Geometric Registration for Deformable Shapes 2.1 ICP + Tangent - PowerPoint PPT Presentation

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

Registration Problem Given Two point cloud data sets P ( model ) and Q ( data ) sampled from surfaces Φ P and Φ Q respectively. Q P data model Assume Φ Q is a part of Φ P . Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem Given Two point cloud data sets P and Q . Goal Register Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms. Q P data model Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Notations P = { p } i Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence → { p } and { q } such that p q i i i i Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence → { p } and { q } such that p q i i i i ∑ → + ⇒ + − 2 p Rp t min Rp t q i i i i R , t i R obtained using SVD of covariance matrix. Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence → { p } and { q } such that p q i i i i ∑ → + ⇒ + − 2 p Rp t min Rp t q i i i i R , t i R obtained using SVD of covariance matrix. = q − t R p Eurographics 2010 Course – Geometric Registration for Deformable Shapes

ICP (Iterated Closest Point) Iterative minimization algorithms (ICP) [Besl 92, Chen 92] 3. Iterate 2. Align corresponding points 1. Build a set of corresponding points Properties • Dense correspondence sets • Converges if starting positions are “close” Eurographics 2010 Course – Geometric Registration for Deformable Shapes

No (explicit) Correspondence Φ P Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Squared Distance Function (F) x Φ P Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Squared Distance Function (F) d Φ P x Φ P = 2 F ( x , ) d Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem → α Rigid transform α that takes points q ( q ) i i Our goal is to solve for, ∑ α Φ min F ( ( q ), ) i P α ∈ q Q i An optimization problem in the squared distance field of P , the model PCD. Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem α = R + rotation ( ) translati on ( t ) Our goal is to solve for, ∑ + Φ min F ( Rq t , ) i P R , t ∈ q Q i Optimize for R and t . Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration in 2D ε θ t x t • Minimize residual error ( , , ) y   θ           = t M M       x 1 2         t     y depends on F + data PCD ( Q ). Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Approximate Squared Distance Ψ For a curve Ψ, d Ψ = + = δ + 2 2 2 2 ( ) x x x x F x, 1 2 1 2 ρ 1 d - 1 [ Pottmann and Hofer 2003 ] Eurographics 2010 Course – Geometric Registration for Deformable Shapes

ICP in Our Framework • Point-to-point ICP (good for large d ) Φ = − ⇒ δ = 2 F x x p ( , ) ( ) 1 P j • Point-to-plane ICP (good for small d )  Φ = ⋅ − ⇒ δ = 2 F n x x p ( , ) ( ( )) 0 P j Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Example d2trees 2D 3D Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Convergence Funnel Translation in x-z plane. Rotation about y-axis. Converges Does not converge Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Convergence Funnel Plane-to-plane ICP distance-field formulation Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Descriptors P = { p } i • closest point → based on Euclidean distance Eurographics 2010 Course – Geometric Registration for Deformable Shapes 20

Descriptors P = { p } i • closest point → based on Euclidean distance P = { p , a , b ,...} i i i • closest point → based on Euclidean distance between point + descriptors (attributes) Eurographics 2010 Course – Geometric Registration for Deformable Shapes 21

(Invariant) Descriptors P = { p } i • closest point → based on Euclidean distance P = { p , a , b ,...} i i i • closest point → based on Euclidean distance between point + descriptors (attributes) Eurographics 2010 Course – Geometric Registration for Deformable Shapes 22

Integral Volume Descriptor 0.20 Relation to mean curvature Eurographics 2010 Course – Geometric Registration for Deformable Shapes

When Objects are Poorly Aligned • Use descriptors for global registrations global alignment → refinement with local (e.g., ICP) Eurographics 2010 Course – Geometric Registration for Deformable Shapes