1) “Geometry of the squared distance function to curves and surfaces,” Pottmann et. al, and 2) “Registration without ICP,”1) Geometry of Distance function…Another example of evolution of SurfaceTrimming ExamplesBlue Quadratic Surface has second order contact with Moulding SurfaceLocal quadratic approximation of distance to curve: distance to the osculating circleTaylor Series expansion around PQuadratic Approx to Distance function for Surfaces2) Registration without ICPFlawed for close points!First Linearize “m” as much as possibleModification to ICPCaveatsResultsComparison to ICPCommentFuture Directions prescribed by authors1) “Geometry of the squared distance function to curves and surfaces,” Pottmann et. al, and 2) “Registration without ICP,” Pottmann et. alSummaries by Joyoni Dey, Oct 22, 20031) Geometry of Distance function…Planar Case: Squared distance function of points in a plane to a curve in the planeNormal plane cuts a parabola on “Moulding” Surface ΦFrenet system instantenously rotatesabout the axis of “osculating circle”.Normal plane rolls along “evolute” Cylinder with rulings parallel to z-axisΛuu^2PAnother example of evolution of SurfaceVertical line may intersect surface at several points:1) Self-intersections need to be trimmed, 2) Multi-valued cases, smallest height points retained (visibility test)Trimming ExamplesSelf-intersection and Multi-valued PPIf the curve is a circle, all points equidistant to center, P. Distance surface will self-intersect at PAnother curve where severalcurve normals can pass through Point P; take smallest distanceBlue Quadratic Surface has second order contact with Moulding SurfaceLocal quadratic approximation of distance to curve: distance to the osculating circle2)^(2^||1 21ρρ−+−= xxdPdd1(x1,x2)CORadius = ρx2QDistance of a local point Q (x1,x2) to the Osculating circle given by d1x1Hence the quadratic approximation of the distance function is:Taylor Series expansion around PFor second order approximation of distance functionis just distance to the tangent plane (dotted line in diagram of previous slide)0≈dQuadratic Approx to Distance function for Surfaces Again for second order approximation of distance functionis just distance to the tangent plane 0≈d2) Registration without ICP Registering a CAD surface model to a cloud of points ICP Algorithm:- Find closest corresponding points - Transform Rigidly (rotation + translation) by m and minimize squared distance:iyixNote: Minimization can be done explicitlyFlawed for close points!Q m(xi)yiRadius = ρxiOkay when points are far away…But when points are close to the surface, ICP takes point-to-point distance CQ when one should take point-to-plane distance RQ.C RConsequences:•Points will tend to cluster around“footpoints” yi•convergence alongtangent plane will be slowFirst Linearize “m” as much as possibleThe rigid-body transform is broken into a helical motion and then a very small rotation + translation. Optimization done only with respect to helical motion.Helical motion xi Æ xi+v(xi)Rotation+translation xi+v(xi) Æ xiModification to ICPStep 1: Find yi, closest points to xi on surfaceStep 2: Minimize distance to tangent planewrt to “linear” part of the transformation –find the c and c (6 parameters again). Explicit again.MinimizeStep 3: Do the small final twist to get the true rotation and translation.Replace old point cloud with rotated point cloudand iterate…CaveatsNote 1:Last step is a small twistNote 2:Degeneracy exists for certain surfaces, such as planes, general cylinders,…,ResultsSynthetic data, added Gaussian NoiseComparison to ICPWhen points are close to the surface, ICP has a residual errorwhile point-to-plane algorithm performs much better. One question I had: The final MS error seems zero for new algorithm. But some residual error still expected of the order of Gaussian noise variance ? Maybe the y-axis scale is too big to showing it.Comment Felt that calling the algorithm non-ICP is not justified because “correspondence” is still obtained.Future Directions prescribed by
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