[{"data":1,"prerenderedAt":397},["ShallowReactive",2],{"method-dellaert2006sqrtsam":3},{"method":4,"reference":51,"equipment":70,"figures":95,"results":96},{"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":22,"limitations":27,"sensors":34,"platform":37,"estimator":39,"association":40,"timeModel":41,"deskew":42,"loopClosure":43,"globalOptimization":44,"mapRepresentation":45,"prior":46,"outputGeometry":47,"compute":48,"codeUrl":49,"codeLicense":42,"relatedVersions":50},"dellaert2006sqrtsam","Dellaert & Kaess, 2006","Square Root SAM","Square Root SAM: Simultaneous Localization and Mapping via Square Root Information Smoothing",2006,"classic","C03","estimation_framework_or_library","本文把平滑（smoothing）視為 EKF 型 SLAM 的替代方案，將資訊矩陣或量測 Jacobian 分解為平方根形式求解。作者主張此類方法精確且更快，可批次或增量使用，較能處理非線性的運動與量測模型，並能以較低成本得到整條軌跡。欄位排序啟發式能間接利用 SLAM 問題在地理上的局部性。","Square-root information smoothing for SLAM factorizes the information matrix or Jacobian, offering an exact and often faster alternative to EKF-SLAM that recovers the full trajectory.","full_text_reviewed","peer_reviewed_published","background","未在營建場域驗證；真實實驗在既有辦公建物室內（軌跡外框約 30 m × 50 m、總長約 190 m、260 組影像），僅與手動對齊的建物平面圖目視比對，無量化幾何精度 [Sec. 8, Fig. 16]。",[20,21],"simulation","completed_building",[23,24,25,26],"Faster yet exact relative to EKF, usable in batch or incremental mode (abstract)","Column ordering heuristics exploit locality of SLAM (abstract)","Incremental smoothing became cheaper than the EKF once about 600 landmarks had been seen in simulation (Sec. 7.2)","Block-structured ordering gave a 15-fold LDL speed-up over colamd alone in the real experiment and reduced non-zeros in R from about 2.8 million (XL ordering) to about 130K in simulation (Sec. 7.1, Sec. 8, Figs. 12 and 13)",[28,29,30,31,32,33],"Batch refactoring performs unnecessary computation when applied incrementally, as noted by kaess2008isam (Sec. I)","Computational complexity grows without bound because the entire trajectory is smoothed (Sec. 9, Fig. 17)","Recovering the joint covariance is expensive, although marginals are cheaper (Sec. 9)","No tight complexity bounds and no comparison with more recent approximate or exact SLAM methods (Sec. 9)","Data association was ignored (Sec. 9, Sec. 10)","In the real experiment the dominant cost was relinearizing the measurement Jacobian (453 evaluations), not factorization (Sec. 8)",[35,36],"8-camera rig (visual point features)","wheel odometry",[20,38],"wheeled UGV (iRobot ATRV-Mini)","square-root information smoothing by factorizing the information matrix (Cholesky or LDL) or the measurement Jacobian (QR), in batch or incremental mode; best performance with Davis' sparse LDL and colamd or symamd ordering applied to the block (pose and landmark) structure; non-linear problems are relinearized and refactorized at each call","real experiment: features matched between successive frames using RANSAC on a trifocal camera arrangement (Sec. 8); the formulation assumes data association is solved (Sec. 2) and the authors state they ignored data association (Sec. 10)","discrete poses","not_applicable","none occurred in the real experiment (Fig. 16 caption); fill-in when closing loops discussed for simulations (Sec. 7.2)","full trajectory and map smoothing","landmarks (simulated landmarks observed with bearing and range; 4383 unknown 3D points in the real experiment)","real experiment: zero-mean priors on height, pitch and roll (standard deviations 0.01 m and 0.02 rad) on the 6-DoF poses, because small floor bumps visibly affect the images in the planar indoor office (Sec. 8); formulation: the first pose x0 is treated as given and fixed at the origin, with a uniform prior over landmarks (Sec. 2; Sec. 3)","entire robot trajectory and map","simulations in MATLAB on a 2 GHz Pentium 4 workstation running Linux (Sec. 7); the real sequence (260 joint images, 17780 measurements, 4383 unknown points) was processed in 11 min 10 s on a 2 GHz Pentium-M laptop with batch SAM invoked every three