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FEAT: Full 3d anisotropic implementation of marching cubes
Also includes surface area calculation algorithm from generated mesh. Convenient output to visualize with `mayavi.mlab`. Efficient Cython implementation.
This commit is contained in:
Josh Warner (Mac)
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#cython: cdivision=True
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#cython: boundscheck=False
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#cython: nonecheck=False
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#cython: wraparound=False
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import numpy as np
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cimport numpy as cnp
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cdef inline double _get_fraction(double from_value, double to_value,
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double level):
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if (to_value == from_value):
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return 0
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return ((level - from_value) / (to_value - from_value))
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def iterate_and_store_3d(cnp.ndarray[double, ndim=3] arr,
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double level, tuple sampling=(1., 1., 1.)):
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"""Iterate across the given array in a marching-cubes fashion,
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looking for volumes with edges that cross 'level'. If such a volume is
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found, appropriate triangulations are added to a growing list of
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triangles to be returned by the function.
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If `sampling` is not provided, vertices are returned in the indexing
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coordinate system (assuming all 3 spatial dimensions sampled equally).
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If `sampling` is provided, vertices will be returned in volume coordinates
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relative to the origin, regularly spaced as specified in each dimension.
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"""
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if arr.shape[0] < 2 or arr.shape[1] < 2 or arr.shape[2] < 2:
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raise ValueError("Input array must be at least 2x2x2.")
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if len(sampling) != 3:
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raise ValueError("`sampling` must be of form (double, double, double)")
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cdef list tri_list = []
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cdef list norm_list = []
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cdef Py_ssize_t n
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cdef bint odd_sampling, plus_z
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plus_z = False
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if ((sampling == (1., 1., 1.)) or
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(sampling == (1., 1., 1)) or
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(sampling == (1., 1, 1.)) or
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(sampling == (1, 1., 1.)) or
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(sampling == (1, 1, 1.)) or
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(sampling == (1., 1, 1)) or
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(sampling == (1, 1., 1)) or
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(sampling == (1, 1, 1))):
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odd_sampling = False
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else:
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odd_sampling = True
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# The plan is to iterate a 2x2x2 cube across the input array. This means
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# the upper-left corner of the cube needs to iterate across a sub-array
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# of size one-less-large in each direction (so we can get away with no
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# bounds checking in Cython). The cube is represented by eight vertices:
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# v1, v2, ..., v8, oriented thus (see Lorensen, Figure 4):
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#
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# v8 ------ v7
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# / | / | y
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# / | / | ^ z
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# v4 ------ v3 | | /
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# | v5 ----|- v6 |/ (note: NOT right handed!)
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# | / | / ----> x
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# |/ | /
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# v1 ------ v2
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#
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# We also maintain the current 2D coordinates for v1, and ensure the array
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# is of type 'double' and is C-contiguous (last index varies fastest).
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# Coords start at (0, 0, 0).
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cdef Py_ssize_t[3] coords
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coords[0] = 0
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coords[1] = 0
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coords[2] = 0
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# Extract doubles from `sampling` for speed
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cdef double[3] sampling2
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sampling2[0] = sampling[0]
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sampling2[1] = sampling[1]
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sampling2[2] = sampling[2]
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# Calculate the number of iterations we'll need
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cdef Py_ssize_t num_cube_steps = ((arr.shape[0] - 1) *
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(arr.shape[1] - 1) *
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(arr.shape[2] - 1))
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cdef unsigned char cube_case = 0
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cdef tuple e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12
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cdef double v1, v2, v3, v4, v5, v6, v7, v8, r0, r1, c0, c1, d0, d1
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cdef Py_ssize_t x0, y0, z0, x1, y1, z1
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e5, e6, e7, e8 = (0, 0, 0), (0, 0, 0), (0, 0, 0), (0, 0, 0)
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for n in range(num_cube_steps):
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# There are 255 unique values for `cube_case`. This algorithm follows
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# the Lorensen paper in vertex and edge labeling, however, it should
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# be noted that Lorensen used a left-handed coordinate system while
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# NumPy uses a proper right handed system. Transforming between these
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# coordinate systems was handled in the definitions of the cube
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# vertices v1, v2, ..., v8.
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#
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# Refer to the paper, figure 4, for cube edge designations e1, ... e12
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# Standard Py_ssize_t coordinates for indexing
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x0, y0, z0 = coords[0], coords[1], coords[2]
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x1, y1, z1 = x0 + 1, y0 + 1, z0 + 1
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if odd_sampling:
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# These doubles are the modified world coordinates; they are only
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# calculated if non-default `sampling` provided.
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r0 = coords[0] * sampling2[0]
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c0 = coords[1] * sampling2[1]
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d0 = coords[2] * sampling2[2]
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r1 = r0 + sampling2[0]
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c1 = c0 + sampling2[1]
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d1 = d0 + sampling2[2]
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else:
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r0, c0, d0, r1, c1, d1 = x0, y0, z0, x1, y1, z1
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# We use a right-handed coordinate system, UNlike the paper, but want
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# to index in agreement - the coordinate adjustment takes place here.
