Suppose you have a sequence of finite sets of 2D points, $\mathbf{S} = S_1, S_2, ..., S_n$, and you have a sequence of points which define a polygonal curve, $C = p_1, p_2, ..., p_n$ . The problem I am trying to solve is to match each $p_i$ with a $p'_i \in S_i$, so that you have a new curve, $C' = p'_1, p'_2, ..., p'_n$, that optimally approximates $C$ under a particular distance measure $d$.
I know that this depends strongly on $d$, but I am hoping to find out first if this is a known/formalized problem in any related context and if so what the problem is referred to as.
After more digging, I have found what I was looking for. A similar problem, is to match a curve with a subset of points in $S$. This problem and several variants have been called the set-chain matching problems [1], or in other places Curve/Point-Set Matching problems (CPSM) [2]. Or in another case, matching a curve with edges in a Graph, has been termed the discrete map matching problem [3].
[1] Wylie, T., & Zhu, B. (2014). Following a curve with the discrete Fréchet distance. Theoretical Computer Science, 556, 34-44.
[2] Accisano, P., & Üngör, A. (2012). Hardness Results on Curve/Point Set Matching with Fr\'echet Distance. arXiv preprint arXiv:1211.2030.
[3]Chen, D., Driemel, A., Guibas, L. J., Nguyen, A., & Wenk, C. (2011, January). Approximate map matching with respect to the Fréchet distance. In 2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX) (pp. 75-83). Society for Industrial and Applied Mathematics.