BALM2 (BALM 2.0)
BALM2 延續以點到平面或邊緣之歐氏距離為殘差的光達 BA，並提出「點簇（point cluster）」概念，把同一特徵上的所有原始點壓縮為一組緊湊參數，使代價、導數與不確定度計算都不需逐點列舉。作者推導封閉形式的 Jacobian 與 Hessian、其零空間與稀疏性，據此建立二階求解器，並利用二階資訊估計由量測雜訊造成的位姿不確定度。論文另示範其用於光達慣性里程計、多光達外參校正與全域建圖。
本頁內容
BALM2 introduces point clusters that summarize all raw points on a feature, enabling a closed-form second-order BA solver over poses that also estimates pose covariance, without enumerating individual points.
技術屬性
欄位內容為文獻擷取紀錄的原文用語（英文），以原文為據；「未查證」表示本研究尚未讀到該資訊，不代表該方法不具備此能力。
| 感測輸入 | 3D LiDAR |
|---|---|
| 原文測試平台 | handheld、UAV、vehicle、simulation |
| 狀態估計 | second-order (Newton/LM-type) batch solver over poses after closed-form elimination of plane/edge parameters; point-cluster coordinates aggregate raw points; LDLT solve (Eigen); pose covariance from second-order information |
| 資料關聯 | raw points to plane or edge features via BALM adaptive voxelization of all points registered with an initial trajectory (root voxel 1 m for Hilti, 2 m for VIRAL and UrbanLoco, at most 3 layers, at least 20 points and threshold 1/25 for the feature test); the incremental ICP trajectory served as the common initialization (Sec. VI-B) |
| 時間表示 | discrete poses (authors discuss possible extension to spline or Gaussian-process trajectories, Sec. VIII-C) |
| 去畸變 | no in-scan motion model inside the BA; real-world scans were deskewed by FAST-LIO2 before BA and its odometry output discarded (Sec. VI-B); the simulation ignored in-frame distortion (Sec. V); continuous-time trajectories are only discussed as an extension (Sec. VIII-C) |
| 迴圈閉合 | none within the method; relies on the supplied initial trajectory |
| 全域最佳化 | batch multi-view BA over all supplied scans |
| 地圖表示 | point clusters per plane/edge feature inside adaptive voxels |
| 先驗資訊 | initial pose trajectory |
| 可輸出幾何 | optimized scan poses with estimated pose covariance; registered point cloud |
| 計算需求 | offline batch; C++ on Ubuntu 20.04, desktop with Intel i7-10750H CPU and 16 GB RAM; LDLT solve with Eigen 3.3.7, termination at 50 iterations or updates below 1e-6 rad and 1e-6 m (Sec. IV; Sec. VI restates the Sec. IV condition with a 200-iteration limit, an internal inconsistency of the paper); solver complexity O(MfMp + MfMp^2 + Mp^3), independent of the number of points (Sec. III-E, Supplementary IV); real-world scans downsampled from 10 Hz to 2 Hz (Sec. VI-B); on the 19 real sequences the double-precision solver needed about one fourth of BAREG's, one sixth of PA's, one eighth of BALM's and one twentieth of EF's optimization time, with single precision a further 40% faster (Sec. VI-B3, Table IV); as a sliding-window LIO back-end the local BA averaged 73 ms per optimization, keeping 10 Hz (Supplementary I-A, Table V) |
使用設備
原文使用的感測器、運算硬體與載具（equipment）。型號保留原文寫法，連結到設備頁中同一型號的歸併名稱；角色依原文用途分為方法輸入、資料集感測器、執行運算平台、參考或真值量測（reference or ground truth）與比較對象設備。
| 類別 | 型號（原文寫法） | 角色 | 資料集 | 原文規格 | 出處 |
|---|---|---|---|---|---|
| LiDAR | Ouster OS0-64 | 資料集感測器 | Hilti SLAM Challenge 2021 | handheld; lidar data of the Hilti dataset | (Liu et al., 2023a, Sec. VI-B) |
| LiDAR | OS1 16-channel (two units)歸入：Ouster OS1-16 | 資料集感測器 | NTU VIRAL | one horizontal and one vertical on a UAV; only the horizontal one used | (Liu et al., 2023a, Sec. VI-B) |