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v1 = arr[x0, y0, z0]
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v2 = arr[x1, y0, z0]
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v3 = arr[x1, y1, z0]
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v4 = arr[x0, y1, z0]
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v5 = arr[x0, y0, z1]
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v6 = arr[x1, y0, z1]
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v7 = arr[x1, y1, z1]
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v8 = arr[x0, y1, z1]
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# Unique triangulation cases
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cube_case = 0
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if (v1 > level): cube_case += 1
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if (v2 > level): cube_case += 2
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if (v3 > level): cube_case += 4
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if (v4 > level): cube_case += 8
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if (v5 > level): cube_case += 16
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if (v6 > level): cube_case += 32
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if (v7 > level): cube_case += 64
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if (v8 > level): cube_case += 128
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if (cube_case != 0 and cube_case != 255):
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# Only do anything if there's a plane intersecting the cube.
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# Cases 0 and 255 are entirely below/above the contour.
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if cube_case > 127:
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cube_case = 255 - cube_case
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# Calculate cube edges, to become triangulation vertices.
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# If we moved in a convenient direction, save 1/3 of the effort by
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# re-assigning prior results.
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if plus_z:
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# Reassign prior calculated edges
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e1 = e5
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e2 = e6
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e3 = e7
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e4 = e8
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else:
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# Calculate edges normally
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if odd_sampling:
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e1 = r0 + _get_fraction(v1, v2, level) * sampling2[0], c0, d0
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e2 = r1, c0 + _get_fraction(v2, v3, level) * sampling2[1], d0
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e3 = r0 + _get_fraction(v4, v3, level) * sampling2[0], c1, d0
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e4 = r0, c0 + _get_fraction(v1, v4, level) * sampling2[1], d0
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else:
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e1 = r0 + _get_fraction(v1, v2, level), c0, d0
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e2 = r1, c0 + _get_fraction(v2, v3, level), d0
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e3 = r0 + _get_fraction(v4, v3, level), c1, d0
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e4 = r0, c0 + _get_fraction(v1, v4, level), d0
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# These must be calculated at each point uunless we implemented a
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# large, growing lookup table for all adjacent values; could save
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# ~30% in terms of runtime at the expense of memory usage and
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# much greater complexity.
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if odd_sampling:
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e5 = r0 + _get_fraction(v5, v6, level) * sampling2[0], c0, d1
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e6 = r1, c0 + _get_fraction(v6, v7, level) * sampling2[1], d1
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e7 = r0 + _get_fraction(v8, v7, level) * sampling2[0], c1, d1
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e8 = r0, c0 + _get_fraction(v5, v8, level) * sampling2[1], d1
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e9 = r0, c0, d0 + _get_fraction(v1, v5, level) * sampling2[2]
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e10 = r1, c0, d0 + _get_fraction(v2, v6, level) * sampling2[2]
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e11 = r0, c1, d0 + _get_fraction(v4, v8, level) * sampling2[2]
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e12 = r1, c1, d0 + _get_fraction(v3, v7, level) * sampling2[2]
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else:
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e5 = r0 + _get_fraction(v5, v6, level), c0, d1
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e6 = r1, c0 + _get_fraction(v6, v7, level), d1
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e7 = r0 + _get_fraction(v8, v7, level), c1, d1
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e8 = r0, c0 + _get_fraction(v5, v8, level), d1
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e9 = r0, c0, d0 + _get_fraction(v1, v5, level)
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e10 = r1, c0, d0 + _get_fraction(v2, v6, level)
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e11 = r0, c1, d0 + _get_fraction(v4, v8, level)
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e12 = r1, c1, d0 + _get_fraction(v3, v7, level)
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# Append appropriate triangles to the growing output `tri_list`
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_append_tris(tri_list, cube_case, e1, e2, e3, e4, e5,
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e6, e7, e8, e9, e10, e11, e12)
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# Advance the coords indices
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if coords[2] < arr.shape[2] - 2:
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coords[2] += 1
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plus_z = True
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elif coords[1] < arr.shape[1] - 2:
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coords[1] += 1
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coords[2] = 0
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plus_z = False
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else:
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coords[0] += 1
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coords[1] = 0
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coords[2] = 0
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plus_z = False
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return tri_list
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def _append_tris(list tri_list, unsigned char case, tuple e1, tuple e2,
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tuple e3, tuple e4, tuple e5, tuple e6, tuple e7, tuple e8,
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tuple e9, tuple e10, tuple e11, tuple e12):
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# Permits recursive use for duplicated planes to conserve code - it's
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# quite long enough as-is.
