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SurfaceDice.py
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import numpy as np
import scipy.ndimage
# neighbour_code_to_normals is a lookup table.
# For every binary neighbour code
# (2x2x2 neighbourhood = 8 neighbours = 8 bits = 256 codes)
# it contains the surface normals of the triangles (called "surfel" for
# "surface element" in the following). The length of the normal
# vector encodes the surfel area.
#
# created by compute_surface_area_lookup_table.ipynb using the
# marching_cube algorithm, see e.g. https://en.wikipedia.org/wiki/Marching_cubes
#
neighbour_code_to_normals = [
[[0, 0, 0]],
[[0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
[[0.125, -0.125, 0.125]],
[[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25]],
[[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25]],
[[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125]],
[[-0.5, 0.0, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[0.5, 0.0, 0.0], [0.5, 0.0, 0.0]],
[[0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.5, 0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, 0.0, -0.5], [0.25, 0.25, 0.25], [-0.125, -0.125, -0.125]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.125, -0.125, -0.125]],
[[0.125, 0.125, 0.125], [0.375, 0.375, 0.375], [0.0, -0.25, 0.25], [-0.25, 0.0, 0.25]],
[[0.125, -0.125, -0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.375, 0.375, 0.375], [0.0, 0.25, -0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
[[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.125, 0.125, 0.125]],
[[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25]],
[[0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
[[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25]],
[[0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125], [-0.25, -0.0, -0.25], [0.25, 0.0, 0.25]],
[[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[-0.375, -0.375, 0.375], [-0.0, 0.25, 0.25], [0.125, 0.125, -0.125], [-0.25, -0.0, -0.25]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.25, 0.25, -0.25], [0.25, 0.25, -0.25], [0.125, 0.125, -0.125], [-0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], [0.125, -0.125, 0.125]],
[[0.0, 0.25, -0.25], [0.375, -0.375, -0.375], [-0.125, 0.125, 0.125], [0.25, 0.25, 0.0]],
[[-0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0]],
[[0.0, 0.5, 0.0], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[0.0, 0.5, 0.0], [0.125, -0.125, 0.125], [-0.25, 0.25, -0.25]],
[[0.0, 0.5, 0.0], [0.0, -0.5, 0.0]],
[[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.125, -0.125, 0.125]],
[[-0.375, -0.375, -0.375], [-0.25, 0.0, 0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
[[0.125, 0.125, 0.125], [0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
[[0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
[[-0.125, 0.125, 0.125], [0.25, -0.25, 0.0], [-0.25, 0.25, 0.0]],
[[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.375, 0.375, -0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
[[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0]],
[[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0]],
[[-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25]],
[[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.375, -0.375, 0.375], [0.0, -0.25, -0.25], [-0.125, 0.125, -0.125], [0.25, 0.25, 0.0]],
[[-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25]],
[[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[-0.25, 0.25, -0.25], [-0.25, 0.25, -0.25], [-0.125, 0.125, -0.125], [-0.125, 0.125, -0.125]],
[[-0.25, 0.0, -0.25], [0.375, -0.375, -0.375], [0.0, 0.25, -0.25], [-0.125, 0.125, 0.125]],
[[0.5, 0.0, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
[[-0.0, 0.0, 0.5], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
[[-0.25, -0.0, -0.25], [-0.375, 0.375, 0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, 0.125]],
[[0.0, 0.0, -0.5], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[-0.0, 0.0, 0.5], [0.0, 0.0, 0.5]],
[[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], [-0.125, 0.125, 0.125]],
[[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
[[0.125, -0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[0.25, 0.0, 0.25], [-0.375, -0.375, 0.375], [-0.25, 0.25, 0.0], [-0.125, -0.125, 0.125]],
[[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
[[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
[[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
[[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.5, 0.0, -0.0], [0.25, -0.25, -0.25], [0.125, -0.125, -0.125]],
[[-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[0.375, -0.375, 0.375], [0.0, 0.25, 0.25], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
[[0.0, -0.5, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[-0.375, -0.375, 0.375], [0.25, -0.25, 0.0], [0.0, 0.25, 0.25], [-0.125, -0.125, 0.125]],
[[-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[0.125, 0.125, 0.125], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
[[0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
[[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
[[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
[[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.125, 0.125, 0.125]],
[[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
