Horn absolute orientation
給定兩座標系中三個以上不共線的對應點，作者提出最小平方意義下的閉式解：平移為一組點的形心與另一組點經旋轉、縮放後之形心的差；若採作者建議的對稱誤差式，尺度為兩組點相對形心的均方根偏差之比，且不需先求旋轉；旋轉以單位四元數表示，為一個 4×4 對稱矩陣最大正特徵值對應的特徵向量。作者也指出，若某一組座標的精度遠高於另一組，採非對稱的尺度式可能較合適；附錄說明可加入權重以反映不同點的量測可信度。
本頁內容
Closed-form least-squares rotation, translation and scale between corresponding point sets using unit quaternions.
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 | 未記錄 |
|---|---|
| 原文測試平台 | 未記錄 |
| 狀態估計 | closed-form least squares (unit quaternion eigenvector) |
| 資料關聯 | known point correspondences |
| 時間表示 | 不適用 |
| 去畸變 | 不適用 |
| 迴圈閉合 | 不適用 |
| 全域最佳化 | 不適用 |
| 地圖表示 | 不適用 |
| 先驗資訊 | 不適用 |
| 可輸出幾何 | 不適用 |
| 計算需求 | 不適用 |
使用設備
尚未收錄此方法的設備紀錄；設備資料仍在分批查證，沒有紀錄不代表原文未使用任何設備。
作者報告的優勢與限制
優勢
- Exact closed-form result preferred to approximate methods (abstract)
- Symmetric treatment of scale (scale section)
- Weights can be incorporated (Appendix A2)
限制
- Assumes known point correspondences and equal measurement quality unless weights are introduced (Appendix A2)
- Requires at least three non-collinear points; two points do not provide enough constraint (Sec. 1.B, 2.A)
- The optimal scale depends on the chosen error term: two asymmetric forms and one symmetric form give different scales; the symmetric form is preferred unless one coordinate set is known much more precisely (Sec. 2.D, 2.E, Appendix A1)
- The rotation is unique only if the most positive eigenvalue of the 4x4 matrix is distinct (Appendix A3)
- (inference, corrected) The reflection failure described in Umeyama (1991) does not apply to this unit-quaternion solution: Horn shows that the composite product preserves cross products, so the result is a rotation, not a reflection (Sec. 3.E, proof in Appendix A6). Umeyama's abstract criticises 'Horn et al.', and Umeyama's reference list contains both this paper and Horn, Hilden & Negahdaripour (1988, orthonormal-matrix method, 10.1364/JOSAA.5.001127); the criticism most plausibly targets the latter and the SVD method of Arun et al.
營建工程相關證據
原文未報告（通用座標轉換方法）
報告的性能數據
性能數據仍在分批查證，目前尚未收錄此方法的報告值。
來源
Horn, 1987
（1987）Closed-form solution of absolute orientation using unit quaternionsJournal of the Optical Society of America A, 4(4):629-642
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