2025-03-19

Optimal Homologous Chain Problem (OHCP)

Given a chain $c\in C_p$, the OHCP seeks to find a chain $x\in C_p$ which minimizes

$$\sum _{i=1}^m{V}_p({\sigma }_i)|x_i|$$

and $x\sim c$. One may generalize to

$$\sum _{i=1}^m{V}_p({\sigma }_i)|x_i|+\lambda \sum _{j=1}^n{V}_{p+1}({\tau }_j)|y_j|$$

which now also minimizes the area of triangles used.

As an optimization problem, we get

$$\begin{array}{rl}\text{minimize}& \sum _{i=1}^m{V}_p({\sigma }_i)|x_i|+\lambda \sum _{j=1}^n{V}_{p+1}({\tau }_j)|y_j|\\ \text{subject to}& x=c+[{\partial }_{p+1}]y\end{array}$$

Now consider the generalized median shape LP for a single input chain $c$:

$$\begin{array}{rl}\text{minimize}& \mu \sum _{i=1}^m{V}_p({\sigma }_i)|t|+\sum _{i=1}^m{V}_p({\sigma }_i)|r_i|+\lambda \sum _{j=1}^n{V}_{p+1}({\tau }_j)|s_j|\\ \text{subject to}& t-c=r+[{\partial }_{p+1}]s\end{array}$$

Thus, our median shape code doubles as an OHCP solver by simply passing in a single input chain. The result is sensitive to $\mu$ and $\lambda$.

Starting Current:

Flat Norm:

Chain Version:

Parameters: $\lambda =0.001$ and $\mu =0.02$

3D Example: Vase

Parameters: $\lambda =0.001$ and $\mu =0.002$