[{"data":1,"prerenderedAt":97},["ShallowReactive",2],{"method-umeyama1991least":3},{"method":4,"reference":36,"equipment":54,"figures":55,"results":56},{"id":5,"label":6,"shortName":7,"title":8,"year":9,"era":10,"cluster":11,"scope":12,"keyIdeaZh":13,"keyIdeaEn":14,"fulltextStatus":15,"publicationStatus":16,"recommendation":17,"constructionRelevance":18,"validationEnvironment":19,"strengths":20,"limitations":24,"sensors":28,"platform":29,"estimator":30,"association":31,"timeModel":32,"deskew":32,"loopClosure":32,"globalOptimization":32,"mapRepresentation":32,"prior":32,"outputGeometry":32,"compute":32,"codeUrl":33,"codeLicense":34,"relatedVersions":35},"umeyama1991least","Umeyama, 1991","Umeyama alignment","Least-squares estimation of transformation parameters between two point patterns",1991,"classic","C10","evaluation_method_or_metric","作者針對 m 維空間中已知對應關係的兩組點，推導使均方誤差最小的相似轉換（旋轉 R、平移 t、尺度 c）閉式解：先求兩組點的平均向量、變異數與交叉共變異矩陣，再對共變異矩陣做奇異值分解，並在其行列式為負時把對角符號矩陣 S 的最後一項設為 -1，以確保得到真正的旋轉而非反射；尺度與平移再由 S、奇異值與平均向量直接算出。作者指出 Arun 與 Horn 的解相當於不論行列式正負都取 S 為單位矩陣，在資料嚴重受擾時可能給出反射；本文解法適用任意維度，而四元數法僅適用三維。數值例中舊解以反射達成零誤差，新解則回傳誤差 0.533 的正常旋轉。","Closed-form least-squares similarity transform (R, t, c) between corresponding point sets in any dimension, via SVD of the cross-covariance with a sign correction that guarantees a proper rotation; Arun and Horn's solution corresponds to ignoring this sign.","full_text_reviewed","peer_reviewed_published","background","not_reported（通用方法）；Hilti-Oxford 以 SE(3) Umeyama 對齊控制點後計分 [zhang2023hiltioxford]。",[],[21,22,23],"Always yields a proper rotation even with corrupted data (abstract, Sec. IV)","Closed form valid in any dimension, whereas the quaternion method is limited to 3D (Sec. IV)","Gives the minimum mean squared error in closed form (Eq. 33)",[25,26,27],"Requires known point correspondences (problem statement, Sec. I)","Unique solution requires rank(Sigma_xy) >= m-1; the author states this holds with more than two distinct points in 2D and more than three non-collinear points in 3D (Sec. IV)","(inference) Plain least squares with equal weights: no robustness to outlier correspondences and no uncertainty output",[],[],"closed-form least squares: SVD of the cross-covariance matrix Sigma_xy = U D V^T; R = U S V^T with S = diag(1,...,1,-1) when det(Sigma_xy) \u003C 0 (or det(U)det(V) = -1 when rank = m-1), c = tr(DS)\u002Fsigma_x^2, t = mu_y - c R mu_x; minimum error sigma_y^2 - tr(DS)^2\u002Fsigma_x^2","known point correspondences","not_applicable",null,"not_verified",[],{"id":5,"kind":37,"shortName":7,"title":8,"authors":38,"year":9,"venue":40,"venueType":41,"publisher":42,"volumeIssuePages":43,"doi":44,"arxivId":33,"url":45,"firstPublicDate":46,"publicationStatus":16,"metadataStatus":47,"fulltextStatus":15,"era":10,"classicReason":48,"codeUrl":33,"cluster":11,"topics":49,"mdpi":50,"verification":51,"label":6,"fulltextRoute":52,"versionRead":53,"addedByCensus":50},"component",[39],"Shinji Umeyama","IEEE Transactions on Pattern Analysis and Machine Intelligence","journal","IEEE","13(4):376-380","10.1109\u002F34.88573","https:\u002F\u002Fdoi.org\u002F10.1109\u002F34.88573","1991-04","metadata_verified","evaluation-calibration method: the SE(3)\u002FSim(3) alignment named explicitly in the evaluation protocols of BAD SLAM, Hilti-Oxford, Tanks and Temples, Oxford Spires (SE(3) for ATE in Sec. 6.1.1, Sim(3) scale for COLMAP outputs in Sec. 5.1.5) and SubT-Tunnel (map-to-surveyed-frame alignment from AprilTags, Sec. III).",[11],false,"confirmed","NTU institutional (curl)","Version of record, IEEE Xplore scanned PDF of TPAMI 13(4):376-380 (text read from page images)",[],[],{"totalRows":57,"groupCount":57,"groups":58,"others":96},1,[59],{"slug":60,"group":61,"sourceId":5,"sourceLabel":6,"table":62,"selfRows":57,"metrics":63,"seqs":69,"entrants":74,"cells":80,"outcomes":86,"locators":89,"hardware":92,"wordings":93,"notes":94},"umeyama1991least-text-sec-iii","umeyama1991least:Text Sec. III","Text Sec. III",[64],{"label":65,"unit":66,"statistic":67,"alignment":68},"least mean squared error e2 of the returned transform","unitless (squared distance)","mean","none",[70],{"dataset":71,"sequence":72,"environment":73},"numerical example (Fig. 1)","3 point pairs in 2D","synthetic",[75,77],{"name":76,"methodId":33,"linkable":50,"proposed":50,"self":50},"Arun and Horn's method (equivalent to S = I)",{"name":78,"methodId":5,"linkable":79,"proposed":79,"self":79},"proposed closed-form solution (Theorem, Eq. 40-43)",true,[81,84],[82,82,82,82,82,82,83,83,82],0,-1,[57,82,82,85,57,57,83,83,82],0.533,[87,88],"other: perfect fit but R = diag(-1, 1) is a reflection, not a rotation (Eq. 63, Fig. 2)","other: proper rotation returned (R with entries 0.832 and 0.555, c = 0.721, t = (-0.800, 0.400)), Eq. 64, Fig. 3",[90,91],"Sec. III, Eq. 63","Sec. III, Eq. 64",[],[],[95],"Numerical example with three 2D point pairs (Fig. 1); least mean squared error of the returned similarity transform.",[],1790510666008]