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if (case == 1):
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# front lower left corner
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tri_list.append([e1, e4, e9])
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elif (case == 2):
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# front lower right corner
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tri_list.append([e10, e2, e1])
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elif (case == 3):
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# front lower plane
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tri_list.append([e2, e4, e9])
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tri_list.append([e2, e9, e10])
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elif (case == 4):
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# front upper right corner
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tri_list.append([e12, e3, e2])
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elif (case == 5):
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# lower left, upper right corners
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_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 6):
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# front right plane
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tri_list.append([e12, e3, e1])
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tri_list.append([e12, e1, e10])
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elif (case == 7):
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# Shelf including v1, v2, v3
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tri_list.append([e3, e4, e12])
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tri_list.append([e4, e9, e12])
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tri_list.append([e12, e9, e10])
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elif (case == 8):
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# front upper left corner
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tri_list.append([e3, e11, e4])
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elif (case == 9):
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# front left plane
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tri_list.append([e3, e11, e9])
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tri_list.append([e3, e9, e1])
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elif (case == 10):
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# upper left, lower right corners
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_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 11):
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# Shelf including v4, v1, v2
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tri_list.append([e3, e11, e2])
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tri_list.append([e11, e10, e2])
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tri_list.append([e11, e9, e10])
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elif (case == 12):
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# front upper plane
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tri_list.append([e11, e4, e12])
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tri_list.append([e2, e4, e12])
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elif (case == 13):
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# Shelf including v1, v4, v3
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tri_list.append([e11, e9, e12])
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tri_list.append([e12, e9, e1])
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tri_list.append([e12, e1, e2])
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elif (case == 14):
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# Shelf including v2, v3, v4
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tri_list.append([e11, e10, e12])
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tri_list.append([e11, e4, e10])
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tri_list.append([e4, e1, e10])
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elif (case == 15):
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# Plane parallel to x-axis through middle
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tri_list.append([e11, e9, e12])
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tri_list.append([e12, e9, e10])
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elif (case == 16):
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# back lower left corner
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tri_list.append([e8, e9, e5])
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elif (case == 17):
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# lower left plane
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tri_list.append([e4, e1, e8])
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tri_list.append([e8, e1, e5])
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elif (case == 18):
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# lower left back, lower right front corners
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 19):
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# Shelf including v1, v2, v5
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tri_list.append([e8, e4, e2])
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tri_list.append([e8, e2, e10])
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tri_list.append([e8, e10, e5])
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elif (case == 20):
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# lower left back, upper right front corners
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 21):
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# lower left plane + upper right front corner, v1, v3, v5
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_append_tris(tri_list, 17, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 22):
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# front right plane + lower left back corner, v2, v3, v5
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 6, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 23):