joint images; at the end forming the information matrix took about 0.6 s and LDL factorization 0.1 s (Sec. 8, Fig. 17)",null,[],{"id":5,"kind":52,"shortName":7,"title":8,"authors":53,"year":9,"venue":56,"venueType":57,"publisher":58,"volumeIssuePages":59,"doi":60,"arxivId":49,"url":61,"firstPublicDate":62,"publicationStatus":16,"metadataStatus":63,"fulltextStatus":15,"era":10,"classicReason":64,"codeUrl":49,"cluster":11,"topics":65,"mdpi":66,"verification":67,"label":6,"fulltextRoute":68,"versionRead":69,"addedByCensus":66},"method",[54,55],"Frank Dellaert","Michael Kaess","The International Journal of Robotics Research","journal","SAGE","25(12):1181-1203","10.1177\u002F0278364906072768","https:\u002F\u002Fapi.crossref.org\u002Fworks\u002F10.1177\u002F0278364906072768","2006-12","metadata_verified","necessary technical node: batch square-root information smoothing that iSAM explicitly extends to the incremental case (kaess2008isam, Sec. I).",[11],false,"corrected","author copy","Authors' pre-publication copy marked 'To appear in the Intl. Journal of Robotics Research' (32 pages, Dellaert06ijrr.pdf); the SAGE version of record (IJRR 25(12):1181-1203) was not accessible in full",[71,77,81,86,91],{"category":72,"model":73,"canonical":73,"role":74,"dataset":49,"specs":75,"locator":76},"camera","FireWire cameras (eight, custom rig; model not reported)","method input","eight cameras distributed equally along a circle and connected to an on-board laptop; rig calibrated in advance; 260 joint images up to 2 m apart","Sec. 8; Fig. 15",{"category":78,"model":79,"canonical":79,"role":74,"dataset":49,"specs":80,"locator":76},"platform","iRobot ATRV-Mini","mobile robot carrying the camera rig",{"category":82,"model":83,"canonical":83,"role":74,"dataset":49,"specs":84,"locator":85},"wheel_or_leg_odometry","odometry provided by the ATRV-Mini robot","standard deviations 0.02 m on x and y and 0.02 rad on yaw","Sec. 8",{"category":87,"model":88,"canonical":88,"role":89,"dataset":49,"specs":90,"locator":85},"compute","2 GHz Pentium-M based laptop","compute for runtime","processed the entire real sequence in 11 min 10 s",{"category":87,"model":92,"canonical":92,"role":89,"dataset":49,"specs":93,"locator":94},"2 GHz Pentium 4 workstation running Linux","MATLAB simulations","Sec. 7.1",[],{"totalRows":97,"groupCount":98,"groups":99,"others":396},56,4,[100,286,329,367],{"slug":101,"group":102,"sourceId":5,"sourceLabel":6,"table":103,"selfRows":104,"metrics":105,"seqs":111,"entrants":138,"cells":150,"outcomes":278,"locators":279,"hardware":281,"wordings":283,"notes":284},"dellaert2006sqrtsam-fig-10-table","dellaert2006sqrtsam:Fig. 10 table","Fig. 10 table",48,[106],{"label":107,"unit":108,"statistic":109,"alignment":110},"computation time averaged over 10 trials","s","mean","none",[112,116,118,120,122,124,126,128,130,132,134,136],{"dataset":113,"sequence":114,"environment":115},"synthetic simulation","M = 200 poses, N = 180 landmarks","simulation (synthetic landmark environments)",{"dataset":113,"sequence":117,"environment":115},"M = 200 poses, N = 500 landmarks",{"dataset":113,"sequence":119,"environment":115},"M = 200 poses, N = 1280 landmarks",{"dataset":113,"sequence":121,"environment":115},"M = 200 poses, N = 2000 landmarks",{"dataset":113,"sequence":123,"environment":115},"M = 500 poses, N = 180 landmarks",{"dataset":113,"sequence":125,"environment":115},"M = 500 poses, N = 500 landmarks",{"dataset":113,"sequence":127,"environment":115},"M = 500 poses, N = 1280 landmarks",{"dataset":113,"sequence":129,"environment":115},"M = 500 poses, N = 2000 landmarks",{"dataset":113,"sequence":131,"environment":115},"M = 1000 poses, N = 180 landmarks",{"dataset":113,"sequence":133,"environment":115},"M = 1000 poses, N = 500 landmarks",{"dataset":113,"sequence":135,"environment":115},"M = 1000 poses, N = 1280 landmarks",{"dataset":113,"sequence":137,"environment":115},"M = 1000 poses, N = 2000 