[[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
[[0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
[[0.125, 0.125, 0.125], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
[[-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[-0.375, -0.375, 0.375], [0.25, -0.25, 0.0], [0.0, 0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.0, -0.5, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[0.375, -0.375, 0.375], [0.0, 0.25, 0.25], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
[[-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[0.5, 0.0, -0.0], [0.25, -0.25, -0.25], [0.125, -0.125, -0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
[[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[0.125, 0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.25, 0.0, 0.25], [-0.375, -0.375, 0.375], [-0.25, 0.25, 0.0], [-0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
[[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], [-0.125, 0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
[[-0.0, 0.0, 0.5], [0.0, 0.0, 0.5]],
[[0.0, 0.0, -0.5], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[-0.25, -0.0, -0.25], [-0.375, 0.375, 0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
[[-0.0, 0.0, 0.5], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
[[0.5, 0.0, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[-0.25, 0.0, -0.25], [0.375, -0.375, -0.375], [0.0, 0.25, -0.25], [-0.125, 0.125, 0.125]],
[[-0.25, 0.25, -0.25], [-0.25, 0.25, -0.25], [-0.125, 0.125, -0.125], [-0.125, 0.125, -0.125]],
[[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[0.375, -0.375, 0.375], [0.0, -0.25, -0.25], [-0.125, 0.125, -0.125], [0.25, 0.25, 0.0]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125]],
[[0.0, -0.25, 0.25], [0.0, -0.25, 0.25]],
[[-0.125, -0.125, 0.125], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
[[-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0]],
[[0.125, 0.125, 0.125], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0]],
[[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
[[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
[[-0.375, 0.375, -0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
[[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.25, -0.25, 0.0], [-0.25, 0.25, 0.0]],
[[0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
[[0.125, 0.125, 0.125], [0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
[[-0.375, -0.375, -0.375], [-0.25, 0.0, 0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
[[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.125, -0.125, 0.125]],
[[0.0, 0.5, 0.0], [0.0, -0.5, 0.0]],
[[0.0, 0.5, 0.0], [0.125, -0.125, 0.125], [-0.25, 0.25, -0.25]],
[[0.0, 0.5, 0.0], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
[[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0]],
[[-0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.0, 0.25, -0.25], [0.375, -0.375, -0.375], [-0.125, 0.125, 0.125], [0.25, 0.25, 0.0]],
[[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], [0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.25, 0.25, -0.25], [0.25, 0.25, -0.25], [0.125, 0.125, -0.125], [-0.125, -0.125, 0.125]],
[[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[-0.375, -0.375, 0.375], [-0.0, 0.25, 0.25], [0.125, 0.125, -0.125], [-0.25, -0.0, -0.25]],
[[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], [0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125], [-0.25, -0.0, -0.25], [0.25, 0.0, 0.25]],
[[0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25]],
[[0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
[[0.125, 0.125, 0.125], [0.125, -0.125, 0.125]],
[[0.125, -0.125, 0.125]],
[[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25]],
[[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.125, 0.125, 0.125]],
[[0.375, 0.375, 0.375], [0.0, 0.25, -0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
[[0.125, -0.125, -0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.125, 0.125, 0.125], [0.375, 0.375, 0.375], [0.0, -0.25, 0.25], [-0.25, 0.0, 0.25]],
[[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125], [0.125, -0.125, -0.125]],
[[-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, 0.0, -0.5], [0.25, 0.25, 0.25], [-0.125, -0.125, -0.125]],
[[0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.5, 0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[-0.125, -0.125, 0.125], [0.125, -0.125, -0.125]],
[[0.0, -0.25, -0.25], [0.0, 0.25, 0.25]],
[[0.125, -0.125, -0.125]],
[[0.5, 0.0, 0.0], [0.5, 0.0, 0.0]],
[[-0.5, 0.0, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125]],
[[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125]],
[[0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
[[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125]],
[[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25]],
[[0.125, 0.125, 0.125], [-0.125, 0.125, 0.125]],
[[-0.125, 0.125, 0.125]],
[[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
[[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
[[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25]],
[[0.125, -0.125, 0.125]],
[[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
[[-0.125, -0.125, 0.125]],
[[0.125, 0.125, 0.125]],
[[0, 0, 0]]]
def compute_surface_distances(mask_gt, mask_pred, spacing_mm):
"""Compute closest distances from all surface points to the other surface.
