在Python中计算体积或表面积的好算法

Question

我正在尝试计算一个三维的numpy数组的体积（或者表面积）。在很多情况下，这些小立方体（体素）在各个方向上的大小是不一样的，我手头有每个方向的像素到厘米的转换系数。

有没有人知道哪里可以找到一个工具包或者软件包来完成这个工作呢？

现在我有一些自己写的代码，但我想升级到一个更专业、更准确的解决方案。

编辑1：这里有一些（不太好的）样本数据。这个数据比典型的球体要小得多。我会在能生成更好的数据时再添加！数据在(self.)tumorBrain.tumor里。

Answer 1

如果你想使用上面Joe的回答，你会遇到：

traits.trait_errors.TraitError: ImageMarchingCubes实例的'input'属性是'只读'的。

下面是需要做的修改，以及如何修复它的对比。

修改后的代码：

import numpy as np
from tvtk.api import tvtk
from tvtk.common import configure_input


def main():
    # Generate some data with anisotropic cells...
    # x,y,and z will range from -2 to 2, but with a
    # different (20, 15, and 5 for x, y, and z) number of steps
    x, y, z = np.mgrid[-2:2:20j, -2:2:15j, -2:2:5j]
    r = np.sqrt(x**2 + y**2 + z**2)

    dx, dy, dz = [np.diff(it, axis=a)[0, 0, 0] for it, a in zip(
        (x, y, z), (0, 1, 2))]

    # Your actual data is a binary (logical) array
    max_radius = 1.5
    data = (r <= max_radius).astype(np.int8)

    ideal_volume = 4.0 / 3 * max_radius**3 * np.pi
    coarse_volume = data.sum() * dx * dy * dz
    est_volume = vtk_volume(data, (dx, dy, dz), (x.min(), y.min(), z.min()))

    coarse_error = 100 * (coarse_volume - ideal_volume) / ideal_volume
    vtk_error = 100 * (est_volume - ideal_volume) / ideal_volume

    print('Ideal volume', ideal_volume)
    print('Coarse approximation', coarse_volume, 'Error', coarse_error, '%')
    print('VTK approximation', est_volume, 'Error', vtk_error, '%')


def vtk_volume(data, spacing=(1, 1, 1), origin=(0, 0, 0)):
    data[data == 0] = -1
    grid = tvtk.ImageData(spacing=spacing, origin=origin)
    grid.point_data.scalars = data.T.ravel()  # It wants fortran order???
    grid.point_data.scalars.name = 'scalars'
    grid.dimensions = data.shape

    iso = tvtk.ImageMarchingCubes()
    configure_input(iso, grid)  # <== will work
    # iso = tvtk.ImageMarchingCubes(input=grid)
    mass = tvtk.MassProperties()
    configure_input(mass, iso)
    # mass = tvtk.MassProperties(input=iso.output)
    return mass.volume


if __name__ == '__main__':
    main()

