This is a numerical implementation of the work in
@article{kurtek2018simplifying,
title={Simplifying transforms for general elastic metrics on the space of plane curves},
author={Kurtek, Sebastian and Needham, Tom},
journal={arXiv preprint arXiv:1803.10894},
year={2018}
}
The transforms are normalized to have a=1, so that the 1-parameter family of metrics considered are the elastic metrics g^{1,b}. Due to non-injectivity and numerical issues outlined in the paper, there are two regimes where the implementation has different performance:
For “large” b values, the transform works in a straightforward and fast way. “Large” depends on how complicated the curves are. Usually b>0.4 works, lower values work for very simple curves, and b>0.5 always works for theoretical reasons. This case is handled by the function main_large_b.
For “small” b values, we use the transform plus a path straightening algorithm to fix the noninjectivity issues. The algorithm is simplistic, but slow in its current form. This case is handled by the function main_small_b.
To run the program, first compile DynamicProgrammingQ.c as a mex file. Then use main_large_b(c1,c2,b) or main_small_b(c1,c2,b). Here, c1 and c2 are discrete parametric curves and b is a positive number (chosen by the user, giving different results).
The code in example provides a basic use case. Example curves are provided in the comp_curves.mat file.
Please feel free to contact me with any questions or to report any issues.