Finds all surface elements "surfels" in the ground truth mask `mask_gt` and
the predicted mask `mask_pred`, computes their area in mm^2 and the distance
to the closest point on the other surface. It returns two sorted lists of
distances together with the corresponding surfel areas. If one of the masks
is empty, the corresponding lists are empty and all distances in the other
list are `inf`
Args:
mask_gt: 3-dim Numpy array of type bool. The ground truth mask.
mask_pred: 3-dim Numpy array of type bool. The predicted mask.
spacing_mm: 3-element list-like structure. Voxel spacing in x0, x1 and x2
direction
Returns:
A dict with
"distances_gt_to_pred": 1-dim numpy array of type float. The distances in mm
from all ground truth surface elements to the predicted surface,
sorted from smallest to largest
"distances_pred_to_gt": 1-dim numpy array of type float. The distances in mm
from all predicted surface elements to the ground truth surface,
sorted from smallest to largest
"surfel_areas_gt": 1-dim numpy array of type float. The area in mm^2 of
the ground truth surface elements in the same order as
distances_gt_to_pred
"surfel_areas_pred": 1-dim numpy array of type float. The area in mm^2 of
the predicted surface elements in the same order as
distances_pred_to_gt
"""
# compute the area for all 256 possible surface elements
# (given a 2x2x2 neighbourhood) according to the spacing_mm
neighbour_code_to_surface_area = np.zeros([256])
for code in range(256):
normals = np.array(neighbour_code_to_normals[code])
sum_area = 0
for normal_idx in range(normals.shape[0]):
# normal vector
n = np.zeros([3])
n[0] = normals[normal_idx, 0] * spacing_mm[1] * spacing_mm[2]
n[1] = normals[normal_idx, 1] * spacing_mm[0] * spacing_mm[2]
n[2] = normals[normal_idx, 2] * spacing_mm[0] * spacing_mm[1]
area = np.linalg.norm(n)
sum_area += area
neighbour_code_to_surface_area[code] = sum_area
# compute the bounding box of the masks to trim
# the volume to the smallest possible processing subvolume
mask_all = mask_gt | mask_pred
bbox_min = np.zeros(3, np.int64)
bbox_max = np.zeros(3, np.int64)
# max projection to the x0-axis
proj_0 = np.max(np.max(mask_all, axis=2), axis=1)
idx_nonzero_0 = np.nonzero(proj_0)[0]
if len(idx_nonzero_0) == 0:
return {"distances_gt_to_pred": np.array([]),
"distances_pred_to_gt": np.array([]),
"surfel_areas_gt": np.array([]),
"surfel_areas_pred": np.array([])}
bbox_min[0] = np.min(idx_nonzero_0)
bbox_max[0] = np.max(idx_nonzero_0)
# max projection to the x1-axis
proj_1 = np.max(np.max(mask_all, axis=2), axis=0)
idx_nonzero_1 = np.nonzero(proj_1)[0]
bbox_min[1] = np.min(idx_nonzero_1)
bbox_max[1] = np.max(idx_nonzero_1)
# max projection to the x2-axis
proj_2 = np.max(np.max(mask_all, axis=1), axis=0)
idx_nonzero_2 = np.nonzero(proj_2)[0]
bbox_min[2] = np.min(idx_nonzero_2)
bbox_max[2] = np.max(idx_nonzero_2)
