ICP seminar

Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University

RISL | Robotics & Intelligent Systems Lab.Seoul National University

A Method for Registration of 3-D Shapes

July , 2016

JunHyuk Shin



1. Introduction

Shape registration: register digitized data from rigid objects with idealized geometric model- Representations of geometric data using shape registration

1) Point sets2) Line segment sets (polylines)3) Implicit curves : 4) Parametric curves : 5) Tringle sets (faceted surfaces)6) Implicit surfaces : 7) Parametric surfaces :

ICP algorithm: ICP algorithm attempts to match two sets of points- One of these sets might be a set of data points, while the other is a reference image- Matching data points sets to samples in the reference set, assigning a definition for mean square error, minimizing that error iteratively



3. Mathematical Preliminaries

A. Closest point problem: computing the closest point to a given point on the various geometric entities1. Point to Basic Geometric Entity Distance - Euclidean distance 1) Point sets

denotation : point for distance : closest point :

2) Line segment sets (polylines)denotation : line for distance : while (, )closest point :




3) Triangle set (polylines)denotation : triangle for distance :

while (, , )

closest point :

2. Point to Parametric Entity Set Distance - single parametric entity where

denotation : entity for distance : while closest point :




- Computations to compute the distance are not closed form and are relative involved 1) Simplex-based approximation method

Curve : polyline for deviate distance estimate of Surface : Tringle set for deviate distance estimate of

2) Newton’s minimization approachScalar object function Minimum of occurs when (vector gradient operator )(curve : / surface : , )Newton’s update formula : where : iterative search minimum - Computational cost is very low in contrast with finding good starting point




3. Point to Implicit Entity Set Distancedenotation : vector-valued multivariate function distance : while =- Computations are not closed form and are relative involved

1) Augmented Lagrange multiplier system of equation

(- # of equations and unknowns for the nonlinear system

: three for planar curves, four for surfaces, five for implicit curves2) Approximate distance algorithm by Taubin

when : only exact if the infinite line with the direction intersects the implicit entity at a point where the normal vector has that same direction




B. Corresponding Point Set Registration: yield the least squares rotation and translation- Quaternion based algorithm : n dimensions for (no reflection)- SVD(singular value decomposition) method : n dimensions for

1. Quaternion based rigid motion computation Definition of Quaternion : with that ( of imaginary parts ())object vector For unit quaternion , is not purely imaginary use (also, composition of rotations corresponds to multiplication of quaternions)




- set unit quaternion , where and and set translation vector : Then complete registration state vector - set with measured data point set and with model point set (

Mean square objective function and minimization procedure 1) Optimal translation vector - The relative coordinates of all the points to their centroids are obtained as with centroid : = , also - By rewriting the objective function in terms of , , , , Optimal translation is given by making the second term to zero :




2) Optimal rotation vector : can be solved from the first term : +

- Equivalently, optimal rotation can be done by maximize : find unit quaternion that maximize - Set matrix for equivalent matrix product ( , then




With cross-covariance (= where )

- has real eigenvalues with with orthogonal unit eigenvector : quaternion is a linear combination of these eigenvectors

- Then and has its maximum when optimal rotation quaternion is the unit eigenvector that corresponds to the largest ei-genvalue of




2. SVD based rigid motion computation : seek a rotation for

1) optimal translation when and , = and 2) optimal rotation - Set covariance matrix and take SVD of : then and get - is diagonal matrix and the trace is maximized if 3) Orientation rectification

if : contains reflection, otherwise - general formula that



4. The Iterative Closest Point Algorithm

A. ICP algorithm statement ICP algorithm operator

- Least square Quaternion operator : ( : mean square point matching error)- Closest point operator : - Position update of data shape point set :

Computational cost

Precondition- Point set P with points from data shape- Point set with points from model shape- Initialize by , , - Registration vector is defined relative to the initial data set

Case Worst

)

)Average




ICP algorithm• Estimate initial registration states• Iterate next steps to converge local minimum

1. Compute the closest points : 2. Compute the registration : 3. Apply the registration :4. Terminate the iteration when change in mean square error falls below threshold

: • Choose the global minimum among found solutions for different initial registration states




B. Convergence Theorem Theorem

: The iterative closest point algorithm always converges monotonically to a local minimum with respect to the mean-square distance objective function

Proof: mean squared error - After operator and registration are applied to get and : : for all the case- After closest point operator, new point set is updated : for each and for all the case Therefore, for all

Because the mean square error sequence is non-increasing and bounded below, the algorithm as stated must converge monotonically to a minimum value. Q.E.D




C. An Accelerated ICP Algorithm- As ICP proceeds, a sequence of registration vec-tors is generated and traces out a path in the registra-tion state space- Different vector sequence : - Angle between two last direction : , and ( : tolerance angle) : consistent direction alignment for the last three registration state vector

- Associated approximate arc length argument value , - Interpolant to the last three data points , ,




- Update logic or : parabola update or : linear update and : maximum allowable update

i.e.

