Spectral surface skeletons demonstrator for branched tubular genus-zero shapes

The first k eigenvectors u_i (with corresponding decreasing ordered eigenvalues<0) of the mesh-Laplacian L are used to determine a shrinked shape of a closed surface X made of tubular-/ bulb-like sub-structures (2D surface in 3D, genus 0). Approximation of the original surface coordinates X by least-squares may be taken to initialize further skeleton construction, e.g., by means of heuristic algorithms that reduce the mesh to one-dimensional connected filaments (curve skeleton), downstream. Leaving out >k eigenvectors means shrinking their contribution to the surface to zero. In this way, the solution to mesh-Laplacian-flow X(t) = a_0 + sum_i { a_i u_i exp( lambda_i t)} (lambda_i<0) is used for modeling strong volume reduction in tubular sub-structures, taking the fastest modes in terms of high negative eigenvalues as already zero (i.e., assuming their degrees of freedom are already relaxed at t=0). The a_i are the coefficients determined by least squares of the initial surface X, and a_0 is the intercept. It is observed from the reference skeleton shape (k=15) that bulb-like ends and tubes are collapsed stronger to quasi-one-dimensional filaments, than the central connecting/branching regions. The parameter k may be selected from the reverse L-shaped curve of volume v(k) of the skeleton or by visual inspection of the positioning of the skeleton filaments in the middle of tubes. It may be interesting to explore the reverse of the procedure, i.e.. to pump up the shape after the initial fit by scaling up some specific eigenvectors. Furthermore, it may be interesting to characterize components of eigenvectors that are dominating the radial contractions towards skeltal filaments. We hypothesize, that longer tubular structures are more radially contracted than shorter ones due to a yet unspecified low-eigenvalue subspace of eigenvectors u_i, i>k. This may be caused by the triangular mesh structure of tubes, that may be represented as larger blocks with only a few off-block entries (pointing to vertices of the interface to the branching-elements) in the Laplacian (i.e. after renumbering). Therefore, after renumbering, block-structures of L can be used to identifiy and modify tubular sub-structures. Moreover, graph-partitioning (e.g. by George Karypis METIS) would probably do a good job to split the structure into sub-structures of minimal number of interface edges. This could help to identify modular sub-structures of - maybe - piecewise-low-variance mean curvature (p-LVMC) as building blocks of a gluing procedure for the creation of new approximate minimum energy shapes. The R-code and two files for test-surfaces (Wavefront .obj-format) are licensed under CC-BY-4.0. Users should fully cite this work in their references, incl. the ZENODO DOI. The two test-surfaces were computed with the help of scripts for Surface Evolver (K. Brakke, The Surface Evolver, Experimental Mathematics vol. 1 no. 2 (1992), 141-165.), derived from:Frickenhaus, S. (2024). Exploration of extreme vesicle shapes and their modular structure. Zenodo. https://doi.org/10.5281/zenodo.11199344

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