Post on 15-Apr-2017
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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 :
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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 :
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
- 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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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)
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
- 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 :
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
3. Mathematical Preliminaries
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
4. The Iterative Closest Point Algorithm
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
4. The Iterative Closest Point Algorithm
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
4. The Iterative Closest Point Algorithm
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 , ,
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
4. The Iterative Closest Point Algorithm
- 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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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 )
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. The Set of Initial Registration
• - 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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. The Set of Initial Registration
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. The Set of Initial Registration
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. Experiment Result
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. Experiment Result
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. Experiment Result
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. Experiment Result
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
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
5. Experiment Result
C. Surface Matching 3) Terrain
: model surface was approximated with 13655 points - 24 initial rotation states and one initial translation states
Robotics & Intelligent System Laboratory Department of Electrical Engineering Seoul National University
RISL | Robotics & Intelligent Systems Lab.Seoul National University
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%