M_{i}(t)\), the gaps are filled using a moving average to create \(M_{i}(t)\). Figure 9 shows the results after applying the interpolation process. \[M_{i}(t)=\frac{1}{N}\sum_{\tau=t-5}^{t+4}\overline{M}(\tau) \tag{14}\]  
The correlation between the baseline envelope in Figure 7 and the contour-based envelope \(M_{i}(t)\) obtained from Figure 9 is calculated using (14), quantifying the similarity with the baseline envelope. For similarity calculation, we use the Pearson correlation, which is simple and widely used [20; 21]. Table 1 shows the results of the correlation calculation. The top three levels with the highest correlation scores are selected as candidate envelopes for Foot, Leg, and Spine. The three envelopes with the highest \(MAC\) values are considered to have significant periodicity, and the corresponding level power values are chosen as parameters to extract the Doppler envelope. In Table 1, the level power values of 3.1927, 6.3853, and 9.578 are selected. \[\text{MAC}_{i}=\frac{|\sum M_{i}(t)\cdot A(t)|}{\sigma_{M_{i}}\cdot\sigma_{A}} \tag{15}\] \begin{table}
\begin{tabular}{l c c c c c c} \hline Level & 3.1927 & 6.3853 & 9.578 & 12.770 & 15.963 & 19.155 \\ \hline MAC & 0.8380 & 0.9671 & 0.9709 & 0.4905 & 0.6309 & 0.6034 \\ \end{tabular}
\end{table} 
Table 1: MAC values between the baseline envelope and the contour-based envelopes. The first row indicates the power level, and the second row shows the corresponding MAC value. Figure 8: Doppler map for normal walking, \(P(v,t)\), and the extracted contours (Contour plot): (a) Doppler map; (b) extracted contours. After profiling the relationship between the level power values and various speeds, as shown in Figure 9, the representative speed values corresponding to the three representative level powers obtained from Table 1 are selected as the representative speed values. In Figure 10a, the 8 representative speed values are selected. Among them, the speed values corresponding to Foot, Leg, and Spine are determined as follows. 1. Foot: Generally, during walking, the foot moves the fastest, so the highest speed is selected. \[\text{Foot}(t)=\text{max}(v_{k})\] (16)
2. Leg: The speed closest to the baseline envelope is selected.