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# Rotated case 14 in the paper
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tri_list.append([e3, e10, e8])
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tri_list.append([e3, e10, e12])
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tri_list.append([e8, e10, e5])
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tri_list.append([e3, e4, e8])
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elif (case == 24):
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# upper front left, lower back left corners
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 25):
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# Shelf including v1, v4, v5
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tri_list.append([e1, e5, e3])
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tri_list.append([e3, e8, e11])
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tri_list.append([e3, e5, e8])
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elif (case == 26):
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# Three isolated corners
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_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 27):
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# Full corner v1, case 9 in paper: (v1, v2, v4, v5)
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tri_list.append([e11, e3, e2])
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tri_list.append([e11, e2, e10])
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tri_list.append([e10, e11, e8])
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tri_list.append([e8, e5, e10])
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elif (case == 28):
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# upper front plane + corner v5
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 12, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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elif (case == 29):
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# special case of 11 in the paper: (v1, v3, v4, v5)
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tri_list.append([e11, e5, e2])
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tri_list.append([e11, e12, e2])
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tri_list.append([e11, e5, e8])
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tri_list.append([e2, e1, e5])
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elif (case == 30):
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# Shelf (v2, v3, v4) and lower left back corner
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_append_tris(tri_list, 14, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
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_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
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e11, e12)
|
||||
elif (case == 31):
|
||||
# Shelf: (v6, v7, v8) by inversion
|
||||
tri_list.append([e11, e12, e10])
|
||||
tri_list.append([e11, e8, e10])
|
||||
tri_list.append([e8, e10, e5])
|
||||
elif (case == 32):
|
||||
# lower right back corner
|
||||
tri_list.append([e6, e5, e10])
|
||||
elif (case == 33):
|
||||
# lower right back, lower left front corners
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 34):
|
||||
# lower right plane
|
||||
tri_list.append([e1, e2, e5])
|
||||
tri_list.append([e2, e6, e5])
|
||||
elif (case == 35):
|
||||
# Shelf: v1, v2, v6
|
||||
tri_list.append([e4, e2, e6])
|
||||
tri_list.append([e4, e9, e6])
|
||||
tri_list.append([e6, e9, e5])
|
||||
elif (case == 36):
|
||||
# upper right front, lower right back corners
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 37):
|
||||
# lower left front, upper right front, lower right back corners
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 38):
|
||||
# Shelf: v2, v3, v6
|
||||
tri_list.append([e3, e1, e5])
|
||||
tri_list.append([e3, e5, e12])
|
||||
tri_list.append([e12, e5, e6])
|
||||
elif (case == 39):
|
||||
# Full corner v2: (v1, v2, v3, v6)
|
||||
tri_list.append([e3, e4, e5])
|
||||
tri_list.append([e4, e9, e5])
|
||||
tri_list.append([e3, e5, e6])
|
||||
tri_list.append([e3, e12, e6])
|
||||
elif (case == 40):
|
||||
# upper left front, lower right back corners
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 41):
|
||||
# front left plane, lower right back corner
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 9, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 42):
|
||||
# lower right plane, upper front left corner
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 34, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 43):
|
||||
# Rotated case 11 in paper
|
||||
tri_list.append([e11, e3, e9])
|
||||
tri_list.append([e3, e9, e6])
|
||||
tri_list.append([e3, e2, e6])
|
||||
tri_list.append([e9, e5, e6])
|
||||
elif (case == 44):
|
||||
# upper front plane, lower right back corner
|
||||
_append_tris(tri_list, 12, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 45):
|
||||
# Shelf: (v1, v3, v4) + lower right back corner
|
||||
_append_tris(tri_list, 13, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 46):
|
||||
# Rotated case 14 in paper
|
||||
tri_list.append([e4, e11, e12])
|
||||
tri_list.append([e4, e12, e5])
|
||||
tri_list.append([e12, e5, e6])
|
||||
tri_list.append([e4, e5, e1])
|
||||
elif (case == 47):
|
||||
# Shelf: (v5, v8, v7) by inversion
|
||||
tri_list.append([e11, e9, e12])
|
||||
tri_list.append([e12, e9, e5])
|
||||
tri_list.append([e12, e5, e6])
|
||||
elif (case == 48):
|
||||
# Back lower plane
|
||||
tri_list.append([e9, e10, e6])
|
||||
tri_list.append([e9, e6, e8])
|
||||
elif (case == 49):
|
||||
# Shelf: (v1, v5, v6)
|
||||
tri_list.append([e4, e8, e6])
|
||||
tri_list.append([e4, e6, e1])
|
||||
tri_list.append([e6, e1, e10])
|
||||
elif (case == 50):
|
||||
# Shelf: (v2, v5, v6)
|
||||
tri_list.append([e8, e6, e2])
|
||||
tri_list.append([e8, e2, e1])
|
||||
tri_list.append([e8, e9, e1])
|
||||
elif (case == 51):
|
||||
# Plane through middle of cube, parallel to x-z axis
|
||||
tri_list.append([e4, e8, e2])
|
||||
tri_list.append([e8, e2, e6])
|
||||
elif (case == 52):
|
||||
# Back lower plane, and front upper right corner
|
||||