| LiDAR | Velodyne HDL 32E歸入：Velodyne HDL-32E | 資料集感測器 | UrbanLoco | car driving on urban streets | (Liu et al., 2023a, Sec. VI-B) |
| LiDAR | MID-100 | 方法輸入 | multi-lidar calibration data of [8] and [9] | three internal lidars L0, L1, L2 with 8.4 deg FoV overlap between adjacent ones; in-factory extrinsics used as ground truth | (Liu et al., 2023a, Supplementary I-B, Fig. 13) |
| LiDAR | AVIA歸入：Livox Avia | 方法輸入 | multi-lidar calibration data of [8] | lidar L3 without FoV overlap with MID-100 | (Liu et al., 2023a, Supplementary I-B, Fig. 13) |
| LiDAR | 16-channel lidar (simulated) | 方法輸入 | authors' simulation | 28,800 points per scan, 100 scans along a 92 m rectangular trajectory in a 30 m x 20 m x 8 m semi-closed space | (Liu et al., 2023a, Sec. V, Fig. 5) |
| GNSS 接收器 | Novatel SPAN-CPT | 參考或真值量測 | UrbanLoco | navigation system combining RTK and precise IMU; ground truth (authors note false sudden jumps) | (Liu et al., 2023a, Sec. VI-B) |
| 全測站 | total station (model not reported) | 參考或真值量測 | Hilti SLAM Challenge 2021 | ground-truth trajectory for some Hilti sequences | (Liu et al., 2023a, Sec. VI-B) |
| 全測站 | Leica Nova MS60 MultiStation | 參考或真值量測 | NTU VIRAL | tracks a crystal prism on the UAV for ground-truth positions | (Liu et al., 2023a, Sec. VI-B) |
| 載具平台 | customized multi-sensor vehicle platform of [8] | 方法輸入 | multi-lidar calibration data of [8] | rotated one full cycle to create co-visible features | (Liu et al., 2023a, Supplementary I-B, Fig. 13) |
| 運算硬體 | Intel i7-10750H | 執行運算平台 | 未標示 | desktop; 16 GB RAM (written '16Gb RAM'); Ubuntu 20.04 | (Liu et al., 2023a, Sec. IV) |
| 其他 | motion capture system (model not reported) | 參考或真值量測 | Hilti SLAM Challenge 2021 | ground-truth trajectory for other Hilti sequences | (Liu et al., 2023a, Sec. VI-B) |
作者報告的優勢與限制
優勢
- Avoids per-point enumeration in cost, derivative and uncertainty evaluation (abstract; Sec. VIII-A)
- Estimates pose uncertainty from measurement noise; normalized NEES stayed close to one for point noise up to 0.3 m over 2000 Monte Carlo runs (abstract; Sec. III-F; Sec. V, Fig. 7)
- Directly minimizes point-to-feature residuals, which the authors argue reinforces map consistency more directly than pose-graph optimization (Sec. VIII-B)
- Default double-precision solver had a lower ATE than every non-BALM2 method on all 19 Hilti, VIRAL and UrbanLoco sequences (its float or edge variants were lower still on a few sequences), average 0.1763 m versus 0.221 m for BALM and 0.268 m for EF, and fewer occupied map cells than every non-BALM2 method (Ours (edge) was lower on Campus2) (Sec. VI-B1, VI-B2, Tables II-III)
- On Hilti UzhArea2, wall points registered with BA-optimized poses formed a plane with 1.7 cm standard deviation versus 15.3 cm with the provided ground-truth poses (Sec. VI-B1, Fig. 10)
- Global BA after MULLS pose-graph output lowered KITTI mean ATE from 1.63 m to 1.34 m (Supplementary I-C, Table VII)
限制
- All scans are optimized at once; authors downsampled 10 Hz data to 2 Hz because 10 Hz was prohibitively costly for all BA methods (Sec. VI-B)
- Higher computation cost than pose-graph optimization; positioned as accuracy refinement from a baseline trajectory (Sec. VIII-B)
- Covariance consistency degrades beyond about 0.3 m point noise, where the first-order approximation no longer holds (Sec. V)
