Robust Point Cloud Registration with Geometry-based Transformation Invariant Descriptor Jianjie Lin, Markus Rickert, Long Wen, Yingbai Hu and Alois Knoll Abstract— This work presents a novel method for point registration in 3D space. The proposed algorithm utilizes transformation-invariant geometry information to estimate the pose of objects based on correspondences between points in two sets. Conventional methods use geometry descriptors to find these correspondences, which can result in a large number of outliers. Most existing algorithms are error-prone when outliers are present. Instead of formulating point registration as a non- convex optimization problem, we propose an intuitive method that filters out spurious correspondences. This is achieved by evaluating three different geometry-based transformation- invariant descriptors for outlier removal. We construct fully connected graphs with the proposed descriptors on corre- spondences, and convert the outlier removal problem into a subgraph isomorphism problem that is solved using a binary clustering approach. The resulting inlier clustering is used to estimate the transformation between the two point sets. The effectiveness of the proposed approach is evaluated on standard 3D data and the 3DMatch scan matching dataset, and compared against existing state-of-the-art methods. Results show that our method effectively reduces outliers and performs similarly to these methods. I. I NTRODUCTION Point cloud registration is a core operation in computer vision and robotics that computes the pose transformation between two-point sets. It has applications in MRI/CAT scan alignment [1], robot manipulation [2] (Fig. 1), ob- ject recognition and localization, 3D reconstruction, etc. A standard approach to point registration without assumed correspondence is based on the iterative closest point (ICP) algorithm [3], [4] by alternatively finding correspondence and solving correspondence-based problems. However, the success of ICP relies heavily on a good initial alignment. The correspondence-based approach is another prevalent paradigm approach. It estimates the transformation  be- tween two point sets by giving the putative correspondences. The transformation  can be analytically solved [5], [6], [7], [8] if the putative correspondences are all correct. In practice, most correspondence point sets contain a large frac- tion of outliers. Even state-of-the-art local descriptors such as FPFH [9] and some deep learning-based methods [10] have difficulties in producing acceptable outlier rates in real-world problems and inevitably produce mismatches. As a result, this leads to a poor estimation of these solvers. RANSAC [11] is probably the most widely used algorithm for estimating transformations assuming the presence of outliers. The aim of RANSAC is to maximize the Consensus Jianjie Lin, Markus Rickert, Long Wen, Yingbai Hu, Alois Knoll are with Robotics, Artificial Intelligence and Real-Time Systems, School of Computation, Information and Technology, Technical University of Munich, Munich, Germany {jianjie.lin, rickert, wenl, yingbai.hu, knoll}@in.tum.de Fig. 1: The picture on the left shows the scanned scenes, and the one on the right illustrates an excerpt of bunny alignment and outlier removal. set by randomly sampling the smallest subset and estimating the transformations using least squares. The transform with maximum consensus is returned as the solution. However, the sampling phase of RANSAC would be inefficient given a large fraction of outliers. The global method [12], [13], [14] is an alternative algorithm that guarantees robustness. Branch-and-bound (B&B) algorithms are used to optimize the parameters of the objective function. However, a general weakness of global algorithms is their high computational cost, especially on data with large size  and high outliers. Contribution: In our previous work, we used the rotation- invariant descriptor [15] to address the object classifica- tion problem. In this work, we further explore the use of geometry-based transformation-invariant descriptors, in- cluding the Triangle Angle Invariant (TAnI), Triangle Dot Product Invariant (TDoI), Triangle Area Invariant (TArI), and Triangle Dot Product Invariant (TDoI), for robust point cloud registration by removing outliers. We construct two fully connected graphs using the proposed descriptors on the putative correspondences and convert the outlier removal into the subgraph isomorphism problem, aiming to find two subgraphs from these two fully connected graphs that have the same topology (isomorphism). The subgraph isomor- phism problem is proven to be NP-complete based on the reduction of the clique problem. In this work, we present a novel approach to reformulate the subgraph matching problem as a clustering approach to enhance the perfor- mance of outlier removal, where one cluster is identified as outliers and the other is denoted as inliers. The remaining two subgraphs contain only inliers, which exhibit the same topology and can be transformed into each other with the estimated transformation matrix. Based on our mathematical