Adding minimum volume oriented bounding ellipsoid matching functionality

[quentin-leboutet]

Add Oriented Bounding Ellipsoid (OBE) support to Open3D

Type

• Bug fix (non-breaking change which fixes an issue): Fixes #
• 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

This PR adds support for Oriented Bounding Ellipsoids (OBE) to Open3D, complementing the existing bounding volume primitives (AABB and OBB). Oriented bounding ellipsoids provide a tighter fit for certain geometries compared to boxes, which can be beneficial for:

• More accurate spatial representations of point clouds and meshes
• Improved collision detection and computational geometry applications
• Better geometric analysis and understanding of 3D shapes

The implementation uses Khachiyan's algorithm to compute the minimum volume enclosing ellipsoid, providing an efficient approximation of the optimal bounding ellipsoid.

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

Core Implementation

New Classes:

• OrientedBoundingEllipsoid (legacy API)
• t::geometry::OrientedBoundingEllipsoid (tensor-based API)

Key Features:

1. Minimum volume ellipsoid computation using Khachiyan's algorithm
   • kernel implementation in cpp/open3d/t/geometry/kernel/MinimumOBE.cpp (tensor-based API)
   • Computes convex hull first for optimal results
   • Canonical orientation mapping for consistent representation
2. Geometry integration:
   • Added GetOrientedBoundingEllipsoid() method to all 3D geometry base classes (Geometry3D.h:74-75)
   • Support for PointCloud, TriangleMesh, LineSet, VoxelGrid, and Octree
3. Ellipsoid mesh generation:
   • TriangleMesh::CreateEllipsoid() - creates ellipsoid mesh primitives
   • LineSet::CreateFromOrientedBoundingEllipsoid() - wireframe representation
4. Transformation operations:
   • Translate, Rotate, Scale operations (BoundingVolume.cpp:75-96)
   • Volume calculation (BoundingVolume.cpp:99-100)
5. Visualization support:
   • Renderer implementation in GeometryRenderer.cpp
   • Shader support in SimpleShader.cpp
   • Filament rendering backend integration
6. Python bindings:
   • Complete Python API through pybind11 (boundingvolume.cpp)
   • create_from_points() static method with robust option
   • Properties: center, R (rotation), radii, color

Try it:

import open3d as o3d
mesh = o3d.io.read_triangle_mesh("<your_mesh>.obj")
mesh.compute_triangle_normals()
ellipsoid = o3d.geometry.OrientedBoundingEllipsoid.create_from_points(o3d.utility.Vector3dVector(mesh.vertices))
print(f"Volume ellipsoid: {ellipsoid.volume()}")
ellipsoid.color = (0, 0.44313725490196076, 0.7725490196078432)
ellipsoid_lines = o3d.geometry.LineSet.create_from_oriented_bounding_ellipsoid(ellipsoid)
o3d.visualization.draw([mesh, ellipsoid_lines])

[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.