Multiply-Connected Quasiconformal Map

This code computes a landmark-matching quasi-conformal map with arclength parameterized boundary condition from a multiply-connected surface to another using the methods in [1] and [2] with modifications.

Any comments and suggestions are welcome.

If you use this code in your own work, please cite the following papers:

[1] G. P. T. Choi, "Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory." Journal of Scientific Computing, 87(3), 70, 2021.

[2] G. P. T. Choi and L. Mahadevan, "Planar Morphometrics Using Teichmuller Maps." Proceedings of the Royal Society A, 474(2217), 20170905, 2018.

Copyright (c) 2023, Gary Pui-Tung Choi

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Usage:

• [map,map_planar] = multiply_connected_quasiconformal_map(v1,f1,bdy1_all,blm1_all,ilm1,v2,f2,bdy2_all,blm2_all,ilm2)

Input:

• v1: nv1 x 2 or 3 vertex coordinates of surface 1
• f1: nf1 x 3 triangulations of surface 1
• bdy1_all: a 1 x k cell array with each cell containing vertex indices of a boundary curve of surface 1
• blm1_all: a 1 x k cell array with each cell containing 1 x b_k vertex indices of boundary landmarks on the k-th boundary curve of surface 1
• ilm1: 1 x i vertex indices of interior landmarks of surface 1
• v2: nv2 x 2 or 3 vertex coordinates of surface 2
• f2: nf2 x 3 triangulations of surface 2
• bdy2_all: a 1 x k cell array with each cell containing vertex indices of a boundary curve of surface 2
• blm2_all: a 1 x k cell array with each cell containing 1 x b_k vertex indices of boundary landmarks on the k-th boundary curve of surface 2
• ilm2: 1 x i vertex indices of interior landmarks of surface 2

Output:

• map: nv1 x 2 or 3 vertex coordinates of the resulting quasiconformal map
• map_planar: nv1 x 2 vertex coordinates of the quasiconformal map on the plane

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Remarks:

• The meshboundaries function can be used for finding the boundaries of the two input surfaces. However, please make sure that the boundaries and the landmarks are in correct order.

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Dependencies:

• Poly-Annulus Conformal Map
• find_inner_circle.m
• Kabsch.m
• interparc.m