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

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem

Given

Two point cloud data sets P (model) and Q (data) sampled from surfaces ΦP and ΦQ respectively.

Assume ΦQ is a part of ΦP. Q P data model

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

} {

i

p P =

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence

i i i i

q p q p → such that } { and } {

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence

i i i i

q p q p → such that } { and } {

∑

− + ⇒ + →

i 2 ,

min

i i t R i i

q t Rp t Rp p

R obtained using SVD of covariance matrix.

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration with known Correspondence

i i i i

q p q p → such that } { and } {

∑

− + ⇒ + →

i 2 ,

min

i i t R i i

q t Rp t Rp p

R obtained using SVD of covariance matrix.

p R t − = q

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

ICP (Iterated Closest Point)

Iterative minimization algorithms (ICP) Properties

• Dense correspondence sets
• Converges if starting positions are “close”
• 1. Build a set of corresponding

points

• 2. Align corresponding points
• 3. Iterate

[Besl 92, Chen 92]

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

No (explicit) Correspondence

P

Φ

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Squared Distance Function (F)

P

Φ x

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

d

P

Φ

Squared Distance Function (F)

2

) , ( d x F

P =

Φ x

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem

∑

∈

Φ

Q q P i

i

q F ) ), ( ( min α

α

An optimization problem in the squared distance field of P, the model PCD.

Rigid transform α that takes points

Our goal is to solve for,

) (

i i

q q α →

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration Problem

∑

∈

Φ +

Q q P i t R

i

t Rq F ) , ( min

,

Optimize for R and t.

) (

• n

translati ) ( rotation t R + = α

Our goal is to solve for,

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Registration in 2D

) , , (

y x t

t θ ε

• Minimize residual error

          =                    

2 1

M M t t

y x

θ

depends on F+ data PCD (Q).

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Approximate Squared Distance

2 2 2 1 2 2 2 1

x x x x ρ ) ( + = + = Ψ

1 1

δ

• x,

F d d [ Pottmann and Hofer 2003 ] For a curve Ψ, Ψ

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

ICP in Our Framework

)) ( ( ) , (

2

= ⇒ − ⋅ = Φ

j

n F δ p x x

P

 1 ) ( ) , (

2

= ⇒ − = Φ

j

F δ p x x

P

• Point-to-plane ICP (good for small d)
• Point-to-point ICP (good for large d)

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

distance-field formulation Plane-to-plane ICP

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Descriptors

20

• closest point → based on Euclidean distance

} {

i

p P =

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Descriptors

21

• closest point → based on Euclidean distance

} {

i

p P = ,...} , , {

i i i

b a p P =

• closest point → based on Euclidean distance between

point + descriptors (attributes)

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

(Invariant) Descriptors

22

• closest point → based on Euclidean distance

} {

i

p P = ,...} , , {

i i i

b a p P =

• closest point → based on Euclidean distance between

point + descriptors (attributes)

Eurographics 2010 Course – Geometric Registration for Deformable Shapes

Integral Volume Descriptor

Relation to mean curvature

0.20

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)