AICS Spherical Hamonics Parametrization

Spherical harmonics parametrization for starlike shapes. Segmented images are converted into triangular meshes using standard marching cubes algorithm from VTK (vtk.org) and centered at the origin. Coordinates of mesh points are converted from cartesian to geographic system (latitude, longitude and altitude). Altitude coordinates are then interpolated over a (lat,lon) grid where each cell has a resolution of π/128.

We used the Python package pyshtools (github.com/SHTOOLS) to expand the equally spaced grid into spherical harmonics using Driscoll and Healy’s sampling theorem [1].

[1] Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

import numpy as np
from aicsshparam import shparam, shtools

# Create a binary test image of a cube with size 32x32x32
img = np.ones((32,32,32), dtype=np.uint8)

# Calculate the spherical harmonics expansion up to order lmax = 2
(coeffs, grid_rec), (image_, mesh, grid, transform) = shparam.get_shcoeffs(image=img, lmax=2)

# Calculate the corresponding reconstruction error
mse = shtools.get_reconstruction_error(grid,grid_rec)

# Print results
print('Coefficients:', coeffs)
print('Error:', mse)

The output coeffs is a dict that constains the sh coefficients values. grid_rec (ndarray) is the reconstructed equally spaced grid. Additional outputs are image_ (ndarray) that is the input image after pre-processing, mesh (ndarray) is a triangular repsentation of image_ centered at origin, grid (ndarray) is the equally spaced grid corresponding to image_, and transform is a tuple of floats that contains the centroid of the shape and the angle applied for alignment. The similar grid_rec and grid are, the better the parametrization is.

To reconstruct the parametrized mesh from grid_rec one could do

# Reconstruct mesh from grid
mesh_rec = aicsshparam.shtools.get_reconstruction_from_grid(grid_rec)

# Save mesh
aicsshparam.shtools.save_polydata(mesh_rec, 'mesh_rec.vtk')

VTK mesh files can be open using Paraview (paraview.org) for further inspection.