- Adding edge features gave no accuracy gain on real data because lidar edge points are noisy (Sec. VI-B1)
- Merging planes to cut feature count reduces time but raises pose RMSE because slightly curved surfaces get merged (Sec. VI-B4, Fig. 13)
- Motion compensation, dynamic-object removal, tighter fusion with IMU or camera and loop closure are left to future integration (Sec. IX)
- Code README warns that large initial pose errors can leave too few detected planes for association and suggests coarse-to-fine re-association (github README)
- (inference) Covariance reflects measurement noise under correct association; calibration, timing or association errors are not represented
營建工程相關證據
以 Hilti 2021 公開資料集評估，其中含 Construction2 工地序列（手持 Ouster OS0-64，軌跡真值來自全測站或動作擷取系統；Sec. VI-B）。在共同初始軌跡下，Construction2 的預設 BALM2 ATE 為 0.0553 m，BALM 為 0.068 m，EF 為 0.086 m（Table II）。此屬公開資料層級的工地證據，並非作者自行於工地部署；地圖品質以無參考之佔用格數（0.1 m 格）評估，未與 TLS 或構件尺寸比對。UzhArea2 牆面點以 BA 位姿配準後平面標準差為 1.7 cm，作者視為達到光達測距雜訊等級（Sec. VI-B1、Fig. 10）。
原文驗證環境：模擬、公開基準、施工中工地、獨立參考量測
報告的性能數據
以下是原文作者報告的性能數值（author-reported results），不是本研究重新量測的結果。每張圖只並列同一個比較組（comparison group，同一張表、同一組實驗設定）內的方法；不同比較組之間的數值不可直接比較，也不構成排名。
本方法共出現在 7 個比較組，合計 64 筆紀錄。以下列出本方法紀錄最多的 4 組，其餘 3 組列在最後，並連到性能比較頁。
Liu et al., 2023a · Table II 本方法 34 筆
表格設定（擷取紀錄原文）：ATE RMSE (m) of multi-view registration; scans deskewed by FAST-LIO2 (odometry output discarded) and downsampled from 10 Hz to 2 Hz; ICP, GICP, NDT from PCL run incrementally against the last 20 scans; the ICP trajectory is the common initialization and adaptive voxelization (root voxel 1 m Hilti, 2 m VIRAL and UrbanLoco) the common association for EF, BALM, PA and Ours (BAREG uses its own); plane features only except Ours (edge); ground truth: Hilti total station or motion capture, VIRAL Leica Nova MS60, UrbanLoco Novatel SPAN-CPT RTK/INS. Column order verified from PDF layout: Ours (float), Ours (edge), Ours (Liu et al., 2023a, Table II)
Absolute trajectory error (RMSE, meters)，Hilti SLAM Challenge 2021 · Basement1
只並列這張表在相同設定下報告的方法；以「本方法：」開頭者為本頁方法。失敗、未執行與未報告以標記呈現，不是 0。
按 Tab 進入圖表後，用上下方向鍵逐一瀏覽各類別，Esc 關閉提示框；也可開啟表格檢視閱讀全部數值。
這些是 Liu et al., 2023a 在此表設定下報告的數值（author-reported results），只能在同一個比較組內對照，不代表方法在其他資料或設定下的表現。
資料來源作者報告值(Liu et al., 2023a, Table II)
| 方法（原文寫法） | 報告值 | 出處 |
|---|---|---|
| ICP (PCL, incremental) | 0.058 m | (Liu et al., 2023a, Table II) |
| GICP (PCL, incremental) | 0.063 m | (Liu et al., 2023a, Table II) |
| NDT (PCL, incremental) | 0.076 m | (Liu et al., 2023a, Table II) |
| EF | 0.047 m | (Liu et al., 2023a, Table II) |
| BALM | 0.042 m | (Liu et al., 2023a, Table II) |
| PA | 0.038 m | (Liu et al., 2023a, Table II) |
| PA (inner) | 0.036 m | (Liu et al., 2023a, Table II) |
| BAREG | 0.04 m | (Liu et al., 2023a, Table II) |
| Ours (float)本方法原文提出 | 0.0359 m | (Liu et al., 2023a, Table II) |
| Ours (edge)本方法原文提出 | 0.0361 m | (Liu et al., 2023a, Table II) |
| Ours本方法原文提出 | 0.0353 m | (Liu et al., 2023a, Table II) |
Wang et al., 2026 · Table III 本方法 12 筆
表格設定（擷取紀錄原文）：Single-robot study on self-collected Mid360 data; spatial BA versus BALM2 (sliding window) and HBA, all on raw odometry without loop-based refinement; z-drift and z-RMSE relative to the z-value of the first frame; MME via MapEval (lower is better) (Wang et al., 2026, Table III)
