open3d | How to calculate the volume of a mesh created by point cloud?

I create a mesh by using 'open3d.geometry.TriangleMesh.create_from_point_cloud_alpha_shape' function and wanted to calculate the volume of it. But a RuntimeError occurs as the following shows:

[Open3D WARNING] [CreateFromPointCloudAlphaShape] invalid tetra in TetraMesh
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
Input In [27], in <cell line: 7>()
      3 point_cloud.points = open3d.utility.Vector3dVector(data_all)
      5 tri_mesh = open3d.geometry.TriangleMesh.create_from_point_cloud_alpha_shape(point_cloud, alpha=10)
----> 7 v = open3d.geometry.TriangleMesh.get_volume(tri_mesh)
      8 print(v)

RuntimeError: [Open3D Error] (double open3d::geometry::TriangleMesh::GetVolume() const) /Users/runner/work/Open3D/Open3D/cpp/open3d/geometry/TriangleMesh.cpp:1220: The mesh is not watertight, and the volume cannot be computed.

I searched online (https://github.com/isl-org/Open3D/pull/3201) and found that the warning message (invalid tetra in TetraMesh) is a common problem and the reason is that some points are inside the mesh and not in the surface. Therefore, I exclude all the points that are not in the surface by calculating each point's distance to the surface.

Then, I recreate the mesh by using 'open3d.geometry.TriangleMesh.create_from_point_cloud_alpha_shape' function. The same issue occurs and the mesh is still not watertight.

Is there any method to solve this problem and calculate the volume?

Thanks!

In order to compute the volume you need a watertight mesh, which means it must have no openings. This isn't something the alpha shape algorithm guarantees; f you pick a smaller alpha, eventually it will just create tiny islands or isolated triangles.
The tutorial has examples of what happens with different alphas:
http://www.open3d.org/html/tutorial/Advanced/surface_reconstruction.html

You can check whether you had success with tri_mesh.is_watertight().

But the core problem here is really how to make the mesh watertight, which is asked and answered in this question.

Maybe picking a larger alpha is good enough for your case, or maybe you would be better served by an alternative algorithm like create_from_point_cloud_poisson (which should be guaranteed watertight, volume would be underestimated if the shape is mostly convex) or create_from_point_cloud_ball_pivoting.

You should plot your geometries and point cloud for visual inspection, and also compare the volume from each method.