Registration and Alignment

Speaker: Liuyu07.12.10

The goal

• Form a 3D model of an object:– Data acquisition– Registration between views– Integration of views

ICP algorithm

References

• A Method for Registration of 3-D Shapes– Paul J. Besl, Member, IEEE, and Neil D. McK

ay– IEEE Transaction on Pattern Analysis and Ma

chine Intelligence,1992

Mathematic Preliminaries• Let t be the triangle defined by the three points

• The distance between and t :

• Let T = {ti} for i =1,…， Nt,, then the distance between and T:

1 1 1 1( , , ),r x y z

3 3 3 3( , , )r x y z

2 2 2 2( , , ),r x y z

p

1 2 3

min( , ) || ||

( 1)d p t ur vr wr

u v w

p

min( , ) ( , )

{1, , } it

d p T d p ti N

References

• Object Modeling by Registration of MultiplyRange Images

– Yang Chen and Gerard Medioni– Robotics and Automation, 1991, Proceedings– In Image and Visual Computer , 1992

Mathematic Preliminaries

• Point to Parametric Entity Distance– the parametric entity– let ,use the Newton`s iteration

method:

• Point to Implicit Entity Distance– Minimize the condition: – Update formula:

( )r u 2( ) ( )f u r u p

p

11 [ ( )( )]t

k k k ku u f u f u

2( ) ,f r r p 0g r

1 02

[ ], 0k k

k k

k

g r g rr r g r

g r

Mathematic Preliminaries

• Corresponding Point Set Registration– Let P = { pi } be a measured data, X be a

model shape, C be the closest point operator: Y = C(P,X),

where Y denote the resulting set of closest points.

Mathematic Preliminaries

• Corresponding Point Set Registration– The least squares registration

(q, d) = φ (P,Y), where q is the registration state vector, and d is the mean square point matching error.

Get q • The formulas to get q :

– where is a unit rotation quaternion to generate the

rotation matrix and is a translation vector.– Minimize the mean square objective function to get

q :

is 3*3 rotation matrix generated by

0 1 2 3 4 5 6[ | ] , [ , , , ] , [ , , ]t t tR T R Tq q q q q q q q q q q q

Rq

Tq

( )RR qRq

2

1

1( ) ( )pN

i R i Tip

f q y R q p qN

Get q• corresponding to the maximum eigenvalue o

f the matrix :

where is the cross-covariance matrix of P and X,

• The translation vector

3

( )( )

( )

TPY

PY TPY PY PY

trQ

tr I

PY

23 31 12[ , , ] ,T Tij PY PY ij

A A A A

Rq

( ) ( ) ( )T Rq mean Y R q mean P

Get d• The mean square point matching error

2

1

1( ) ( )pN

i R i Tip

d f q y R q p qN

ICP Algorithm Statement• Input: the point set P = { pi } from the data shape and the m

odel shape X( with Nx supporting geometric primitives) , a tolerance т

• Initialization of the iteration : P0= P, and k=0;

• then start the iteration:– Computer the closest points : Yk = C( Pk, X )(cost:o( Np*logNx) )

– Compute the registration : (qk,, dk) = φ ( Pk , Yk )(cost O( Np )

– Apply the registration : Pk+1 = qk( P0 ) (cost: O( Np ) )

– Terminate the iteration when dk – dk+1 < т.

0 [1,0,0,0,0,0,0]tq

An Accelerated ICP Algorith• For q

is the angular tolerance

• For d : a linear approximation and a parabolic interpolant to the last three datas d1(v) = a1*v+b1;

d2(v) = a2*v^2 + b2 *v + c2;

1k k kq q q

1 1

1

costk k

kk k

q qq q

1k k kq q q

1,k kand

Resuts

• curve

Results

• triangular

Results

• triangular

Thank you!