# Scale-Invariant Fast Functional Registration

arXiv: https://arxiv.org/abs/2209.12763 · HTML: https://mlsanity.com/paper/2209.12763/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Builds on (most cited first, 19 of 19)

- [ScanNet: Richly-annotated 3D Reconstructions of Indoor Scenes](https://mlsanity.com/paper/1702.04405/) (2017)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [KPConv: Flexible and Deformable Convolution for Point Clouds](https://mlsanity.com/paper/1904.08889/) (2019)
- [PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes](https://mlsanity.com/paper/1711.00199/) (2017)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [3DMatch: Learning Local Geometric Descriptors from RGB-D Reconstructions](https://mlsanity.com/paper/1603.08182/) (2016)
- [TEASER: Fast and Certifiable Point Cloud Registration](https://mlsanity.com/paper/2001.07715/) (2020)
- [PointNetLK: Robust & Efficient Point Cloud Registration using PointNet](https://mlsanity.com/paper/1903.05711/) (2019)
- [The Perfect Match: 3D Point Cloud Matching with Smoothed Densities](https://mlsanity.com/paper/1811.06879/) (2018)
- [Go-ICP: A Globally Optimal Solution to 3D ICP Point-Set Registration](https://mlsanity.com/paper/1605.03344/) (2016)
- [DeepGMR: Learning Latent Gaussian Mixture Models for Registration](https://mlsanity.com/paper/2008.09088/) (2020)
- [Guaranteed Outlier Removal for Point Cloud Registration with Correspondences](https://mlsanity.com/paper/1711.10209/) (2017)
- [DPDist : Comparing Point Clouds Using Deep Point Cloud Distance](https://mlsanity.com/paper/2004.11784/) (2020)
- [A Practical Maximum Clique Algorithm for Matching with Pairwise Constraints](https://mlsanity.com/paper/1902.01534/) (2019)
- [PHASER: a Robust and Correspondence-free Global Pointcloud Registration](https://mlsanity.com/paper/2102.02767/) (2021)
- [LiTAMIN2: Ultra Light LiDAR-based SLAM using Geometric Approximation applied with KL-Divergence](https://mlsanity.com/paper/2103.00784/) (2021)
- [Continuous Direct Sparse Visual Odometry from RGB-D Images](https://mlsanity.com/paper/1904.02266/) (2019)
- [A New Framework for Registration of Semantic Point Clouds from Stereo and RGB-D Cameras](https://mlsanity.com/paper/2012.03683/) (2020)
- [Nonparametric Continuous Sensor Registration](https://mlsanity.com/paper/2001.04286/) (2020)

## Built on by (most cited first, 2 of 2)

- [Fast Ergodic Search with Kernel Functions](https://mlsanity.com/paper/2403.01536/) (2024)
- [Volumetric Ergodic Control](https://mlsanity.com/paper/2511.11533/) (2025)

## Abstract (arXiv)

Functional registration algorithms represent point clouds as functions (e.g. spacial occupancy field) avoiding unreliable correspondence estimation in conventional least-squares registration algorithms. However, existing functional registration algorithms are computationally expensive. Furthermore, the capability of registration with unknown scale is necessary in tasks such as CAD model-based object localization, yet no such support exists in functional registration. In this work, we propose a scale-invariant, linear time complexity functional registration algorithm. We achieve linear time complexity through an efficient approximation of L2-distance between functions using orthonormal basis functions. The use of orthonormal basis functions leads to a formulation that is compatible with least-squares registration. Benefited from the least-square formulation, we use the theory of translation-rotation-invariant measurement to decouple scale estimation and therefore achieve scale-invariant registration. We evaluate the proposed algorithm, named FLS (functional least-squares), on standard 3D registration benchmarks, showing FLS is an order of magnitude faster than state-of-the-art functional registration algorithm without compromising accuracy and robustness. FLS also outperforms state-of-the-art correspondence-based least-squares registration algorithm on accuracy and robustness, with known and unknown scale. Finally, we demonstrate applying FLS to register point clouds with varying densities and partial overlaps, point clouds from different objects within the same category, and point clouds from real world objects with noisy RGB-D measurements.