--- kind: concept slug: iterative-closest-point title: Iterative closest point (ICP) summary: >- Iterative closest point aligns two point clouds by alternating correspondences with a closed-form rigid fit (SVD). Variants differ in correspondences, residuals and weights. Classic ICP assumes equal point counts, which partial views break. Learned matching (PRNet) and semantic weighting (SuMa++) respond; SoGraB uses ICP only to pre-align. tags: [point-cloud, registration, lidar, slam, correspondence, deformation] papers: ["1910.12240", "2105.11320", "2411.19408"] related: [image-registration, local-feature-matching, simultaneous-localization-and-mapping, lidar-inertial-odometry, unsupervised-keypoint-detection, surfel-mapping, chamfer-distance, benchmark-validity] status: draft updated_at: 2026-10-25 --- ## Overview ICP finds the rigid motion (rotation and translation) that lines up one point cloud with another. It repeats two steps: choose correspondences between the clouds under the current motion, then solve for the best rigid motion for those correspondences [arxiv:1910.12240 §3.1]. The closed-form fit is shared by most variants: centre both sets, form the cross-covariance, take its SVD, and read the rotation and translation off the factors [arxiv:1910.12240 §3.1]. The variants differ in how they choose correspondences, which residual they minimise and how they weight each residual. The invariant that all of them rely on is the closed-form fit for a fixed correspondence. ## Variants ### Classic ICP with nearest-neighbour correspondences The baseline form alternates a nearest-neighbour mapping with the closed-form fit [arxiv:1910.12240 §3.1]. Its stated formulation assumes both clouds have the same number of points, M = N [arxiv:1910.12240 §3.1]. Partial views violate this: usually N ≠ M and only subsets of the two clouds match [arxiv:1910.12240 §3.2]. On the ModelNet40 partial-to-partial split, the PRNet authors report a rotation MSE of 1134.552 for ICP, against 10.235 for PRNet [arxiv:1910.12240 tab. 1]. These are author-reported results on synthetic motions. The paper does not describe the ICP baseline's settings beyond its text [arxiv:1910.12240 §4.1]. Card: [PRNet: Self-Supervised Learning for Partial-to-Partial Registration](/papers/1910.12240). ### Frame-to-model point-to-plane ICP with robust and semantic weights SuMa++ aligns each new LiDAR scan to a surfel map rather than to the previous scan [arxiv:2105.11320 §III-F]. Each residual is the distance along the model surface normal at the projected point, so the solve is a Gauss-Newton step on a point-to-plane objective [arxiv:2105.11320 §III-F]. Each residual is weighted by a Huber weight, by a semantic compatibility term, and by an indicator for whether the surfel is stable [arxiv:2105.11320 §III-F]. Where a point's label disagrees with a movable-class surfel, the semantic term lowers its influence, so moving objects count less in the pose estimate [arxiv:2105.11320 §III-E] [arxiv:2105.11320 §III-F]. The paper's KITTI odometry result is for the full system. The average translational error on training sequences 00–10 falls from 0.83% (SuMa) to 0.70% (SuMa++), author-reported [arxiv:2105.11320 §IV-B]. The result tables compare whole systems and do not isolate the effect of the semantic ICP weights [arxiv:2105.11320 §IV]. Card: [SuMa++: Efficient LiDAR-based Semantic SLAM](/papers/2105.11320). ### Learned iterative matching PRNet keeps the iteration and the closed-form fit but replaces the nearest-neighbour step with a learned, soft matching between keypoints [arxiv:1910.12240 §3.2]. Each pass samples a near-one-hot matching, solves the motion, moves the source and repeats; training runs three passes [arxiv:1910.12240 §3.2]. The network is trained self-supervised on synthetic motions [arxiv:1910.12240 §4.1]. On partial ModelNet40 the authors report a rotation MSE of 10.235, against 45.005 for DCP-v2 [arxiv:1910.12240 tab. 1]. On real Stanford Bunny scans the model was fine-tuned on 100 synthetic examples per scan; the paper gives qualitative results only [arxiv:1910.12240 §4.2]. Card: [PRNet: Self-Supervised Learning for Partial-to-Partial Registration](/papers/1910.12240). ### ICP as a pre-alignment step for a deformation score SoGraB uses ICP for a different purpose. It does not estimate motion for its own sake. It aligns the grasped object's point cloud to the pre-grasp cloud, and a Density-Aware Chamfer Distance between the aligned clouds then measures how much the object deformed [arxiv:2411.19408 §3.1] [arxiv:2411.19408 §3.2]. The ICP starts from a pose estimate taken from the robot's kinematics [arxiv:2411.19408 §3.2]. For symmetric objects with large deformation, the paper aligns centres of mass and principal axes instead [arxiv:2411.19408 §3.2]. The paper says ICP approximates the transform reliably for moderate deformation, given a good initial estimate [arxiv:2411.19408 §3.2]. The reported results are grasp outcomes and scores, with no measure of alignment accuracy [arxiv:2411.19408 §5]. The paper does not name the ICP type. The released code uses point-to-point ICP, with a distance threshold of 10000 and fixed initial matrices chosen by gripper angle, and it has no centre-of-mass or principal-axis step. The card lists these differences [card: Paper vs code](/papers/2411.19408). Card: [SoGraB: A Visual Method for Soft Grasping Benchmarking and Evaluation](/papers/2411.19408). ## When to use which | Situation | Use | Why | |---|---|---| | Two clouds with near-complete overlap, no labels, no training budget | Classic ICP with nearest-neighbour matching | Needs no training. The equal-size assumption is what the paper's formulation rests on [arxiv:1910.12240 §3.1]. | | Partial views of one object class, with synthetic training data available | Learned iterative matching ([PRNet](/papers/1910.12240)) | Reports far lower error than ICP on partial views [arxiv:1910.12240 tab. 1]. Requires training, and the reported numbers are on synthetic data [arxiv:1910.12240 §4.1]. | | Sequential LiDAR mapping where a segmenter supplies labels and moving objects corrupt the map | Point-to-plane frame-to-model ICP with semantic weights ([SuMa++](/papers/2105.11320)) | The motivation is that moving objects give wrong associations [arxiv:2105.11320 §III-E]. The gain is reported for the full system, so budget for the segmenter and the surfel map too [arxiv:2105.11320 §IV-B]. | | Aligning a deformed object between two scans, with a rough pose from kinematics and moderate deformation | ICP pre-alignment before a deformation distance ([SoGraB](/papers/2411.19408)) | The paper uses this setup [arxiv:2411.19408 §3.2]. Its reliability is claimed only for moderate deformation with a good start [arxiv:2411.19408 §3.2]. | | Real scans with no fine-tuning budget | Not settled by these sources | PRNet's real-scan result needs fine-tuning and has no numeric metric [arxiv:1910.12240 §4.2]. SuMa++ is evaluated on KITTI only [arxiv:2105.11320 §IV]. Test any variant on your own data. | | Strongly deformed or symmetric objects | Not settled by these sources | SoGraB's fallback is centroid and principal-axis alignment [arxiv:2411.19408 §3.2], which the released code does not implement. Check alignment on your own data. | ## Implementation sketch The common core is the loop below. Variants change the marked lines: the correspondence step, the residual and the weights. ```pseudocode # Classic ICP loop [arxiv:1910.12240 §3.1] # Input: source X (N, 3), target Y (M, 3), initial R (3, 3), t (3,) # Output: rotation R, translation t def icp(X, Y, R, t, iters): for _ in range(iters): # Correspondence step: nearest target point per moved source point # (learned variants replace this with a soft matching) [arxiv:1910.12240 §3.2] Z = nearest_neighbour(Y, R @ X + t) # Weights: 1 here; SuMa++ multiplies in Huber and semantic terms [arxiv:2105.11320 §III-F] w = ones(N) # Closed-form fit: centre, cross-covariance, SVD [arxiv:1910.12240 §3.1] dR, dt = weighted_rigid_fit(X, Z, w) R, t = dR @ R, dR @ t + dt return R, t ``` For point-to-plane variants, the fit is not closed form. It solves a weighted normal-equation system for a small pose increment at each step [arxiv:2105.11320 §III-F]. Stop criteria and iteration counts are not stated in the text of the PRNet or SoGraB papers, so set them on your own data. ## Contradictions None known between these three papers. Their numbers are not comparable: one reports synthetic partial ModelNet40 results [arxiv:1910.12240 tab. 1], another reports KITTI LiDAR odometry for a full mapping system [arxiv:2105.11320 §IV-B], and the third reports grasp scores on printed objects [arxiv:2411.19408 §5]. ## Papers - [PRNet: Self-Supervised Learning for Partial-to-Partial Registration](/papers/1910.12240): learned soft matching in an iterative registration loop for partial clouds, with classic ICP as a baseline. - [SuMa++: Efficient LiDAR-based Semantic SLAM](/papers/2105.11320): frame-to-model point-to-plane ICP whose residuals are weighted by semantic labels and surfel stability. - [SoGraB: A Visual Method for Soft Grasping Benchmarking and Evaluation](/papers/2411.19408): ICP pre-alignment, from a kinematic start, before a density-aware Chamfer distance scores deformation.