15. Marching cubes

Marching cubes is an algorithm to extract a 2D surface mesh from a 3D volume. This can be conceptualized as a 3D generalization of isolines on topographical or weather maps.

It works by iterating across the volume, looking for regions which cross the level of interest. If such regions are found, triangulations are generated and added to an output mesh. The final result is a set of vertices and a set of triangular faces.

The algorithm requires a data volume and an isosurface value. For example, in CT imaging Hounsfield units of +700 to +3000 represent bone. So, one potential input would be a reconstructed CT set of data and the value +700, to extract a mesh for regions of bone or bone-like density.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d.art3d import Poly3DCollection

from skimage import measure
from skimage.draw import ellipsoid

15.1. Generate a level set about zero of two identical ellipsoids in 3D

ellip_base = ellipsoid(6, 10, 16, levelset=True)
ellip_double = np.concatenate((ellip_base[:-1, ...],
                               ellip_base[2:, ...]), axis=0)

15.2. Use marching cubes to obtain the surface mesh of these ellipsoids

verts, faces, normals, values = measure.marching_cubes_lewiner(ellip_double, 0)
c:\program files\python37\lib\site-packages\ipykernel_launcher.py:1: FutureWarning: marching_cubes_lewiner is deprecated in favor of marching_cubes. marching_cubes_lewiner will be removed in version 0.19
  """Entry point for launching an IPython kernel.

15.3. Display resulting triangular mesh using Matplotlib

fig = plt.figure(figsize=(10, 10))
ax = fig.add_subplot(111, projection='3d')

# Fancy indexing: `verts[faces]` to generate a collection of triangles
mesh = Poly3DCollection(verts[faces])

ax.set_xlabel("x-axis: a = 6 per ellipsoid", fontsize=16)
ax.set_ylabel("y-axis: b = 10", fontsize=16)
ax.set_zlabel("z-axis: c = 16", fontsize=16)

ax.set_xlim(0, 24)  # a = 6 (times two for 2nd ellipsoid)
ax.set_ylim(0, 20)  # b = 10
ax.set_zlim(0, 32)  # c = 16