_append_tris(tri_list, 48, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 53):
|
||||
# Shelf (v1, v5, v6) and front upper right corner
|
||||
_append_tris(tri_list, 49, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 54):
|
||||
# Rotated case 11 from paper (v2, v3, v5, v6)
|
||||
tri_list.append([e1, e9, e3])
|
||||
tri_list.append([e9, e3, e6])
|
||||
tri_list.append([e9, e8, e6])
|
||||
tri_list.append([e12, e3, e6])
|
||||
elif (case == 55):
|
||||
# Shelf: (v4, v8, v7) by inversion
|
||||
tri_list.append([e4, e8, e6])
|
||||
tri_list.append([e4, e6, e3])
|
||||
tri_list.append([e6, e3, e12])
|
||||
elif (case == 56):
|
||||
# Back lower plane + upper left front corner
|
||||
_append_tris(tri_list, 48, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 57):
|
||||
# Rotated case 14 from paper (v4, v1, v5, v6)
|
||||
tri_list.append([e3, e11, e8])
|
||||
tri_list.append([e3, e8, e10])
|
||||
tri_list.append([e10, e6, e8])
|
||||
tri_list.append([e3, e1, e10])
|
||||
elif (case == 58):
|
||||
# Shelf: (v2, v6, v5) + upper left front corner
|
||||
_append_tris(tri_list, 50, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 59):
|
||||
# Shelf: (v3, v7, v8) by inversion
|
||||
tri_list.append([e2, e6, e8])
|
||||
tri_list.append([e8, e2, e3])
|
||||
tri_list.append([e8, e3, e11])
|
||||
elif (case == 60):
|
||||
# AMBIGUOUS CASE: parallel planes (front upper, back lower)
|
||||
_append_tris(tri_list, 48, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 12, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 61):
|
||||
# Upper back plane + lower right front corner by inversion
|
||||
_append_tris(tri_list, 63, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 62):
|
||||
# Upper back plane + lower left front corner by inversion
|
||||
_append_tris(tri_list, 63, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 63):
|
||||
# Upper back plane
|
||||
tri_list.append([e11, e12, e6])
|
||||
tri_list.append([e11, e8, e6])
|
||||
elif (case == 64):
|
||||
# Upper right back corner
|
||||
tri_list.append([e12, e7, e6])
|
||||
elif (case == 65):
|
||||
# upper right back, lower left front corners
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 66):
|
||||
# upper right back, lower right front corners
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 67):
|
||||
# lower front plane + upper right back corner
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 3, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 68):
|
||||
# upper right plane
|
||||
tri_list.append([e3, e2, e6])
|
||||
tri_list.append([e3, e7, e6])
|
||||
elif (case == 69):
|
||||
# Upper right plane, lower left front corner
|
||||
_append_tris(tri_list, 68, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 70):
|
||||
# Shelf: (v2, v3, v7)
|
||||
tri_list.append([e1, e3, e7])
|
||||
tri_list.append([e1, e10, e7])
|
||||
tri_list.append([e7, e10, e6])
|
||||
elif (case == 71):
|
||||
# Rotated version of case 11 in paper (v1, v2, v3, v7)
|
||||
tri_list.append([e10, e7, e4])
|
||||
tri_list.append([e4, e3, e7])
|
||||
tri_list.append([e10, e4, e9])
|
||||
tri_list.append([e7, e10, e6])
|
||||
elif (case == 72):
|
||||
# upper left front, upper right back corners
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 73):
|
||||
# front left plane, upper right back corner
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 9, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 74):
|
||||
# Three isolated corners, exactly case 7 in paper
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 75):
|
||||
# Shelf: (v1, v2, v4) + upper right back corner
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 11, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 76):
|
||||
# Shelf: (v4, v3, v7)
|
||||
tri_list.append([e4, e2, e6])
|
||||
tri_list.append([e4, e11, e7])
|
||||
tri_list.append([e4, e7, e6])
|
||||
elif (case == 77):
|
||||
# Rotated case 14 in paper (v1, v4, v3, v7)
|
||||
tri_list.append([e11, e9, e1])
|
||||
tri_list.append([e11, e1, e6])
|
||||
tri_list.append([e1, e6, e2])
|
||||
tri_list.append([e11, e6, e7])
|
||||
elif (case == 78):
|
||||
# Full corner v3: (v2, v3, v4, v7)
|
||||
tri_list.append([e1, e4, e7])
|
||||
tri_list.append([e1, e7, e6])
|
||||
tri_list.append([e4, e11, e7])
|
||||
tri_list.append([e1, e10, e6])
|
||||
elif (case == 79):
|
||||
# Shelf: (v6, v5, v8) by inversion
|
||||
tri_list.append([e9, e11, e10])
|
||||
tri_list.append([e11, e7, e10])
|
||||
tri_list.append([e7, e10, e6])
|
||||
elif (case == 80):
|
||||
# lower left back, upper right back corners (v5, v7)
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 81):
|
||||
# lower left plane, upper right back corner
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 17, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 82):
|
||||
# isolated corners (v2, v5, v7)
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 83):
|
||||
# Shelf: (v1, v2, v5) + upper right back corner
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 19, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 84):
|
||||
# upper right plane, lower left back corner
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 68, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 85):
|
||||
# AMBIGUOUS CASE: upper right and lower left parallel planes
|
||||
_append_tris(tri_list, 17, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 68, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 86):
|
||||
# Shelf: (v2, v3, v7) + lower left back corner
|
||||
_append_tris(tri_list, 70, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 87):
|
||||
# Upper left plane + lower right back corner, by inversion
|
||||
_append_tris(tri_list, 119, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 88):
|
||||
# Isolated corners v4, v5, v7
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 89):
|
||||
# Shelf: (v1, v4, v5) + isolated corner v7
|
||||
_append_tris(tri_list, 25, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 90):
|
||||
# Four isolated corners v2, v4, v5, v7
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 64, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 91):
|
||||
# Three isolated corners, v3, v6, v8 by inversion
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 92):
|
||||
# Shelf (v4, v3, v7) + isolated corner v5
|
||||