landmarks",[139,141,144,146,148],{"name":140,"methodId":49,"linkable":66,"proposed":66,"self":66},"none (no factorization; measures overhead)",{"name":142,"methodId":5,"linkable":143,"proposed":143,"self":143},"batch square-root SAM, ldl (Davis sparse LDL)",true,{"name":145,"methodId":5,"linkable":143,"proposed":143,"self":143},"batch square-root SAM, chol (MATLAB built-in Cholesky)",{"name":147,"methodId":5,"linkable":143,"proposed":143,"self":143},"batch square-root SAM, mfqr (multifrontal QR)",{"name":149,"methodId":5,"linkable":143,"proposed":143,"self":143},"batch square-root SAM, qr (MATLAB built-in QR)",[151,155,158,161,164,166,168,169,171,173,175,177,179,181,183,185,187,189,191,193,195,197,199,201,203,205,207,209,211,213,215,217,219,221,223,225,227,229,231,233,235,237,239,241,243,245,248,250,252,254,256,259,261,263,265,267,270,272,274,276],[152,152,152,153,154,152,152,154,152],0,0.031,-1,[156,152,152,157,154,152,152,154,152],1,0.062,[159,152,152,160,154,152,152,154,152],2,0.092,[162,152,152,163,154,152,152,154,152],3,0.868,[98,152,152,165,154,152,152,154,152],1.685,[152,152,156,167,154,152,152,154,152],0.034,[156,152,156,157,154,152,152,154,152],[159,152,156,170,154,152,152,154,152],0.094,[162,152,156,172,154,152,152,154,152],1.19,[98,152,156,174,154,152,152,154,152],1.256,[152,152,159,176,154,152,152,154,152],0.036,[156,152,159,178,154,152,152,154,152],0.068,[159,152,159,180,154,152,152,154,152],0.102,[162,152,159,182,154,152,152,154,152],1.502,[98,152,159,184,154,152,152,154,152],1.21,[152,152,162,186,154,152,152,154,152],0.037,[156,152,162,188,154,152,152,154,152],0.07,[159,152,162,190,154,152,152,154,152],0.104,[162,152,162,192,154,152,152,154,152],1.543,[98,152,162,194,154,152,152,154,152],1.329,[152,152,98,196,154,152,152,154,152],0.055,[156,152,98,198,154,152,152,154,152],0.176,[159,152,98,200,154,152,152,154,152],0.247,[162,152,98,202,154,152,152,154,152],2.785,[98,152,98,204,154,152,152,154,152],11.92,[152,152,206,157,154,152,152,154,152],5,[156,152,206,208,154,152,152,154,152],0.177,[159,152,206,210,154,152,152,154,152],0.271,[162,152,206,212,154,152,152,154,152],3.559,[98,152,206,214,154,152,152,154,152],8.43,[152,152,216,178,154,152,152,154,152],6,[156,152,216,218,154,152,152,154,152],0.175,[159,152,216,220,154,152,152,154,152],0.272,[162,152,216,222,154,152,152,154,152],5.143,[98,152,216,224,154,152,152,154,152],6.348,[152,152,226,188,154,152,152,154,152],7,[156,152,226,228,154,152,152,154,152],0.181,[159,152,226,230,154,152,152,154,152],0.279,[162,152,226,232,154,152,152,154,152],5.548,[98,152,226,234,154,152,152,154,152],6.908,[152,152,236,190,154,152,152,154,152],8,[156,152,236,238,154,152,152,154,152],0.401,[159,152,236,240,154,152,152,154,152],0.523,[162,152,236,242,154,152,152,154,152],10.297,[98,152,236,244,154,152,152,154,152],42.986,[152,152,246,247,154,152,152,154,152],9,0.109,[156,152,246,249,154,152,152,154,152],0.738,[159,152,246,251,154,152,152,154,152],0.945,[162,152,246,253,154,152,152,154,152],12.112,[98,152,246,255,154,152,152,154,152],77.849,[152,152,257,258,154,152,152,154,152],10,0.124,[156,152,257,260,154,152,152,154,152],0.522,[159,152,257,262,154,152,152,154,152],0.746,[162,152,257,264,154,152,152,154,152],14.151,[98,152,257,266,154,152,152,154,152],35.719,[152,152,268,269,154,152,152,154,152],11,0.126,[156,152,268,271,154,152,152,154,152],0.437,[159,152,268,273,154,152,152,154,152],0.657,[162,152,268,275,154,152,152,154,152],15.914,[98,152,268,277,154,152,152,154,152],25.611,[],[280],"Fig. 10 (tabulated values); Sec. 7.1",[282],"MATLAB on a 2 GHz Pentium 4 workstation running Linux",[],[285],"Batch square-root SAM in synthetic environments; time averaged over 10 trials for trajectory length M and N landmarks",{"slug":287,"group":288,"sourceId":5,"sourceLabel":6,"table":289,"selfRows":98,"metrics":290,"seqs":301,"entrants":308,"cells":313,"outcomes":322,"locators":323,"hardware":325,"wordings":326,"notes":327},"dellaert2006sqrtsam-text-sec-8","dellaert2006sqrtsam:Text Sec.8","Text