Basic ICP plot Accelerated ICP plot

ICP Plot- Basic ICP : plot indicates a consis-tent direction of update for all but the first few iteration- Accelerated ICP : after acceleration steps of jumpy behavior on , most quantities get close to final value → 50 ICP iteration is accelerated to 15 or 20 iterations



5. The Set of Initial Registration

Global Minimum Reaching Problem- ICP algorithm must converge monotonically to a local minimum from any given rota-tion and translation, but it may or may not converge on the desired global minimum : characterize the partitioning of the registration state space into local minima wells- Consider 6-D state space : subset of the space and specified by “inside or on unit 3 sphere”

*equivalence classFor non-pathological shape and given point set : already registeredConsider any initial state of (will converge to a local minimum)Then there are finite number of local minima natural partitioning of into equivalence class , where every value of that converges to local minima is a member of the class

and if ( : set of all local minima )




• - How to construct an initial set of states for any given object that guarantees a correct global minimum?

• - How to construct an initial set of states that guarantees all shapes in a given class of shapes when those shapes converge to the correct respective global minimum?

: By using a sufficient dense uniform sampling of quaternion on the unit sphere com-bined with a sufficiently dense sampling of translation vectors occupying the total vol-ume about the shape determine complete finite set of local minima with sufficiently small error

A. Initial States for Global Matching- Global matching : Point set covers a significant portion of the model shape for sufficiently large factor 1) Translation state : ICP algorithm insensitive to the initial translation state for global minimum case




2) Rotation state : ICP algorithm insensitive to the initial translation state for global minimum case

*dispersion value : square roots of the eigenvalues of : square roots of the eigenvalues of

*Condition set of eigenvalues a) and : match using only eigenvectors of and b) and : initial rotation states only expanded for rotations about the non-ambiguous axis c) , : must use fine sampling of quaternion states that cover the entire surface of the northern hemisphere of the unit 4 sphere must be used uniformly (

General rule of thumb : more complicated the object, the more initial states required




B. A Counter Example- Proposed registration has problems registering “sea urchins” type shapes due to limited # of initial rotation states- almost equal eigenvalues of the covariance matrix and small shape features at a fine scale relative to the overall shapes are difficult to register correctly

C. Local Shape Matching: entire set of data points matches a subset of the model shape - More than one initial translation must be used

- Time complexity in the worst case becomes as opposed to global matching’s ( : # of initial translation, : # of initial rotation)* # of initial translation state esitmate

Quantity : ratio of size of the data and model shape where is the general quantity measurementsThen : approximate complexity proportionality factor of



5. Experiment Result

A. Point Set Matching

: matching a set of 8 points against a set of 11 points, using an ICP algorithm with 6 iteration- One initial rotation and translation state




B. Curve Matching

: rotated and translated curve used a polyline description with 64 points - Gaussian noise is then applied to each point of the polyline - 12 initial rotation states and 6 initial translation states : 72 initial registration states used




C. Surface Matching 1) A Bezier Surface Patch

: subset of 138 noisy points was selected for local matching - 24 initial rotation states and 6 initial translation states : 144 initial registration states used




C. Surface Matching 2) The NRCC African Mask

: model surface was approximated with 2546 points - 6 iterations were used on each of the 24 initial state vectors




2) The NRCC African Mask

- extra data points for no possible surface correspondence : ICP algorithm is not designed to handle this data, but minor misgrouping have minor effect




C. Surface Matching 3) Terrain

: model surface was approximated with 13655 points - 24 initial rotation states and one initial translation states



5. Conclusion & Limitation

ICP algorithm register a data shape with points to a model shape with - Any type of shape can be incorporated- Accelerated ICP algorithm converges to a local minimum quickly- Global shape matching can be achieved using sufficient dense sampling of quaternions- Local shape matching by combining a sufficiently dense sampling of translation

Pros & Cons- Pros : No requirement of preprocessing 3-D data point, such as smoothing.. etc.- Cons : Susceptible to gross statistical outliers.. etc.

Limitation- For global shape matching, sea urchin or planetoid shape cannot be registered : feature extraction technique are preferable (or error weighting on model feature?)- Cost of local matching can get quite large for small occlusion percentages, e.g., 10%

ICP seminar

Engineering

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