_append_tris(tri_list, 76, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 16, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 93):
|
||||
# Lower right plane + isolated corner v8 by inversion
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 34, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 94):
|
||||
# Isolated corners v1, v6, v8 by inversion
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 95):
|
||||
# Isolated corners v6, v8 by inversion
|
||||
_append_tris(tri_list, 32, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 96):
|
||||
# back right plane
|
||||
tri_list.append([e7, e12, e5])
|
||||
tri_list.append([e5, e10, e12])
|
||||
elif (case == 97):
|
||||
# back right plane + isolated corner v1
|
||||
_append_tris(tri_list, 96, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 98):
|
||||
# Shelf: (v2, v6, v7)
|
||||
tri_list.append([e1, e7, e5])
|
||||
tri_list.append([e7, e1, e12])
|
||||
tri_list.append([e1, e12, e2])
|
||||
elif (case == 99):
|
||||
# Rotated case 14 in paper: (v1, v2, v6, v7)
|
||||
tri_list.append([e9, e2, e7])
|
||||
tri_list.append([e9, e2, e4])
|
||||
tri_list.append([e2, e7, e12])
|
||||
tri_list.append([e7, e9, e5])
|
||||
elif (case == 100):
|
||||
# Shelf: (v3, v6, v7)
|
||||
tri_list.append([e3, e7, e5])
|
||||
tri_list.append([e3, e5, e2])
|
||||
tri_list.append([e2, e5, e10])
|
||||
elif (case == 101):
|
||||
# Shelf: (v3, v6, v7) + isolated corner v1
|
||||
_append_tris(tri_list, 100, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 102):
|
||||
# Plane bisecting left-right halves of cube
|
||||
tri_list.append([e1, e3, e7])
|
||||
tri_list.append([e1, e7, e5])
|
||||
elif (case == 103):
|
||||
# Shelf: (v4, v5, v8) by inversion
|
||||
tri_list.append([e3, e7, e5])
|
||||
tri_list.append([e3, e5, e4])
|
||||
tri_list.append([e4, e5, e9])
|
||||
elif (case == 104):
|
||||
# Back right plane + isolated corner v4
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 96, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 105):
|
||||
# AMBIGUOUS CASE: back right and front left planes
|
||||
_append_tris(tri_list, 96, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 9, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 106):
|
||||
# Shelf: (v2, v6, v7) + isolated corner v4
|
||||
_append_tris(tri_list, 98, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 8, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 107):
|
||||
# Back left plane + isolated corner v3 by inversion
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 111, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 108):
|
||||
# Rotated case 11 from paper: (v4, v3, v7, v6)
|
||||
tri_list.append([e4, e10, e7])
|
||||
tri_list.append([e4, e10, e2])
|
||||
tri_list.append([e4, e11, e7])
|
||||
tri_list.append([e7, e10, e5])
|
||||
elif (case == 109):
|
||||
# Back left plane + isolated corner v2 by inversion
|
||||
_append_tris(tri_list, 111, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 110):
|
||||
# Shelf: (v1, v5, v8) by inversion
|
||||
tri_list.append([e1, e5, e7])
|
||||
tri_list.append([e1, e7, e11])
|
||||
tri_list.append([e1, e11, e4])
|
||||
elif (case == 111):
|
||||
# Back left plane
|
||||
tri_list.append([e11, e9, e7])
|
||||
tri_list.append([e9, e7, e5])
|
||||
elif (case == 112):
|
||||
# Shelf: (v5, v6, v7)
|
||||
tri_list.append([e9, e10, e12])
|
||||
tri_list.append([e9, e12, e7])
|
||||
tri_list.append([e9, e7, e8])
|
||||
elif (case == 113):
|
||||
# Exactly case 11 from paper: (v1, v5, v6, v7)
|
||||
tri_list.append([e1, e8, e12])
|
||||
tri_list.append([e1, e8, e4])
|
||||
tri_list.append([e8, e7, e12])
|
||||
tri_list.append([e12, e1, e10])
|
||||
elif (case == 114):
|
||||
# Full corner v6: (v2, v6, v7, v5)
|
||||
tri_list.append([e1, e9, e7])
|
||||
tri_list.append([e1, e7, e12])
|
||||
tri_list.append([e1, e12, e2])
|
||||
tri_list.append([e9, e8, e7])
|
||||
elif (case == 115):
|
||||
# Shelf: (v3, v4, v8)
|
||||
tri_list.append([e2, e4, e8])
|
||||
tri_list.append([e2, e12, e7])
|
||||
tri_list.append([e2, e8, e7])
|
||||
elif (case == 116):
|
||||
# Rotated case 14 in paper: (v5, v6, v7, v3)
|
||||
tri_list.append([e9, e2, e7])
|
||||
tri_list.append([e9, e2, e10])
|
||||
tri_list.append([e9, e8, e7])
|
||||
tri_list.append([e2, e3, e7])
|
||||
elif (case == 117):
|
||||
# upper left plane + isolated corner v2 by inversion
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 119, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 118):
|
||||
# Shelf: (v1, v4, v8)
|
||||
tri_list.append([e1, e3, e7])
|
||||
tri_list.append([e7, e1, e8])
|
||||
tri_list.append([e1, e8, e9])
|
||||
elif (case == 119):
|
||||
# Upper left plane
|
||||
tri_list.append([e4, e3, e7])
|
||||
tri_list.append([e4, e8, e7])
|
||||
elif (case == 120):
|
||||
# Shelf: (v1, v2, v3) + isolated corner v8
|
||||
_append_tris(tri_list, 7, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 121):
|
||||
# Front right plane + isolated corner v8
|
||||
_append_tris(tri_list, 6, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 122):
|
||||
# Isolated corners v1, v3, v8
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 123):
|
||||
# Isolated corners v3, v8
|
||||
_append_tris(tri_list, 4, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 124):
|
||||
# Front lower plane + isolated corner v8
|
||||
_append_tris(tri_list, 3, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 125):
|
||||
# Isolated corners v2, v8
|
||||
_append_tris(tri_list, 2, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127)
|
||||
elif (case == 126):
|
||||
# Isolated corners v1, v8
|
||||
_append_tris(tri_list, 1, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
_append_tris(tri_list, 127, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10,
|
||||
e11, e12)
|
||||
elif (case == 127):
|
||||
# Isolated corner v8
|
||||
tri_list.append([e11, e7, e8])
|
||||
|
||||
return
|
||||
@@ -0,0 +1,200 @@
|
||||
import numpy as np
|
||||
from . import _marching_cubes
|
||||
|
||||
|
||||
def marching_cubes(volume, level, sampling=(1., 1., 1.)):
|
||||
"""
|
||||
Marching cubes algorithm to find iso-valued surfaces in 3d volumetric data
|
||||
|
||||
Parameters
|
||||
----------
|
||||
volume : (M, N, P) array of doubles
|
||||
Input data volume to find isosurfaces. Will be cast to `np.float64` if
|
||||
not provided in this format.