MME，self-collected (Mid360) · Garage
只並列這張表在相同設定下報告的方法；以「本方法：」開頭者為本頁方法。失敗、未執行與未報告以標記呈現，不是 0。
按 Tab 進入圖表後，用上下方向鍵逐一瀏覽各類別，Esc 關閉提示框；也可開啟表格檢視閱讀全部數值。
這些是 Wang et al., 2026 在此表設定下報告的數值（author-reported results），只能在同一個比較組內對照，不代表方法在其他資料或設定下的表現。
資料來源作者報告值(Wang et al., 2026, Table III)
| 方法（原文寫法） | 報告值 | 出處 |
|---|---|---|
| HBA | -6.83 | (Wang et al., 2026, Table III) |
| BALM2本方法 | -6.93 | (Wang et al., 2026, Table III) |
| Ours (spatial BA)原文提出 | -6.86 | (Wang et al., 2026, Table III) |
Liu et al., 2023a · Table IV 本方法 6 筆
指標Optimization time (total, unit not stated)
表格設定（擷取紀錄原文）：Total optimization time of the BA methods on the Table II inputs (pairwise methods excluded); the table does not state the time unit; desktop Intel i7-10750H, 16 GB RAM (Sec. IV) (Liu et al., 2023a, Table IV)
Optimization time (total, unit not stated)，Hilti SLAM Challenge 2021 · Construction2
只並列這張表在相同設定下報告的方法；以「本方法：」開頭者為本頁方法。失敗、未執行與未報告以標記呈現，不是 0。
按 Tab 進入圖表後，用上下方向鍵逐一瀏覽各類別，Esc 關閉提示框；也可開啟表格檢視閱讀全部數值。
這些是 Liu et al., 2023a 在此表設定下報告的數值（author-reported results），只能在同一個比較組內對照，不代表方法在其他資料或設定下的表現。
資料來源作者報告值(Liu et al., 2023a, Table IV)
| 方法（原文寫法） | 報告值 | 出處 |
|---|---|---|
| EF硬體：desktop Intel i7-10750H, 16 GB RAM | 1415.18 | (Liu et al., 2023a, Table IV) |
| BALM硬體：desktop Intel i7-10750H, 16 GB RAM | 412 | (Liu et al., 2023a, Table IV) |
| PA硬體：desktop Intel i7-10750H, 16 GB RAM | 335.7 | (Liu et al., 2023a, Table IV) |
| PA (inner)硬體：desktop Intel i7-10750H, 16 GB RAM | 313.23 | (Liu et al., 2023a, Table IV) |
| BAREG硬體：desktop Intel i7-10750H, 16 GB RAM | 231.48 | (Liu et al., 2023a, Table IV) |
| Ours (float)本方法原文提出硬體：desktop Intel i7-10750H, 16 GB RAM | 33.04 | (Liu et al., 2023a, Table IV) |
| Ours (edge)本方法原文提出硬體：desktop Intel i7-10750H, 16 GB RAM | 47.34 | (Liu et al., 2023a, Table IV) |
| Ours本方法原文提出硬體：desktop Intel i7-10750H, 16 GB RAM | 47.12 | (Liu et al., 2023a, Table IV) |
Liu et al., 2023b · Table VIII 本方法 6 筆
表格設定（擷取紀錄原文）：MulRan DCC sequences with LIO-SAM loop-closed input ('Initial'); original BA, reduced block-diagonal BA and proposed hierarchical BA; RMSE of ATE and total optimization time; hardware not reported (Liu et al., 2023b, Table VIII)
RMSE of ATE，MulRan · DCC01
只並列這張表在相同設定下報告的方法；以「本方法：」開頭者為本頁方法。失敗、未執行與未報告以標記呈現，不是 0。
按 Tab 進入圖表後，用上下方向鍵逐一瀏覽各類別，Esc 關閉提示框；也可開啟表格檢視閱讀全部數值。
這些是 Liu et al., 2023b 在此表設定下報告的數值（author-reported results），只能在同一個比較組內對照，不代表方法在其他資料或設定下的表現。
資料來源作者報告值(Liu et al., 2023b, Table VIII)
| 方法（原文寫法） | 報告值 | 出處 |
|---|---|---|
| Initial (LIO-SAM with loop closure) | 5.67 m | (Liu et al., 2023b, Table VIII (version of record)) |
| Original BA本方法 | 5.17 m | (Liu et al., 2023b, Table VIII (version of record)) |
| Reduced BA | 5.66 m | (Liu et al., 2023b, Table VIII (version of record)) |
| Proposed原文提出 | 5.19 m | (Liu et al., 2023b, Table VIII (version of record)) |
其他比較組
來源
Liu et al., 2023a
（2023）Efficient and Consistent Bundle Adjustment on Lidar Point CloudsIEEE Transactions on Robotics, 39(6):4366-4386
DOI 10.1109/tro.2023.3311671arXiv 2209.08854程式碼
同儕審查已出版已讀全文近十年
相關版本
- 預印本：arXiv 2209.08854 (v2, 2024-06-16) https://arxiv.org/abs/2209.08854
- 程式碼釋出：BALM 2.0 in hku-mars/BALM https://github.com/hku-mars/BALM
程式碼：https://github.com/hku-mars/BALM（授權：GPL-2.0 (LICENSE file)）。有公開程式碼不等於已被重現，也不代表目前版本與論文版本相同。