Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples of samplng patterns Range mages and range surfaces Gven a range mage, we can perform a prelmnary reconstructon known as a range surface. 3 4
essellaton threshold o avod prematurely aggressve reconstructon, a tessellaton threshold s employed: Regstraton Any surface reconstructon algorthm strves to use all of the detal n the range data. o preserve ths detal, the range data must be precsely regstered. Accurate regstraton may requre: Calbrated scanner postonng Software optmzaton Both 5 6 Regstraton Problem: gven two overlappng range scans, what s the rgd transformaton,, that mnmzes the dstance between them. Least squares error How do we measure ths dstance? If we thnk of surfaces, we can pose a least squares problem n ntegral form, somethng lke: E = p ( u, v) q ( u, v) dudv where p(u,v) and q(u,v) are correspondng ponts on P and Q, respectvely. Alternatvely, we can wrte out a sampled verson of ths: E = p q where p and q are correspondng samples on P and Q, respectvely. 7 8
Soluton to least squares problem A dervaton due to Horn shows that there s a closed form soluton to the problem of fndng the that mnmzes: E = p q hs soluton s for the class of s that permt scale, rotaton, and translaton. We ll just allow the latter two (rgd body transformatons): E = p ( Rq + t) q Soluton to least squares problem o solve, we frst compute the centrod of each pont set: Horn showed that the best rotaton satsfes: In other words: p = p q = q 1. Convert the ponts nto vectors relatve to ther centrods.. Fnd a rotaton that makes correspondng vectors have dot products as close to 1 as possble. argmax ( p p) R( q q) R p 9 Centrod relatve Common orgn 10 Soluton to least squares problem o solve for ths rotaton, you can construct a 3x3 matrx: and then solve: M = ( p p)( q q) Correspondences: closest ponts So, we now have a closed form soluton for gven correspondng p and q. How do we get these correspondences?? One soluton s to fnd the nearest ponts to q that le on P. R= M( M M) 1/ whch amounts to solvng an egenvalue problem for a 3x3 matrx. ote that the resultng p can le on faces, edges, and vertces of P. he optmal translaton s then just: t= p Rq Alternatves nclude nearest pont: along the drecton of the normal at q along a fxed drecton 11 1
Iterated Closest Pont (ICP) After fndng the best based on these correspondences, we wll have brought the surfaces closer together, but not all the way. How do we go the rest of the way? Iterate! untl E s small Identfy nearest ponts Compute the optmal end untl hs procedure, called Iterated Closest Pont (ICP), was developed by Besl and McKay. Sprngs that slow convergence One shortcomng of the ICP method s slow convergence. We can thnk of a least squares soluton as: 1. ackng a bunch of sprngs between ponts. Requrng ther rest lengths to be zero 3. Solvng for the lowest energy confguraton If many of the ponts are near each other, but should slde past each other, the sprngs wll resst: Q: What knds of practcal problems do you thnk you wll encounter when algnng two range scans? Q: how mght you speed ths up? 13 14 Sldng sprngs Chen and Medon proposed an alternate error functon that does not penalze sldng. In partcular, at each closest pont, p, the normal defnes a tangent plane: n ( x p ) = 0 Sgned dstance from ths plane s smply: d ( x) = n ( x p ) Sldng sprngs here s no known closed soluton for n ths case, but t can be solved quckly n a few lnear subteratons. he algorthm otherwse proceeds as ICP. Result: faster convergence. hs was the method of choce for the Dgtal Mchelangelo Project. he error functon can now be wrtten n terms of square dstances from planes: E = n ( q p) q x n 15 16
Error accumulaton Consder a set of scans around an object. Wth each parwse regstraton you get a least squares optmal transformaton. Wll ths transformaton brng the range data nto perfect algnment? What happens when you come full crcle and compare scan -1 to scan 0? Global regstraton he problem now becomes: fnd the set of transformatons that smultaneously mnmzes dstances between range scans. hs s sometmes called the global regstraton problem. One soluton s to defne a new global error functon and solve for the best j n: E = M M jk p j k p j j k k where: M s the number of scans jk s the number of ponts n correspondence between scans j and k j s the transformaton for scan j p j s the -th pont from the j-th scan Can ntalze wth parwse ICP and then perform a large, global, non-lnear ICP. 17 18 Global regstraton For the Dgtal Mchelangelo Project, Kar Pull developed a smpler, faster verson of global ICP. One suggested approach: Perform parwse regstraton. Save a sub-sampled best set of parwse correspondences. untl convergence Select next scan j Compute the optmal j w.r.t. E j : Global regstraton Pull modfes ths to: keep the orgnal parwse transforms, jk substtute p k wth jk p j he error functon at each step s then: E M jk = p p j j j k j k j k end untl M jk E = p p j j j k k k 19 0
on-rgd regstraton Bblography Calbratng scanners can be extremely dffcult. he DMP scanner was not 100% calbrated. How to compensate? Soluton: fold non-lnear scanner parameters nto some of the regstraton procedures. Q: Is there an analagous problem n computer vson? Besl, P.J. and McKay, H.D., A method for regstraton of 3-D shapes, IEEE ransactons on Pattern Analyss and Machne Intellgence, Feb. 199, (4), pp. 39--56. Chen, Y. and Medon, G., Object modelng by regstraton of multple range mages, Image and Vson Computng, 10(3), Aprl, 199, pp. 145-155. Horn BKP, Hlden HM, egahdarpour S, "Closed-form Soluton of Absolute Orentaton Usng Orthonormal Matrces," Journal of the Optcal Socety of Amerca, Seres A, 5, 7, 1988, pp 117-1135. K. Pull. Multvew Regstraton for Large Data Sets, Int.Conf. on 3D Dgtal Imagng and Modelng, Ottawa, pp.160-168, 1999 1