# print("bounding box min = {}".format(bbox_min))
# print("bounding box max = {}".format(bbox_max))
# crop the processing subvolume.
# we need to zeropad the cropped region with 1 voxel at the lower,
# the right and the back side. This is required to obtain the "full"
# convolution result with the 2x2x2 kernel
cropmask_gt = np.zeros((bbox_max - bbox_min) + 2, np.uint8)
cropmask_pred = np.zeros((bbox_max - bbox_min) + 2, np.uint8)
cropmask_gt[0:-1, 0:-1, 0:-1] = mask_gt[bbox_min[0]:bbox_max[0] + 1,
bbox_min[1]:bbox_max[1] + 1,
bbox_min[2]:bbox_max[2] + 1]
cropmask_pred[0:-1, 0:-1, 0:-1] = mask_pred[bbox_min[0]:bbox_max[0] + 1,
bbox_min[1]:bbox_max[1] + 1,
bbox_min[2]:bbox_max[2] + 1]
# compute the neighbour code (local binary pattern) for each voxel
# the resultsing arrays are spacially shifted by minus half a voxel in each axis.
# i.e. the points are located at the corners of the original voxels
kernel = np.array([[[128, 64],
[32, 16]],
[[8, 4],
[2, 1]]])
neighbour_code_map_gt = scipy.ndimage.filters.correlate(cropmask_gt.astype(np.uint8), kernel, mode="constant",
cval=0)
neighbour_code_map_pred = scipy.ndimage.filters.correlate(cropmask_pred.astype(np.uint8), kernel, mode="constant",
cval=0)
# create masks with the surface voxels
borders_gt = ((neighbour_code_map_gt != 0) & (neighbour_code_map_gt != 255))
borders_pred = ((neighbour_code_map_pred != 0) & (neighbour_code_map_pred != 255))
# compute the distance transform (closest distance of each voxel to the surface voxels)
if borders_gt.any():
distmap_gt = scipy.ndimage.morphology.distance_transform_edt(~borders_gt, sampling=spacing_mm)
else:
distmap_gt = np.Inf * np.ones(borders_gt.shape)
if borders_pred.any():
distmap_pred = scipy.ndimage.morphology.distance_transform_edt(~borders_pred, sampling=spacing_mm)
else:
distmap_pred = np.Inf * np.ones(borders_pred.shape)
# compute the area of each surface element
surface_area_map_gt = neighbour_code_to_surface_area[neighbour_code_map_gt]
surface_area_map_pred = neighbour_code_to_surface_area[neighbour_code_map_pred]
# create a list of all surface elements with distance and area
distances_gt_to_pred = distmap_pred[borders_gt]
distances_pred_to_gt = distmap_gt[borders_pred]
surfel_areas_gt = surface_area_map_gt[borders_gt]
surfel_areas_pred = surface_area_map_pred[borders_pred]
# sort them by distance
if distances_gt_to_pred.shape != (0,):
sorted_surfels_gt = np.array(sorted(zip(distances_gt_to_pred, surfel_areas_gt)))
distances_gt_to_pred = sorted_surfels_gt[:, 0]
surfel_areas_gt = sorted_surfels_gt[:, 1]
if distances_pred_to_gt.shape != (0,):
sorted_surfels_pred = np.array(sorted(zip(distances_pred_to_gt, surfel_areas_pred)))
distances_pred_to_gt = sorted_surfels_pred[:, 0]
surfel_areas_pred = sorted_surfels_pred[:, 1]
return {"distances_gt_to_pred": distances_gt_to_pred,
"distances_pred_to_gt": distances_pred_to_gt,
"surfel_areas_gt": surfel_areas_gt,
"surfel_areas_pred": surfel_areas_pred}
def compute_average_surface_distance(surface_distances):
distances_gt_to_pred = surface_distances["distances_gt_to_pred"]
distances_pred_to_gt = surface_distances["distances_pred_to_gt"]
surfel_areas_gt = surface_distances["surfel_areas_gt"]
surfel_areas_pred = surface_distances["surfel_areas_pred"]
average_distance_gt_to_pred = np.sum(distances_gt_to_pred * surfel_areas_gt) / np.sum(surfel_areas_gt)
average_distance_pred_to_gt = np.sum(distances_pred_to_gt * surfel_areas_pred) / np.sum(surfel_areas_pred)
return (average_distance_gt_to_pred, average_distance_pred_to_gt)
def compute_robust_hausdorff(surface_distances, percent):
distances_gt_to_pred = surface_distances["distances_gt_to_pred"]
distances_pred_to_gt = surface_distances["distances_pred_to_gt"]
surfel_areas_gt = surface_distances["surfel_areas_gt"]
surfel_areas_pred = surface_distances["surfel_areas_pred"]
if len(distances_gt_to_pred) > 0:
surfel_areas_cum_gt = np.cumsum(surfel_areas_gt) / np.sum(surfel_areas_gt)
idx = np.searchsorted(surfel_areas_cum_gt, percent / 100.0)
perc_distance_gt_to_pred = distances_gt_to_pred[min(idx, len(distances_gt_to_pred) - 1)]
else:
perc_distance_gt_to_pred = np.Inf
if len(distances_pred_to_gt) > 0:
surfel_areas_cum_pred = np.cumsum(surfel_areas_pred) / np.sum(surfel_areas_pred)
idx = np.searchsorted(surfel_areas_cum_pred, percent / 100.0)
perc_distance_pred_to_gt = distances_pred_to_gt[min(idx, len(distances_pred_to_gt) - 1)]
else:
perc_distance_pred_to_gt = np.Inf
return max(perc_distance_gt_to_pred, perc_distance_pred_to_gt)
def compute_surface_overlap_at_tolerance(surface_distances, tolerance_mm):
distances_gt_to_pred = surface_distances["distances_gt_to_pred"]
distances_pred_to_gt = surface_distances["distances_pred_to_gt"]
surfel_areas_gt = surface_distances["surfel_areas_gt"]
surfel_areas_pred = surface_distances["surfel_areas_pred"]
rel_overlap_gt = np.sum(surfel_areas_gt[distances_gt_to_pred <= tolerance_mm]) / np.sum(surfel_areas_gt)
rel_overlap_pred = np.sum(surfel_areas_pred[distances_pred_to_gt <= tolerance_mm]) / np.sum(surfel_areas_pred)
return (rel_overlap_gt, rel_overlap_pred)
def compute_surface_dice_at_tolerance(surface_distances, tolerance_mm):
distances_gt_to_pred = surface_distances["distances_gt_to_pred"]
distances_pred_to_gt = surface_distances["distances_pred_to_gt"]
surfel_areas_gt = surface_distances["surfel_areas_gt"]
surfel_areas_pred = surface_distances["surfel_areas_pred"]
overlap_gt = np.sum(surfel_areas_gt[distances_gt_to_pred <= tolerance_mm])
overlap_pred = np.sum(surfel_areas_pred[distances_pred_to_gt <= tolerance_mm])
surface_dice = (overlap_gt + overlap_pred) / (
np.sum(surfel_areas_gt) + np.sum(surfel_areas_pred))
return surface_dice
def compute_dice_coefficient(mask_gt, mask_pred):
"""Compute soerensen-dice coefficient.
compute the soerensen-dice coefficient between the ground truth mask `mask_gt`
and the predicted mask `mask_pred`.
Args:
mask_gt: 3-dim Numpy array of type bool. The ground truth mask.
mask_pred: 3-dim Numpy array of type bool. The predicted mask.
