Smoothness Penalty

P=1Tt=1T((xt+1xt)(xtxt1))2(xt+1xt)2+(xtxt1)2P=\frac{1}{T} \sum_{t=1}^T \frac{\left(\left(x_{t+1}-x_t\right)-\left(x_t-x_{t-1}\right)\right)^2}{\left(x_{t+1}-x_t\right)^2+\left(x_t-x_{t-1}\right)^2}

From this we get nice properties such as

  • enforced angle smoothness
  • enforced constant velocity
  • partly enforced noise cancellation