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Iterative Closest Point (ICP) Algorithm. - · PDF fileto speed-up closest point selection K-d...

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

    Yaroslav Halchenko

    CS @ NJIT

    Iterative Closest Point (ICP) Algorithm. p. 1

    file:[email protected]

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    Iterative Closest Point (ICP) Algorithm. p. 2

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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

    0.00

    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

    0 100 200 300 400 500 6000

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    2nd norm1st norm

    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

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