Sec.8",[291,294,296,298],{"label":292,"unit":108,"statistic":293,"alignment":110},"total processing time for the entire sequence (11 min 10 s)","not_reported",{"label":295,"unit":108,"statistic":293,"alignment":110},"time to form the information matrix A^T A (hessian)",{"label":297,"unit":108,"statistic":293,"alignment":110},"time to factorize the information matrix (LDL)",{"label":299,"unit":300,"statistic":293,"alignment":110},"speed-up of LDL execution time from block-structured ordering relative to colamd alone","x",[302,306],{"dataset":303,"sequence":304,"environment":305},"authors' office sequence","entire sequence","completed building (indoor office, bounding box about 30 m by 50 m)",{"dataset":303,"sequence":307,"environment":305},"end of sequence (about 15K x 15K information matrix)",[309,311],{"name":310,"methodId":5,"linkable":143,"proposed":143,"self":143},"incremental (repeated batch) square-root SAM",{"name":312,"methodId":5,"linkable":143,"proposed":143,"self":143},"block-structured ordering heuristic",[314,316,318,320],[152,152,152,315,154,152,152,154,152],670,[152,156,156,317,154,156,152,154,152],0.6,[152,159,156,319,154,156,152,154,152],0.1,[156,162,152,321,154,152,154,154,152],15,[],[85,324],"Sec. 8; Fig. 17",[88],[],[328],"Real indoor office sequence: ATRV-Mini with eight cameras and odometry, 260 joint images, about 190 m trajectory; batch square-root SAM every three joint images with LDL and block-structured colamd ordering",{"slug":330,"group":331,"sourceId":5,"sourceLabel":6,"table":332,"selfRows":159,"metrics":333,"seqs":337,"entrants":340,"cells":349,"outcomes":358,"locators":359,"hardware":363,"wordings":364,"notes":365},"dellaert2006sqrtsam-text-figs-11-13","dellaert2006sqrtsam:Text Figs.11-13","Text Figs.11-13",[334],{"label":335,"unit":336,"statistic":293,"alignment":110},"number of non-zeros (approximate)","count",[338],{"dataset":113,"sequence":339,"environment":20},"M = 1000, N = 500",[341,343,345,347],{"name":342,"methodId":49,"linkable":66,"proposed":66,"self":66},"XL ordering (states then landmarks)",{"name":344,"methodId":5,"linkable":143,"proposed":143,"self":143},"colamd ordering",{"name":346,"methodId":5,"linkable":143,"proposed":143,"self":143},"block-structured colamd ordering",{"name":348,"methodId":49,"linkable":66,"proposed":66,"self":66},"EKF filtering covariance matrix (entries)",[350,352,354,356],[152,152,152,351,154,152,154,154,152],2800000,[156,152,152,353,154,152,154,154,152],250000,[159,152,152,355,154,156,154,154,152],130000,[162,152,152,357,154,159,154,154,152],500000,[],[360,361,362],"Sec. 7.1; Fig. 12 caption","Sec. 7.1; Fig. 13 caption","Fig. 13 caption",[],[],[366],"Synthetic 1000-step random walk in a 500-landmark Manhattan world (Fig. 9): non-zeros in the Cholesky factor R for different column orderings, compared with the filtering covariance matrix",{"slug":368,"group":369,"sourceId":5,"sourceLabel":6,"table":370,"selfRows":159,"metrics":371,"seqs":377,"entrants":382,"cells":385,"outcomes":389,"locators":390,"hardware":392,"wordings":393,"notes":394},"dellaert2006sqrtsam-text-sec-7-2","dellaert2006sqrtsam:Text Sec.7.2","Text Sec.7.2",[372,375],{"label":373,"unit":374,"statistic":293,"alignment":110},"number of landmarks seen when smoothing every step becomes cheaper than the EKF","landmarks",{"label":376,"unit":108,"statistic":293,"alignment":110},"time per factorization at the end of the run",[378,380],{"dataset":113,"sequence":379,"environment":20},"500 steps, 2000-landmark environment",{"dataset":113,"sequence":381,"environment":20},"end of run, N = 1100 landmarks seen",[383],{"name":384,"methodId":5,"linkable":143,"proposed":143,"self":143},"incremental square-root SAM (LDL)",[386,388],[152,152,152,387,154,152,154,154,152],600,[152,156,156,317,154,152,152,154,152],[],[391],"Sec. 7.2; Fig. 14",[282],[],[395],"Incremental square-root SAM versus a standard EKF, 500 time steps in a synthetic environment with 2000 landmarks (sparse LDL with symamd ordering)",[],1790510661953]