Resampling

Algorithm : Upsampling
Input : $\left\{ v\lbrack k\rbrack = \left( x\lbrack k\rbrack,y\lbrack k\rbrack,t\lbrack k\rbrack \right),0 \leq k \leq n \right\}$ and a target rate sampling_min_output_rate, Interpolate time and position between each $k$ by linearly adding interpolated values
This is done by adding linearly interpolated values so that the output stream is $$\left\{ v\lbrack 0\rbrack,\underset{\text{ interpolated}}{\underbrace{u_{0}\lbrack 1\rbrack,\ldots,u_{0}\left\lbrack n_{0} - 1 \right\rbrack}},v\lbrack 1\rbrack,\underset{\text{ interpolated}}{\underbrace{u_{1}\lbrack 1\rbrack,\ldots,u_{1}\left\lbrack n_{1} - 1 \right\rbrack}},v\lbrack 2\rbrack,\ldots \right\}$$ Each $n_{i}$ is the minimum integer such that $$\begin{aligned} & \frac{t\lbrack i\rbrack - t\lbrack i - 1\rbrack}{n_{i}} < \Delta_{\text{target}} \\ \Leftrightarrow & n_{i} = \left\lceil \frac{t\lbrack i + 1\rbrack - t\lbrack i\rbrack}{\Delta_{\text{target}}} \right\rceil \end{aligned}$$ and the linear interpolation means $$u_{j}\lbrack k\rbrack = \left( 1 - \frac{k}{n_{j}} \right)v\lbrack j\rbrack + \frac{k}{n_{i}}v\lbrack j + 1\rbrack$$
Output : $\left\{ \left( x\prime\lbrack k\prime\rbrack,y\prime\lbrack k\prime\rbrack,t\prime\lbrack k\prime\rbrack \right),0 \leq k; \leq n\prime \right\}$ the upsampled position and times. This verifies $$\forall k\prime,t\prime\lbrack k\prime\rbrack - t\prime\lbrack k\prime - 1\rbrack < \Delta_{\text{target }} = \frac{1}{\text{sampling_min_output_rate}}$$
Remark : As this is a streaming algorithm, we only calculate this interpolation with respect to the latest stroke position.