我所做更改的对比

2a3,4
> from tvtk.common import configure_input
> 
6c8
<     # x,y,and z will range from -2 to 2, but with a 
---
>     # x,y,and z will range from -2 to 2, but with a
8c10
<     x,y,z = np.mgrid[-2:2:20j, -2:2:15j, -2:2:5j]
---
>     x, y, z = np.mgrid[-2:2:20j, -2:2:15j, -2:2:5j]
11c13,14
<     dx, dy, dz = [np.diff(it, axis=a)[0,0,0] for it, a in zip((x,y,z),(0,1,2))]
---
>     dx, dy, dz = [np.diff(it, axis=a)[0, 0, 0] for it, a in zip(
>         (x, y, z), (0, 1, 2))]
24,26c27,30
<     print 'Ideal volume', ideal_volume
<     print 'Coarse approximation', coarse_volume, 'Error', coarse_error, '%'
<     print 'VTK approximation', est_volume, 'Error', vtk_error, '%'
---
>     print('Ideal volume', ideal_volume)
>     print('Coarse approximation', coarse_volume, 'Error', coarse_error, '%')
>     print('VTK approximation', est_volume, 'Error', vtk_error, '%')
> 
28c32
< def vtk_volume(data, spacing=(1,1,1), origin=(0,0,0)):
---
> def vtk_volume(data, spacing=(1, 1, 1), origin=(0, 0, 0)):
31c35
<     grid.point_data.scalars = data.T.ravel() # It wants fortran order???
---
>     grid.point_data.scalars = data.T.ravel()  # It wants fortran order???
35,36c39,44
<     iso = tvtk.ImageMarchingCubes(input=grid)
<     mass = tvtk.MassProperties(input=iso.output)
---
>     iso = tvtk.ImageMarchingCubes()
>     configure_input(iso, grid)  # <== will work
>     # iso = tvtk.ImageMarchingCubes(input=grid)
>     mass = tvtk.MassProperties()
>     configure_input(mass, iso)
>     # mass = tvtk.MassProperties(input=iso.output)
39c47,49
< main()
---
> 
> if __name__ == '__main__':
>     main()

更改的具体列表：

• 通过2to3工具转换，使其可以在Python 3中运行
• 根据autopep8修复代码，使其符合PEP8规范（包括语法、长度、空格的调整）
• 导入configure_imput，因为TVTK中的这些更改（Github代码更改）
• 修改ImageMarchingCubes构造函数中的input=参数
• 修改MassProperties构造函数中的input=参数
• 将对main()的调用包裹在一个直接调用中（以防止在导入时执行）#最佳实践

Answer 2

这些体素应该是比较简单的规则多面体，对吧？你只需要计算每个体素的体积，然后把它们加起来就行了。

Answer 3

一个选择是使用 VTK。 （这里我将使用 tvtk 的 Python 绑定来进行演示…）

在某些情况下，获取等值面内的区域会更准确一些。

另外，关于表面积， tvtk.MassProperties 也可以计算表面积。 这个表面积的值是 mass.surface_area（在下面的代码中有 mass 对象）。

import numpy as np
from tvtk.api import tvtk

def main():
    # Generate some data with anisotropic cells...
    # x,y,and z will range from -2 to 2, but with a 
    # different (20, 15, and 5 for x, y, and z) number of steps
    x,y,z = np.mgrid[-2:2:20j, -2:2:15j, -2:2:5j]
    r = np.sqrt(x**2 + y**2 + z**2)

    dx, dy, dz = [np.diff(it, axis=a)[0,0,0] for it, a in zip((x,y,z),(0,1,2))]

    # Your actual data is a binary (logical) array
    max_radius = 1.5
    data = (r <= max_radius).astype(np.int8)

    ideal_volume = 4.0 / 3 * max_radius**3 * np.pi
    coarse_volume = data.sum() * dx * dy * dz
    est_volume = vtk_volume(data, (dx, dy, dz), (x.min(), y.min(), z.min()))

    coarse_error = 100 * (coarse_volume - ideal_volume) / ideal_volume
    vtk_error = 100 * (est_volume - ideal_volume) / ideal_volume

    print 'Ideal volume', ideal_volume
    print 'Coarse approximation', coarse_volume, 'Error', coarse_error, '%'
    print 'VTK approximation', est_volume, 'Error', vtk_error, '%'

def vtk_volume(data, spacing=(1,1,1), origin=(0,0,0)):
    data[data == 0] = -1
    grid = tvtk.ImageData(spacing=spacing, origin=origin)
    grid.point_data.scalars = data.T.ravel() # It wants fortran order???
    grid.point_data.scalars.name = 'scalars'
    grid.dimensions = data.shape

    iso = tvtk.ImageMarchingCubes(input=grid)
    mass = tvtk.MassProperties(input=iso.output)
    return mass.volume

main()

这样就得到了：

Ideal volume 14.1371669412
Coarse approximation 14.7969924812 Error 4.66731094565 %
VTK approximation 14.1954890878 Error 0.412544796894 %