Iterative Closest Point (ICP)Algorithm.L1 solution. . .

Yaroslav Halchenko

CS @ NJIT

Iterative Closest Point (ICP) Algorithm. p. 1

file:yh42@njit.edu

Registration

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Iterative Closest Point (ICP) Algorithm. p. 2

Registration

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Iterative Closest Point (ICP) Algorithm. p. 3

Iterative Closest Point

ICP is a straightforward method [Besl 1992] to align twofree-form shapes (model X , object P ):

Initial transformation

Iterative procedure to converge to local minima1. p P find closest point x X2. Transform Pk+1 Q(Pk) to minimize distances

between each p and x3. Terminate when change in the error falls below a

preset threshold

Choose the best among found solutions for differentinitial positions

Iterative Closest Point (ICP) Algorithm. p. 4

Specifics of Original ICP

Converges to local minima

Based on minimizing squared-error

Suggests Accelerated ICP

Iterative Closest Point (ICP) Algorithm. p. 5

ICP Refinements

Different methods/strategies

to speed-up closest point selectionK-d trees, dynamic cachingsampling of model and object points

to avoid local minimaremoval of outliersstochastic ICP, simulated annealing, weightinguse other metrics (point-to-surface vs -point)use additional information besides geometry(color, curvature)

Iterative Closest Point (ICP) Algorithm. p. 6

ICP Refinements

Different methods/strategies

to speed-up closest point selectionK-d trees, dynamic cachingsampling of model and object points

to avoid local minimaremoval of outliersstochastic ICP, simulated annealing, weightinguse other metrics (point-to-surface vs -point)use additional information besides geometry(color, curvature)

All closed-form solutions are for squared-error ondistances

Iterative Closest Point (ICP) Algorithm. p. 6

Found on the Web

Tons of papers/reviews/articles

No publicly available Matlab code

Registration Magic Toolkit(http://asad.ods.org/RegMagicTKDoc) - fullfeatured registration toolkit with modified ICP

Iterative Closest Point (ICP) Algorithm. p. 7

Implemented in This Work

Original ICP Method [Besl 1992]

Choice for caching of computed distances

Iterative Closest Point (ICP) Algorithm. p. 8

Absolute Distances or L1 norm

Why bother?

More stable to presence of outliers

Better statistical estimator in case of non-gaussiannoise (sparse, high-kurtosis)

might help to avoid local minimas

Iterative Closest Point (ICP) Algorithm. p. 9

Absolute Distances or L1 norm

Why bother?

More stable to presence of outliers

Better statistical estimator in case of non-gaussiannoise (sparse, high-kurtosis)

might help to avoid local minimas

How?

use some parametric approximation for y = |x| anddo non-linear optimization

present this as a convex linear programming problem

Iterative Closest Point (ICP) Algorithm. p. 9

LP: Formulation

Absolute Values y = |x|

x y and x y while minimizing y

Euclidean Distance ~v =

v2x + v2y

3.543.54

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4.582.00

1.344.82

5.003.541.34

0.004.82

~v

|rx~v| ~v, |ry~v| ~v

Iterative Closest Point (ICP) Algorithm. p. 10

LP: Rigid Transformation

Arguments: rotation matrix R and translation vector ~tRigid Transformation:

~p = R~p + ~t

Iterative Closest Point (ICP) Algorithm. p. 11

LP: Rigid Transformation

Arguments: rotation matrix R and translation vector ~tRigid Transformation:

~p = R~p + ~t

Problem: How to ensure that R is rotation matrix?Solution: Take a set of support vectors in objectspace and specify their length explicitly.

~pj ~pk ~pj ~pk = 0 ~pi, ~pj P

Iterative Closest Point (ICP) Algorithm. p. 11

LP

~p = R~p + ~t

~pi ~xi di = 0 i, s.t. ~pi P, ~xi X

~pj ~pk ~pj ~pk = 0 ~pi, ~pj P

Objective: minimize C =

i di

Iterative Closest Point (ICP) Algorithm. p. 12

LP: Problems

Contraction (shrinking):

~pj ~pk ~pj ~pk = 0

is actually

~pj ~pk ~pj ~pk 0

R matrix needs to be normalized to the nearestorthonormal matrix due to our x LPapproximation even if no contraction occurred.

Iterative Closest Point (ICP) Algorithm. p. 13

LP: Results

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Iterative Closest Point (ICP) Algorithm. p. 14

LP: Results

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Iterative Closest Point (ICP) Algorithm. p. 15

LP: Conclusions

Presented problem is suitable to minimize L1 errorinstead of L2 error commonly used.

Using L1 norm improved solution in the presence ofstrong outliers.

Iterative Closest Point (ICP) Algorithm. p. 16

RegistrationRegistrationIterative Closest PointSpecifics of Original ICPICP RefinementsICP Refinements

Found on the WebImplemented in This WorkAbsolute Distances or $L_1$ normAbsolute Distances or $L_1$ norm

LP: FormulationLP: Rigid TransformationLP: Rigid Transformation

LPLP: ProblemsLP: ResultsLP: ResultsLP: Conclusions