Geometrc Regstraton for Deformable Shapes 2.1 ICP + Tangent Space optmzaton for Rgd Motons
Regstraton Problem Gven Two pont cloud data sets P (model) and Q (data) sampled from surfaces Φ P and Φ Q respectvely. Q P data model Assume Φ Q s a part of Φ P. Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton Problem Gven Two pont cloud data sets P and Q. Goal Regster Q aganst P by mnmzng the squared dstance between the underlyng surfaces usng only rgd transforms. Q P data model Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Notatons P = { p } Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton wth known Correspondence { p }and{ q }such that p q Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton wth known Correspondence { p }and{ q }such that p q p Rp + t mn R, t Rp + t q 2 R obtaned usng SVD of covarance matrx. Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton wth known Correspondence { p }and{ q }such that p q p Rp + t mn R, t Rp + t q 2 R obtaned usng SVD of covarance matrx. t = q R p Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
ICP (Iterated Closest Pont) Iteratve mnmzaton algorthms (ICP) [Besl 92, Chen 92] 1. Buld a set of correspondng ponts 2. Algn correspondng ponts 3. Iterate Propertes Dense correspondence sets Converges f startng postons are close Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
No (explct) Correspondence Φ P Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Squared Dstance Functon (F) x Φ P Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Squared Dstance Functon (F) x d Φ P F( x, Φ P ) = d 2 Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton Problem Rgd transform α that takes ponts q α ( q ) Our goal s to solve for, mn α q Q F ( α( q ), Φ ) P An optmzaton problem n the squared dstance feld of P, the model PCD. Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Regstraton Problem α = rotaton ( R ) + translaton( t) Our goal s to solve for, mn R, t q Q F ( Rq + t, Φ ) P Optmze for R and t. Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes Regstraton n 2D ),, ( y t x t θ ε Mnmze resdual error = 1 M 2 M t t y x θ depends on F + data PCD (Q).
Approxmate Squared Dstance For a curve Ψ, Ψ d F( x, + 2 2 2 2 Ψ) = x1 + x2 = δ1x1 x2 d-ρ1 Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes [ Pottmann and Hofer 2003 ]
Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes ICP n Our Framework 0 )) ( ( ), ( 2 = = Φ j n F δ p x x P 1 ) ( ), ( 2 = = Φ j F δ p x x P Pont-to-plane ICP (good for small d) Pont-to-pont ICP (good for large d)
Example d2trees 2D 3D Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Convergence Funnel Translaton n x-z plane. Rotaton about y-axs. Converges Does not converge Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Convergence Funnel Plane-to-plane ICP Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes dstance-feld formulaton
Descrptors P = { p } closest pont based on Eucldean dstance Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Descrptors P = { p } closest pont based on Eucldean dstance P = { p, a, b,...} closest pont based on Eucldean dstance between pont + descrptors (attrbutes) Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
(Invarant) Descrptors P = { p } closest pont based on Eucldean dstance P = { p, a, b,...} closest pont based on Eucldean dstance between pont + descrptors (attrbutes) Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
Integral Volume Descrptor 0.20 Relaton to mean curvature Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes
When Objects are Poorly Algned Use descrptors for global regstratons global algnment refnement wth local (e.g., ICP) Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes