Registration beyond Points: General Affine Subspace Alignment via Geodesic Distance on Grassmann Manifold

2025-07-23

AI Summary

Existing methods only measure proximity of features on the affine Grassmann manifold but fail to explicitly model distances as functions of rigid transformations, resulting in non-differentiable, non-optimizable distance metrics and poor applicability to non-point features (e.g., lines, planes) in 3D registration. This paper introduces the first general affine subspace alignment framework grounded in the Grassmann geodesic distance. We derive, for the first time, a differentiable, unambiguous cost function parameterized directly by subspace bases and expressed in closed form with respect to rigid transformations—enabling globally optimal optimization. Furthermore, we integrate geometric constraints with a maximum-inlier objective to design a deterministic branch-and-bound (BnB) optimizer. Our method significantly improves convergence and registration accuracy across diverse vision tasks involving lines, planes, and higher-dimensional affine subspaces. The implementation is open-sourced and supports registration of affine subspaces of arbitrary dimension.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Motion & Tracking
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Affine Grassmannian has been favored for expressing proximity between lines and planes due to its theoretical exactness in measuring distances among features. Despite this advantage, the existing method can only measure the proximity without yielding the distance as an explicit function of rigid body transformation. Thus, an optimizable distance function on the manifold has remained underdeveloped, stifling its application in registration problems. This paper is the first to explicitly derive an optimizable cost function between two Grassmannian features with respect to rigid body transformation ($mathbf{R}$ and $mathbf{t}$). Specifically, we present a rigorous mathematical proof demonstrating that the bases of high-dimensional linear subspaces can serve as an explicit representation of the cost. Finally, we propose an optimizable cost function based on the transformed bases that can be applied to the registration problem of any affine subspace. Compared to vector parameter-based approaches, our method is able to find a globally optimal solution by directly minimizing the geodesic distance which is agnostic to representation ambiguity. The resulting cost function and its extension to the inlier-set maximizing ac{BnB} solver have been demonstrated to improve the convergence of existing solutions or outperform them in various computer vision tasks. The code is available on https://github.com/joomeok/GrassmannRegistration.

Problem

• Derive optimizable cost function for Grassmannian features
• Measure geodesic distance for affine subspace registration
• Improve convergence in computer vision tasks

Innovation

• Optimizable cost function on Grassmann manifold
• Rigid body transformation explicit representation
• Geodesic distance minimizes representation ambiguity