Add Normal Distributions Transform (NDT) registration

[Xiang-Zeng]

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

Open3D provides ICP, Colored ICP, Generalized ICP, RANSAC, and FGR registration
pipelines, but does not currently expose a 3D Normal Distributions Transform
(NDT) registration method. NDT is a common LiDAR / point-cloud registration
method that optimizes a source-to-target rigid transform against voxelized
Gaussian target distributions instead of nearest-neighbor point
correspondences.

Upstream check:

• Local source search under cpp, python, docs, and examples found no
  existing NDT registration implementation.
• GitHub API search for active Open3D PRs matching NDT registration returned
  total_count: 0 on 2026-07-08.

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

This PR adds a legacy CPU registration API:

• NormalDistributionsTransformOption
• RegistrationNDT
• Python binding: open3d.pipelines.registration.registration_ndt

The implementation builds a target voxel Gaussian map once per registration
call, accepts either center-voxel or six-neighbor voxel residuals, regularizes
small covariance eigenvalues, rejects residuals by squared Mahalanobis distance,
and solves the rigid update using Open3D's existing Eigen SE(3) utilities.

The returned RegistrationResult includes fitness, Euclidean RMSE against the
accepted voxel means, the estimated transform, and representative target-point
indices for accepted voxel residuals.

This implementation is based on the Normal Distributions Transform (NDT) registration algorithm originally proposed by Biber and Straßer [1]. The concrete implementation follows the open‑source reference code by Gao [2], while being adapted to Open3D's registration API and coding style.

[1] P. Biber and W. Straßer, "The Normal Distributions Transform: A New Approach to Laser Scan Matching," Proceedings of the 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003), vol. 3, pp. 2743–2748, IEEE, 2003.
[2] Xiang Gao, SLAM Technology in Autonomous Driving and Robotics: From Theory to Practice (in Chinese), Publishing House of Electronics Industry, 2023. Code available at: https://github.com/gaoxiang12/slam_in_autonomous_driving/blob/master/src/ch7/ndt_3d.cc

Added:

• C++ tests for invalid option validation, synthetic known-transform recovery,
  and improvement over initial alignment.
• Python tests for option binding, synthetic known-transform recovery, and
  invalid option validation.
• Python example: examples/python/pipelines/ndt_registration.py
• Tutorial page: docs/tutorial/pipelines/ndt_registration.rst

Local verification:

env -u https_proxy -u http_proxy -u HTTPS_PROXY -u HTTP_PROXY \
  /home/qi/.local/bin/cmake -S . -B build \
  -DCMAKE_EXPORT_COMPILE_COMMANDS=ON \
  -DBUILD_UNIT_TESTS=ON \
  -DBUILD_PYTHON_MODULE=ON \
  -DBUILD_CUDA_MODULE=OFF \
  -DBUILD_SYCL_MODULE=OFF \
  -DBUILD_ISPC_MODULE=OFF \
  -DBUILD_GUI=OFF \
  -DBUILD_WEBRTC=OFF \
  -DBUILD_EXAMPLES=OFF \
  -DBUILD_BENCHMARKS=OFF \
  -DBUILD_SHARED_LIBS=OFF

env -u https_proxy -u http_proxy -u HTTPS_PROXY -u HTTP_PROXY \
  /home/qi/.local/bin/cmake --build build --target tests pybind --parallel 2

./build/bin/tests --gtest_filter='*NormalDistributionsTransform*'
# [  PASSED  ] 3 tests.

PYTHONPATH=/home/qi/ws/ndt/Open3D/build/lib/python_package \
  python3 -m pytest python/test/pipelines/test_normal_distributions_transform.py -q
# 3 passed

python3 -m py_compile \
  python/test/pipelines/test_normal_distributions_transform.py \
  examples/python/pipelines/ndt_registration.py

Real-data validation: examples/python/pipelines/ndt_registration.py

The Python test was run against the local build package via PYTHONPATH; no
global Open3D package was installed or overwritten.

[ssheorey]

Hi @Xiang-Zeng thanks for submitting this great new feature!
Please take a look at the failing CI checks (please ignore if they are due to unrelated changes - the main branch is in some major refactoring at the moment).
Also please provide the citation of the paper that is implemented here (add to docs and also here in the PR).

[Xiang-Zeng]

Hi @ssheorey ,
Thank you for the positive feedback! I will go through the failing CI checks and separate issues caused by this PR.
The Codacy/Cppcheck warnings about missing #include files (e.g., <Eigen/Core>, , ) seems to be false positives.
Some remaining failures appear to be related to GitHub attestation/OIDC permissions in forked PR workflows (Run actions/attest@v4 Error: missing "id-token" permission. Please add "permissions: id-token: write" to your workflow.).
I added the paper citation to the documentation and updated the PR description.