问题
Is there an "alpha shape" function in 3 dimensions in python, other than the CGAL python bindings?
Alternatively, is there a way to extend the example below into 3D?
2D example: draw a smooth polygon around data points in a scatter plot, in matplotlib
I'm currently calculating volume using this ConvexHull example, but for my purposes the volumes are inflated due to the "convex" constraint.
Thanks,
回答1:
I wrote some code for finding alpha shape surface. I hope this helps.
from scipy.spatial import Delaunay
import numpy as np
from collections import defaultdict
def alpha_shape_3D(pos, alpha):
"""
Compute the alpha shape (concave hull) of a set of 3D points.
Parameters:
pos - np.array of shape (n,3) points.
alpha - alpha value.
return
outer surface vertex indices, edge indices, and triangle indices
"""
tetra = Delaunay(pos)
# Find radius of the circumsphere.
# By definition, radius of the sphere fitting inside the tetrahedral needs
# to be smaller than alpha value
# http://mathworld.wolfram.com/Circumsphere.html
tetrapos = np.take(pos,tetra.vertices,axis=0)
normsq = np.sum(tetrapos**2,axis=2)[:,:,None]
ones = np.ones((tetrapos.shape[0],tetrapos.shape[1],1))
a = np.linalg.det(np.concatenate((tetrapos,ones),axis=2))
Dx = np.linalg.det(np.concatenate((normsq,tetrapos[:,:,[1,2]],ones),axis=2))
Dy = -np.linalg.det(np.concatenate((normsq,tetrapos[:,:,[0,2]],ones),axis=2))
Dz = np.linalg.det(np.concatenate((normsq,tetrapos[:,:,[0,1]],ones),axis=2))
c = np.linalg.det(np.concatenate((normsq,tetrapos),axis=2))
r = np.sqrt(Dx**2+Dy**2+Dz**2-4*a*c)/(2*np.abs(a))
# Find tetrahedrals
tetras = tetra.vertices[r<alpha,:]
# triangles
TriComb = np.array([(0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3)])
Triangles = tetras[:,TriComb].reshape(-1,3)
Triangles = np.sort(Triangles,axis=1)
# Remove triangles that occurs twice, because they are within shapes
TrianglesDict = defaultdict(int)
for tri in Triangles:TrianglesDict[tuple(tri)] += 1
Triangles=np.array([tri for tri in TrianglesDict if TrianglesDict[tri] ==1])
#edges
EdgeComb=np.array([(0, 1), (0, 2), (1, 2)])
Edges=Triangles[:,EdgeComb].reshape(-1,2)
Edges=np.sort(Edges,axis=1)
Edges=np.unique(Edges,axis=0)
Vertices = np.unique(Edges)
return Vertices,Edges,Triangles
回答2:
You are looking for a "concave hull". The marching cube algorithm can be used to find such a hull. You can find a full example here.
Limitations: This approach works well if your data comes from a volumetric dataset or if you have a cloud of points that can easily be converted into a volumetric data set (voxel-like). This can be done relatively easily with a dense set of points using, for example, a spatial indexer like the scipy cKDTree, but you might end up scratching your head a bit to get a good result if you have a sparse cloud of points.
来源:https://stackoverflow.com/questions/26303878/alpha-shapes-in-3d