Geometric Registration for Deformable Shapes. 2.1 ICP + Tangent Space optimization for Rigid Motions

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1 Geometrc Regstraton for Deformable Shapes 2.1 ICP + Tangent Space optmzaton for Rgd Motons

2 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

3 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

4 Notatons P = { p } Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

5 Regstraton wth known Correspondence { p }and{ q }such that p q Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

6 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

7 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

8 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

9 No (explct) Correspondence Φ P Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

10 Squared Dstance Functon (F) x Φ P Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

11 Squared Dstance Functon (F) x d Φ P F( x, Φ P ) = d 2 Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

12 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

13 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

14 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).

15 Approxmate Squared Dstance For a curve Ψ, Ψ d F( x, Ψ) = x1 + x2 = δ1x1 x2 d-ρ1 Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes [ Pottmann and Hofer 2003 ]

16 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)

17 Example d2trees 2D 3D Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

18 Convergence Funnel Translaton n x-z plane. Rotaton about y-axs. Converges Does not converge Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

19 Convergence Funnel Plane-to-plane ICP Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes dstance-feld formulaton

20 Descrptors P = { p } closest pont based on Eucldean dstance Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

21 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

22 (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

23 Integral Volume Descrptor 0.20 Relaton to mean curvature Eurographcs 2010 Course Geometrc Regstraton for Deformable Shapes

24 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