|
||||
level : float
|
||||
Contour value to search for isosurfaces in `volume`.
|
||||
sampling : length-3 tuple of floats
|
||||
Voxel spacing in spatial dimensions corresponding to numpy array
|
||||
indexing dimensions (M, N, P) as in `volume`.
|
||||
|
||||
Returns
|
||||
-------
|
||||
vert_list : list
|
||||
Every entry in this list is a unique vertex on the isosurface.
|
||||
tri_list : list
|
||||
Every entry in this list is a length-3 list of integers. These
|
||||
represent triangular faces; the integers in each sub-list correspond
|
||||
to vertices held in `vert_list`.
|
||||
|
||||
Notes
|
||||
-----
|
||||
The marching cubes algorithm is implemented as described in [1]_.
|
||||
A simple explanation is available here::
|
||||
|
||||
http://www.essi.fr/~lingrand/MarchingCubes/algo.html
|
||||
|
||||
There are several known ambiguous cases in the marching cubes algorithm.
|
||||
Using point labeling as in [1]_, Figure 4, as shown:
|
||||
|
||||
v8 ------ v7
|
||||
/ | / | y
|
||||
/ | / | ^ z
|
||||
v4 ------ v3 | | /
|
||||
| v5 ----|- v6 |/ (note: NOT right handed!)
|
||||
| / | / ----> x
|
||||
|/ | /
|
||||
v1 ------ v2
|
||||
|
||||
Most notably, if v4, v8, v2, and v6 are all >= `level` (or any
|
||||
generalization of this case) two parallel planes are generated by this
|
||||
algorithm, separating v4 and v8 from v2 and v6. An equally valid
|
||||
interpretation would be a single connected thin surface enclosing all
|
||||
four points. This is the best known ambiguity, though there are others.
|
||||
|
||||
This algorithm does not attempt to resolve such ambiguities; it is a naive
|
||||
implementation of marching cubes as in [1]_, but may be a good beginning
|
||||
for work with more recent techniques (Dual Marching Cubes, Extended
|
||||
Marching Cubes, Cubic Marching Squares, etc.).
|
||||
|
||||
Because of interactions between neighboring cubes, the isosurface(s)
|
||||
generated by this algorithm are NOT guaranteed to be closed, particularly
|
||||
for complicated contours. Furthermore, this algorithm does not guarantee
|
||||
a single contour will be returned. Indeed, ALL isosurfaces which cross
|
||||
`level` will be found, regardless of connectivity.
|
||||
|
||||
The output is a triangular mesh consisting of a set of unique vertices and
|
||||
connecting triangles. The order of these vertices and triangles in the
|
||||
output list is determined by the position of the smallest ``x,y,z`` (in
|
||||
lexicographical order) coordinate in the contour. This is a side-effect
|
||||
of how the input array is traversed, but can be relied upon.
|
||||
|
||||
To quantify the area of an isosurface generated by this algorithm, pass
|
||||
the output directly into `skimage.measure.mesh_surface_area`.
|
||||
|
||||
Regarding visualization of algorithm output, the ``mayavi`` package
|
||||
is recommended. To contour a volume named `myvolume` about the level 0.0:
|
||||
|
||||
>>> from mayavi import mlab
|
||||
>>> verts, tris = marching_cubes(myvolume, 0.0, (1., 1., 2.))
|
||||
>>> mlab.triangular_mesh([vert[0] for vert in verts],
|
||||
[vert[1] for vert in verts],
|
||||
[vert[2] for vert in verts],
|
||||
tris)
|
||||
>>> mlab.show()
|
||||
|
||||
References
|
||||
----------
|
||||
.. [1] Lorensen, William and Harvey E. Cline. Marching Cubes: A High
|
||||
Resolution 3D Surface Construction Algorithm. Computer Graphics
|
||||
(SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163-170).
|
||||
|
||||
See Also
|
||||
--------
|
||||
skimage.measure.mesh_surface_area
|
||||
|
||||
"""
|
||||
# Check inputs
|
||||
if volume.ndim != 3:
|
||||
raise ValueError("Input volume must be 3d.")
|
||||
if volume.dtype.kind == 'f':
|
||||
volume = volume.astype(np.float)
|
||||
else:
|
||||
from skimage.util import img_as_float
|
||||
# If incorrect type provided, convert BOTH contour value and input
|
||||
# volume using same method
|
||||
level = img_as_float(np.array(level, dtype=volume.dtype))[0]
|
||||
volume = img_as_float(volume)
|
||||
|
||||
# Extract raw triangles using marching cubes in Cython
|
||||
# Returns a list of length-3 lists, each sub-list containing three
|
||||
# tuples. The tuples hold (x, y, z) coordinates for triangle vertices.
|
||||
# Note: this algorithm is fast, but returns degenerate "triangles" which
|
||||
# have repeated vertices - and equivalent vertices are redundantly
|
||||
# placed in every triangle they connect with.
|
||||
raw_tris = _marching_cubes.iterate_and_store_3d(volume, float(level),
|
||||
sampling)
|
||||
|
||||
# Find and collect unique vertices, storing triangle verts as indices.
|
||||
# Removes much redundancy and eliminates degenerate "triangles".
|
||||
vert_list, tri_list = _unpack_unique_verts(raw_tris)
|
||||
|
||||
return vert_list, tri_list
|
||||
|
||||
|
||||
def _unpack_unique_verts(trilist):
|
||||
"""
|
||||
Converts a list of lists of tuples corresponding to triangle vertices into
|
||||
a unique vertex list, and a list of triangle faces w/indices corresponding
|
||||
to entries of the vertex list.
|
||||
|
||||
"""
|
||||
idx = 0
|
||||
vert_index = {}
|
||||
vert_list = []
|
||||
tri_list = []
|
||||
|
||||
# Iterate over triangles
|
||||
for i in range(len(trilist)):
|
||||
templist = []
|
||||
|
||||
# Only parse vertices from non-degenerate triangles
|
||||
if not ((trilist[i][0] == trilist[i][1]) or
|
||||
(trilist[i][0] == trilist[i][2]) or
|
||||
(trilist[i][1] == trilist[i][2])):
|
||||
|
||||
# Iterate over vertices within each triangle
|
||||
for j in range(3):
|
||||
vert = trilist[i][j]
|
||||
|
||||
# Check if a new unique vertex found
|
||||
if vert not in vert_index:
|
||||
vert_index[vert] = idx
|
||||
templist.append(idx)
|
||||
vert_list.append(vert)
|
||||
idx += 1
|
||||
else:
|
||||
templist.append(vert_index[vert])
|
||||
|
||||
tri_list.append(templist)
|
||||
|
||||
return vert_list, tri_list
|
||||
|
||||
|
||||
def mesh_surface_area(verts, tris):
|
||||
"""
|
||||
Compute surface area, given vertices & triangular faces
|
||||
|
||||
Parameters
|
||||
----------
|
||||
verts : list
|
||||
List of length-3 NumPy arrays containing vertex coordinates.
|
||||
Units in each dimension should be consistent.
|
||||
|
||||
tris : list
|
||||
List of length-3 lists of integers, referencing vertex coordinates as
|
||||
provided in `verts`
|
||||
|
||||
Returns
|
||||
-------
|
||||
area : float
|
||||
Surface area of mesh. Units in coordinates maintained, but squared.
|
||||
|
||||
Notes
|
||||
-----
|
||||
The arguments expected by this function are the exact outputs from
|
||||
`skimage.measure.marching_cubes`. For unit correct output, ensure correct
|
||||
`spacing` was passed to `skimage.measure.marching_cubes`.
|
||||
|
||||
See Also
|
||||
--------
|
||||
skimage.measure.marching_cubes
|
||||
|
||||
"""
|
||||
# Define two vector arrays `a` and `b` from triangle vertices
|
||||
actual_verts = np.array([[verts[i] for i in tri] for tri in tris])
|
||||
a = actual_verts[:, 0, :] - actual_verts[:, 1, :]
|
||||
b = actual_verts[:, 0, :] - actual_verts[:, 2, :]
|
||||
del actual_verts
|
||||
|
||||
# Area of triangle = 1/2 * Euclidean norm of cross product
|
||||
return ((np.cross(a, b) ** 2).sum(axis=1) ** 0.5).sum() / 2.
|
||||
Reference in New Issue
Block a user