Registration of Point Cloud Data from a Geometric Optimization Perspective

Niloy J. Mitra1, Natasha Gelfand1, Helmut Pottmann2, Leonidas J. Guibas1

1 Stanford University 2 Vienna University of Technology

Registration of PCD

Registration Problem

GivenTwo point cloud data sets P (model) and Q (data) sampled from surfaces P and Q respectively.

Q P

modeldata

Assume Q is a part of P.

Registration of PCD

Registration Problem

GivenTwo point cloud data sets P and Q.

GoalRegister Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.

Q P

data model

Registration of PCD

Registration Problem

• use of second order accurate approximation to the squared distance field.

• no explicit closest point information needed.

• proposed algorithm has good convergence properties.

GivenTwo point cloud data sets P and Q.

GoalRegister Q against P by minimizing the squared distance between the underlying surfaces using only rigid transforms.

Contributions

Registration of PCD

Related Work

Iterated Closest Point (ICP) point-point ICP [ Besl-McKay ] point-plane ICP [ Chen-Medioni ]

Matching point clouds based on flow complex [ Dey et al. ] based on geodesic distance [ Sapiro and Memoli ]

MLS surface for PCD [ Levin ]

Registration of PCD

Notations

}{ ipP

Registration of PCD

Notations

P

Registration of PCD

Squared Distance Function (F)

xP

Registration of PCD

Squared Distance Function (F)

x

2d),( PxF

dP

Registration of PCD

Registration Problem

Qq

Pi

i

qF )),((

min

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

. )( ii qq Rigid transform that takes points

Our goal is to solve for,

Registration of PCD

Registration Problem

Qq

PitR

i

tRqF ),(,

min

Optimize for R and t.

)()( tR ntranslatio rotation

Our goal is to solve for,

Registration of PCD

Overview of Our Approach

Construct approximate such that, to second order.

Linearize .

Solve

to get a linear system.

Apply to data PCD (Q) and iterate.

),(),( PP xFxF ),( PxF

Qq

PitR

i

tRqF ),(,

min

Registration of PCD

Registration in 2D

F Ey Dx CyBxy Ax )( 22 xF

TxQ ][][ 1yx 1yx

• Quadratic Approximant

Registration of PCD

Registration in 2D

)sin(

1)cos(

xt yxx

yt y xy

TxQxF ][][ 1yx 1yx )( • Quadratic Approximant

• Linearize rigid transform

),( tR

of function ] tt [ yx

Registration of PCD

Registration in 2D

xt yxx

yt y xy

• Quadratic Approximant

• Linearize rigid transform

),,(),( yx tt F

QqPi

i

tRq

• Residual error

TxQxF ][][ 1yx 1yx )(

Registration of PCD

Registration in 2D

),,( yx tt• Minimize residual error

21 MM

t

t

y

x

Depends on F+ and data PCD (Q).

Registration of PCD

Registration in 2D

),,( yx tt• Minimize residual error

• Solve for R and t.

• Apply a fraction of the computed motion

• F+ valid locally

• Step size determined by Armijo condition

• Fractional transforms [Alexa et al. 2002]

Registration of PCD

Registration in 3D

• Quadratic Approximant

• Linearize rigid transform

),,,,,( zyx ttt

• Residual error

T]1zyx[]1zyx[ )( xQxF

Minimize to get a linear system

Registration of PCD

Approximate Squared Distance

Two methods for estimating F

1. d2Tree based computation

2. On-demand computation

)F(x, P valid in the neighborhood of x

Registration of PCD

F(x, P) using d2Tree

A kd-tree like data structure for storing approximants of the squared distance function.

Each cell (c) stores a quadratic approximant as a matrix Qc.

Efficient to query.

[ Leopoldseder et al. 2003]

Registration of PCD

F(x, P) using d2Tree

A kd-tree like data structure for storing approximants of the squared distance function.

Each cell (c) stores a quadratic approximant as a matrix Qc.

Efficient to query.

Simple bottom-up construction

Pre-computed for a given PCD.

Closest point information implicitly embedded in the squared distance function.

Registration of PCD

Example d2trees

2D 3D

Registration of PCD

Approximate Squared Distance

22

21

22

21 xxxx

ρ)( 1

1

-

x, Fd

d

[ Pottmann and Hofer 2003 ]

For a curve

Registration of PCD

Approximate Squared Distance

22

21

22

21 xxxx

ρ)( 1

1

-

x, Fd

d

[ Pottmann and Hofer 2003 ]

For a curve

23

22

21

23

22

21 xxxxx

ρx

ρ)( 21

21

--

x, Fd

d

d

d

For a surface

Registration of PCD

On-demand Computation

Given a PCD, at each point p we pre-compute,

• a local frame

• normal

• principal direction of curvatures

• radii of principal curvature (and

)2e and e( 1

)n(

Registration of PCD

On-demand Computation

Given a PCD, at each point p we pre-compute,

• a local frame

• normal

• principal direction of curvatures

• radii of principal curvature (and

)2e and e( 1

)n(

Estimated from a PCD using local analysis

• covariance analysis for local frame

• quadric fitting for principal curvatures

Registration of PCD

On-demand Computation

Given a point x,

nearest neighbor (p) computed using approximate nearest neighbor (ANN) data structure

where j =d/(d-j) if d < 0

0 otherwise.

222

21 ))(())(())((),( pxpxpxxF nee 21P

Registration of PCD

Iterated Closest Point (ICP)

Iterate

1. Find correspondence between P and Q.

• closest point (point-to-point).

• tangent plane of closest point (point-to-plane).

2. Solve for the best rigid transform given the correspondence.

Registration of PCD

ICP in Our Framework

0))((),( 2 j nF pxx P

1)(),( 2 j F pxx P

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

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

Registration of PCD

Convergence Properties

Gradient decent over the error landscapeGauss -Newton Iteration

Zero residue problem (model and data PCD-s match)Quadratic Convergence

For fractional steps, Armijo condition usedDamped Gauss-Newton IterationLinear convergence

can be improved by quadratic motion approximation (not currently used)

Registration of PCD

Convergence Funnel

Set of all initial poses of the data PCD with respect to the model PCD that is successfully aligned using the algorithm.

Desirable properties

• broad

• stable

Registration of PCD

Convergence Funnel

Translation in x-z plane. Rotation about y-axis.

Converges

Does not converge

Registration of PCD

Convergence Funnel

Our algorithmPlane-to-plane ICP

Registration of PCD

Convergence Rate I

Bad Initial Alignment

Registration of PCD

Convergence Rate II

Good Initial Alignment

Registration of PCD

Partial Alignment

Starting Position

Registration of PCD

Partial Alignment

After 6 iterations

Registration of PCD

Partial Alignment

After 6 iterationsDifferent sampling density

Registration of PCD

Future Work

Partial matching

Global registration

Non-rigid transforms

Registration of PCD

Questions?