$$\begin{array}{rcl} h(\theta) &=& \int_a^b\| B_\theta(s_t) - \tilde{f}(t)\|^2 \sqrt{\tilde{f}_x'(t)^2 +\tilde{f}_y'(t)^2}\,dt\\ &\approx& \sum_{i=1}^N \| B_\theta(s_{t_i}) - \tilde{f}(t_i)\|^2 \sqrt{\tilde{f}_x'(t_i)^2 +\tilde{f}_y'(t_i)^2}\,w_i. \end{array} $$