# 2D Discrete Inverse Spectral Problem

### Goal

To calculate an approximate solution to the classic (Laplace-Beltrami) inverse spectral problem for discrete (genus 0) surfaces.

### Content

Included is a suite of MATLAB codes implementing the naive direct gradient descent approach.

test_script.m is the top-level script that generates results.

### People

Project envisioned, advised, and supervised by Prof. Etienne Vouga and Prof. Keenan Crane

Some codes here (on mesh optimization and a demo of spherical harmonics) are not mine.

### Testing Procedures

1. use conformalized mean curvature flow (cMCF) to get a spherical mesh with a target set of conformal factors from a target mesh

2. use BFGS descent search for some conformal factors that achieve a spectrum similar to the one desired

3. embed the metric to a resulting mesh from the sphere by optimizing edge lengths obtained from the factors

4. compare with the target mesh, target spectrum, and cMCF conformal factors

### Results

(cheating) optimize for cMCF spectrum instead

Tests with Spot the cow

smoothed cow without bi-laplacian regularization

original spot with regularization

Tests with bunny

recursive run

finer mesh recursive run

Before and After (with minor smoothing)

even finer mesh recursive run

“Mesh-free” Spherical Harmonic Basis Solution

From now on we have number of eigenvalues used = number of free SH basis function coefficient = n, LB operator expanded in 961 SH basis functions

PL spectrum as target: n = 36 n = 49 n = 64

SH spectrum as target (cheating): n = 49

Results were adjusted up to SO(3) to mod out the rigid rotation ambiguity

### Ways to go

1. (ongoing) without prior knowledge of the target mesh, we will have to start from a uniform (coarse) spherical mesh and develop a suitable adaptive refinement scheme

2. (banging my head) why does high frequency data matter in the FEM/hat function basis? the current way involves optimizing for them and then penalize for its noisyness via regularization, which seems very silly…

3. in practice the inverse problem would not be about the Laplace-Beltrami operator (need to consider bending energy of thin shell etc.)

4. in practice higher frequencies will most definitly be prohibitively noisy

5. can we guess the topology beforehand? would higher genus surfaces work in similar fashion despite planarity/hyperbolicity? (e.g. there are known non-trivial isospectral hyperbolic (g>5) surfaces…) (Yes. Reuter, Wolter, Peinecke 2006 ~ first 500 eigenvalues)