Returns:
the dice coeffcient as float. If both masks are empty, the result is NaN
"""
volume_sum = mask_gt.sum() + mask_pred.sum()
if volume_sum == 0:
return np.NaN
volume_intersect = (mask_gt & mask_pred).sum()
return 2 * volume_intersect / volume_sum
def compute_iou_coefficient(mask_gt, mask_pred):
smooth = 1e-5
intersection = (mask_gt & mask_pred).sum()
union = (mask_gt | mask_pred).sum()
return (intersection + smooth) / (union + smooth)
# %% Some Simple Tests
# single pixels, 2mm away
mask_gt = np.zeros((128, 128, 128), np.uint8)
mask_pred = np.zeros((128, 128, 128), np.uint8)
mask_gt[50, 60, 70] = 1
mask_pred[50, 60, 72] = 1
surface_distances = compute_surface_distances(mask_gt, mask_pred, spacing_mm=(3, 2, 1))
print("surface dice at 1mm: {}".format(compute_surface_dice_at_tolerance(surface_distances, 1)))
print("volumetric dice: {}".format(compute_dice_coefficient(mask_gt, mask_pred)))
# %% two cubes. cube 1 is 100x100x100 mm^3 and cube 2 is 102x100x100 mm^3
mask_gt = np.zeros((100, 100, 100), np.uint8)
mask_pred = np.zeros((100, 100, 100), np.uint8)
spacing_mm = (2, 1, 1)
mask_gt[0:50, :, :] = 1
mask_pred[0:51, :, :] = 1
surface_distances = compute_surface_distances(mask_gt, mask_pred, spacing_mm)
print("surface dice at 1mm: {}".format(compute_surface_dice_at_tolerance(surface_distances, 1)))
print("volumetric dice: {}".format(compute_dice_coefficient(mask_gt, mask_pred)))
print("")
print("expected average_distance_gt_to_pred = 1./6 * 2mm = {}mm".format(1. / 6 * 2))
print("expected volumetric dice: {}".format(2. * 100 * 100 * 100 / (100 * 100 * 100 + 102 * 100 * 100)))
# %% test empty mask in prediction
mask_gt = np.zeros((128, 128, 128), np.uint8)
mask_pred = np.zeros((128, 128, 128), np.uint8)
mask_gt[50, 60, 70] = 1
# mask_pred[50,60,72] = 1
surface_distances = compute_surface_distances(mask_gt, mask_pred, spacing_mm=(3, 2, 1))
print("average surface distance: {} mm".format(compute_average_surface_distance(surface_distances)))
print("hausdorff (100%): {} mm".format(compute_robust_hausdorff(surface_distances, 100)))
print("hausdorff (95%): {} mm".format(compute_robust_hausdorff(surface_distances, 95)))
print("surface overlap at 1mm: {}".format(compute_surface_overlap_at_tolerance(surface_distances, 1)))
print("surface dice at 1mm: {}".format(compute_surface_dice_at_tolerance(surface_distances, 1)))
print("volumetric dice: {}".format(compute_dice_coefficient(mask_gt, mask_pred)))
# %% test empty mask in ground truth
mask_gt = np.zeros((128, 128, 128), np.uint8)
mask_pred = np.zeros((128, 128, 128), np.uint8)
# mask_gt[50,60,70] = 1
mask_pred[50, 60, 72] = 1
surface_distances = compute_surface_distances(mask_gt, mask_pred, spacing_mm=(3, 2, 1))
print("average surface distance: {} mm".format(compute_average_surface_distance(surface_distances)))
print("hausdorff (100%): {} mm".format(compute_robust_hausdorff(surface_distances, 100)))
print("hausdorff (95%): {} mm".format(compute_robust_hausdorff(surface_distances, 95)))
print("surface overlap at 1mm: {}".format(compute_surface_overlap_at_tolerance(surface_distances, 1)))
print("surface dice at 1mm: {}".format(compute_surface_dice_at_tolerance(surface_distances, 1)))
print("volumetric dice: {}".format(compute_dice_coefficient(mask_gt, mask_pred)))
# %% test both masks empty
mask_gt = np.zeros((128, 128, 128), np.uint8)
mask_pred = np.zeros((128, 128, 128), np.uint8)
# mask_gt[50,60,70] = 1
# mask_pred[50,60,72] = 1
surface_distances = compute_surface_distances(mask_gt, mask_pred, spacing_mm=(3, 2, 1))
print("average surface distance: {} mm".format(compute_average_surface_distance(surface_distances)))
print("hausdorff (100%): {} mm".format(compute_robust_hausdorff(surface_distances, 100)))
print("hausdorff (95%): {} mm".format(compute_robust_hausdorff(surface_distances, 95)))
print("surface overlap at 1mm: {}".format(compute_surface_overlap_at_tolerance(surface_distances, 1)))
print("surface dice at 1mm: {}".format(compute_surface_dice_at_tolerance(surface_distances, 1)))
print("volumetric dice: {}".format(compute_dice_coefficient(mask_gt, mask_pred)))