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Iterative Closest Point (ICP) Algorithm. L 1 solution... Yaroslav Halchenko CS @ NJIT Iterative Closest Point (ICP) Algorithm. p. 1 Registration Max= 250 Min= 0 Max= 254 Min= 0 250 250 50 50 200 200 100 100 150 150 0.1 150

SLIDE 1

Iterative Closest Point (ICP) Algorithm.

L1 solution...

Yaroslav Halchenko CS @ NJIT

Iterative Closest Point (ICP) Algorithm. – p. 1

SLIDE 2

Registration

50 100 150 200 250 Max= 250 Min= 0 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 Max= 254 Min= 0 50 100 150 200 250 50 100 150 200 250 50 100 150 200 250 Max= 254 Min= 0 50 100 150 200 250 50 100 150 200 250

−0.05 0.05 0.1 −0.05 0.05 0.05 0.1

Iterative Closest Point (ICP) Algorithm. – p. 2

SLIDE 3

Registration

−0.05 0.05 0.1 −0.05 0.05 0.05 0.1

Iterative Closest Point (ICP) Algorithm. – p. 3

SLIDE 4

Iterative Closest Point

ICP is a straightforward method [Besl 1992] to align two free-form shapes (model X, object P): Initial transformation Iterative procedure to converge to local minima

1. ∀p ∈ P find closest point x ∈ X
2. Transform Pk+1 ← Q(Pk) to minimize distances

between each p and x

3. Terminate when change in the error falls below a

preset threshold Choose the best among found solutions for different initial positions

Iterative Closest Point (ICP) Algorithm. – p. 4

SLIDE 5

Specifics of Original ICP

Converges to local minima Based on minimizing squared-error Suggests ‘Accelerated ICP’

Iterative Closest Point (ICP) Algorithm. – p. 5

SLIDE 6

ICP Refinements

Different methods/strategies to speed-up closest point selection K-d trees, dynamic caching sampling of model and object points to avoid local minima removal of outliers stochastic ICP, simulated annealing, weighting use other metrics (point-to-surface vs -point) use additional information besides geometry (color, curvature)

Iterative Closest Point (ICP) Algorithm. – p. 6

SLIDE 7

ICP Refinements

Different methods/strategies to speed-up closest point selection K-d trees, dynamic caching sampling of model and object points to avoid local minima removal of outliers stochastic ICP, simulated annealing, weighting use other metrics (point-to-surface vs -point) use additional information besides geometry (color, curvature) All closed-form solutions are for squared-error on distances

Iterative Closest Point (ICP) Algorithm. – p. 6

SLIDE 8

Found on the Web

Tons of papers/reviews/articles No publicly available Matlab code Registration Magic Toolkit (http://asad.ods.org/RegMagicTKDoc) - full featured registration toolkit with modified ICP

Iterative Closest Point (ICP) Algorithm. – p. 7

SLIDE 9

Implemented in This Work

Original ICP Method [Besl 1992] Choice for caching of computed distances

Iterative Closest Point (ICP) Algorithm. – p. 8

SLIDE 10

Absolute Distances or L1 norm

Why bother? More stable to presence of outliers Better statistical estimator in case of non-gaussian noise (sparse, high-kurtosis) might help to avoid local minima’s

Iterative Closest Point (ICP) Algorithm. – p. 9

SLIDE 11

Absolute Distances or L1 norm

Why bother? More stable to presence of outliers Better statistical estimator in case of non-gaussian noise (sparse, high-kurtosis) might help to avoid local minima’s How? use some parametric approximation for y = |x| and do non-linear optimization present this as a convex linear programming problem

Iterative Closest Point (ICP) Algorithm. – p. 9

SLIDE 12

LP: Formulation

Absolute Values y = |x|

x ≤ y and −x ≤ y while minimizing y

Euclidean Distance v =

x + v2 y

× × × ×

× × ×

3.54 3.54 0.00

4.58 2.00 1.34 4.82

× ×

5.00 3.54 1.34

× × ×0.00

4.82

|rx v| ≤ v, |ry v| ≤ v

Iterative Closest Point (ICP) Algorithm. – p. 10

SLIDE 13

LP: Rigid Transformation

Arguments: rotation matrix R and translation vector t Rigid Transformation:

p = R p + t

Iterative Closest Point (ICP) Algorithm. – p. 11

SLIDE 14

LP: Rigid Transformation

Arguments: rotation matrix R and translation vector t Rigid Transformation:

p = R p + t Problem: How to ensure that R is rotation matrix? “Solution”: Take a set of “support” vectors in object space and specify their length explicitly.

pj − ˙ pk − pj − pk = 0

pj ∈ P

Iterative Closest Point (ICP) Algorithm. – p. 11

SLIDE 15

LP

p = R p + t

pi − xi − di = 0 ∀i, s.t. pi ∈ P, xi ∈ X

pj − ˙ pk − pj − pk = 0

pj ∈ P Objective: minimize C =

i di

Iterative Closest Point (ICP) Algorithm. – p. 12

SLIDE 16

LP: Problems

Contraction (shrinking):

pj − ˙ pk − pj − pk = 0 is actually

pj − ˙ pk − pj − pk ≤ 0 R matrix needs to be “normalized” to the nearest

rthonormal matrix due to our x LP

approximation even if no contraction occurred.

Iterative Closest Point (ICP) Algorithm. – p. 13

SLIDE 17

LP: Results

−1 −0.5 0.5 1 −1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

Iterative Closest Point (ICP) Algorithm. – p. 14

SLIDE 18

LP: Results

100 200 300 400 500 600 0.05 0.1 0.15 0.2 0.25 Rotation (R) # of outliers 100 200 300 400 500 600 0.1 0.2 0.3 0.4 Translation (t) # of outliers 2nd norm 1st norm

Iterative Closest Point (ICP) Algorithm. – p. 15

SLIDE 19

LP: Conclusions

Presented problem is suitable to minimize L1 error instead of L2 error commonly used. Using L1 norm improved solution in the presence of strong outliers.