fix alpha shape reconstruction if alpha too small for point scale

[rxba]

Type

• Bug fix (non-breaking change which fixes an issue): Fixes Unable to perform alpha surface reconstruction on Apple Silicon #6086
• New feature (non-breaking change which adds functionality). Resolves #
• Breaking change (fix or feature that would cause existing functionality to not work as expected) Resolves #

Motivation and Context

Surface reconstruction with alpha shapes by calling create_from_point_cloud_alpha_shape fails if the alpha value is too small for the internal constraint to become true.

Open3D/cpp/open3d/geometry/SurfaceReconstructionAlphaShape.cpp

Lines 105 to 110 in db00e33

	double r = std::sqrt(dx * dx + dy * dy + dz * dz - 4 * a * c) /
	(2 * std::abs(a));
	if (r <= alpha) {
	mesh->triangles_.push_back(TriangleMesh::GetOrderedTriangle(
	tetra(0), tetra(1), tetra(2)));

There is currently no check if the reconstructed mesh is empty, thus the following call to

Open3D/cpp/open3d/geometry/SurfaceReconstructionAlphaShape.cpp

Lines 165 to 168 in db00e33

	auto tmesh2 = t::geometry::vtkutils::ComputeNormals(
	tmesh, /*vertex_normals=*/true, /*face_normals=*/false,
	/*consistency=*/true, /*auto_orient_normals=*/true,
	/*splitting=*/false);

leads to a non-descriptive unordered_map key error.

To reproduce, an easy test is to attempt surface reconstruction with the Bunny and the Armadillo Mesh.
While the bunny mesh works fine with the tutorial alpha shape values (e.g. 0.03), the Armadillo mesh fails.
This is a point scale issue, i.e. max_bound (Armadillo) is [62.1 96.4 56.8] and max_bound (Bunny) is [0.061 0.187 0.058]. If the alpha is large enough (e.g. 10), or the points are scaled down, it works again for the Armadillo.

To reproduce the error:

bunny = o3d.data.BunnyMesh()
arma = o3d.data.ArmadilloMesh()
for m in [bunny, arma]:
    mesh = o3d.io.read_triangle_mesh(m.path)
    pcd = mesh.sample_points_poisson_disk(300)
    alpha = 0.03
    mesh = o3d.geometry.TriangleMesh.create_from_point_cloud_alpha_shape(pcd, alpha)
    mesh.compute_vertex_normals()
    o3d.visualization.draw_geometries([mesh], mesh_show_back_face=True)

Checklist:

• I have run python util/check_style.py --apply to apply Open3D code style
  to my code.
• This PR changes Open3D behavior or adds new functionality.
  • Both C++ (Doxygen) and Python (Sphinx / Google style) documentation is
    updated accordingly.
  • I have added or updated C++ and / or Python unit tests OR included test
    results (e.g. screenshots or numbers) here.
• I will follow up and update the code if CI fails.
• For fork PRs, I have selected Allow edits from maintainers.

Description

To prevent the error, I've added a check if the mesh is empty after the alpha shape reconstruction is completed. If it is empty, a warning is logged and the empty mesh gets returned.

[update-docs]

Thanks for submitting this pull request! The maintainers of this repository would appreciate if you could update the CHANGELOG.md based on your changes.

[ssheorey]

Thanks @rxba looks good!