Download - 1 Numerical geometry of non-rigid shapes Iterative closest point algorithms Iterative closest point algorithms Tutorial 4 © Maks Ovsjanikov tosca.cs.technion.ac.il/book.

1Numerical geometry of non-rigid shapes Iterative closest point algorithms

Iterative closest point algorithmsTutorial 4

© Maks Ovsjanikovtosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapesStanford University, Winter 2009


Outline

Closed-form solution of absolute orientation using unit quaternions. Horn, B. K. P., Journal of the Optical Society of America, Vol. 4 1987

1.Problem Formulation.

2.Intuition.

3.Some Math.

Robust Global Registration Gelfand N., Mitra N., Guibas L., Pottmann H., Proceedings of SGP, 2005

Formulation.

1.Problem and Motivation.

2.Multi-Scale point descriptors.

3.Global Matching – Branch and Bound.

4.Results.



Problem Formulation:

1. Given two sets points: in . Find the rigid transform:

and that minimizes:





and that minimizes:





and that minimizes:

2. In practice, have many points. Perform this step many times. Want a fast way to

get an exact solution. Representation is important.


Given two sets points: in . Find the rigid transform:

and that minimizes:

Finding the translation part .1. Let and : the centroids of and

and let and . Then .

2. where:

minimized if



and that minimizes:

Need to find the rotation .1. Several ways to represent rotation: Orthonormal matrix, Euler angle, Gibbs vector,

quaternions, etc. Which one is most useful in this case?

2. Plan of action:

Want to minimize

Since rotation preserves lengths. Need to maximize:

Question: What unit vector maximizes if is symmetric?

Answer: Eigenvector corresponding to the maximum eigenvalue of .



and that minimizes:

Need to find the rotation .2. Plan of action:

Need to maximize:

3. If represent using orthonormal matrices, then get:

Would prefer: , where is a vector and - matrices

Quaternions allow us to express rotation in such form.


Quaternions (two slides intro).

1. Vectors with 4 components, 3 of which are imaginary:

2. Multiplication is defined using the rules:

from which follows:

In matrix form:

3. Dot product:

Matrices associated with unit quaternions are orthonormal.


Quaternions (two slides intro).

4. Conjugation: . Matrix corresponding to the conjugate is

the transpose of the original.

5. Regular vectors can be represented with purely imaginary quaternions:

6. Rotation can be represented with composite product by a unit quaternion :

where .

Conversely, every unit quaternion corresponds to a rotation.


Back to Alignment.

1. Want to maximize:

2. In quaternion form:

3. The optimal rotation quaternion is the eigenvector corresponding to the maximum

eigenvalue of:


Summary

Efficient algorithm to find the optimal translation and rotation in the

rigid alignment, provided that the correspondences are known.

Uniform scaling can be found similarly:

Quaternions provide a simple and efficient way to represent rotations.

Other methods: SVD, Orthogonal Matrices, Dual Quaternions


Estimating 3-D rigid body transformations: a comparison of four major

algorithms. Eggert et al., Machine Vision and Applications, Vol. 9 1997


Outline


1.Problem Formulation.

2.Intuition.

3.Some Math.

Robust Global Registration Gelfand N., Mitra N., Guibas L., Pottmann H., Proceedings of SGP, 2005

Formulation.

1.Problem and Motivation.

2.Multi-Scale point descriptors.

3.Global Matching – Branch and Bound.

4.Results.Images by N. Gelfand


Robust Global Registration Gelfand, Mitra, Guibas, Pottmann, Proceedings of SGP, 2005

Problem:

Given:

Two shapes P and Q which partially overlap.

Using only rigid transforms, register Q against P by minimizing the squared distance between them.

Goal:

No assumptions about initial positions!


Approaches

• Rigid transform can be specified with small number of parameters– Try all possible transform bases– Find the one that aligns the most points

• Guaranteed to find the correct transform– But can be costly


Approaches

• Use neighborhood geometric information

• Descriptors should be– Invariant under transform– Local– Cheap

• Improves correspondence search or used for feature extraction

[Mokh 01, Huber 03, Li 05]


Registration Problem

• Why it’s hard– Unknown areas of overlap– Have to solve the correspondence problem

• Why it’s easy– Rigid transform is specified by small number of points– Prominent features are easy to identify

We only need to align a few points correctly.


Method Overview

1. Use descriptors to identify features– Integral volume descriptor

2. Build correspondence search space– Few correspondences for each feature

3. Efficiently explore search space– Distance error metric– Pruning algorithms


Integral Descriptors

•

• Multiscale

• Inherent smoothing

f(x)p

Br(p)

[Manay 04]


Integral Volume Descriptor

•

• Related to mean curvature.

• Provably robust to noise.

0.20


Descriptor Computation

• Approximate using a voxel grid

– Convolution of occupancy grid with ball


Feature Identification

• Pick as features points with rare descriptor values– Rare in the data rare in the model few correspondences

• Works for any descriptor

0

10

20

30

40

50

60

70

80

90

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Descriptor value

# o

ccu

ren

ces

Features


Multiscale Algorithm

• Features should be persistent over scale change

r=0.5 r=1 r=2 r=4 r=8

Y Y Y N N

N N Y Y Y

N N Y N N


• Search the whole model for correspondences– Range query for descriptor values– Cluster and pick representatives

Correspondence Space

PQ


Evaluating Correspondences

• Coordinate root mean squared distance

– Requires best aligning transform– Looks at correspondences individually


Rigidity Constraint

• Pair-wise distances between features and correspondences should be the same

PQ


Evaluating Correspondences

• Distance root mean squared distance

– Depends only on internal distances


Search Algorithm

• Few features, each with few potential correspondences– Minimize dRMS– Exhaustive search still too expensive

PQ


Branch and Bound.

• Find global minimum of a function.

Search Space:


Branch and Bound.

• Find global minimum of a function.

Search Space:

1.Split (Branch).

2.Bound.

3.Prune:If min2 > max1, can remove branch 2.

min1

max1

min2

max2


• Branch and bound– Initial bound using greedy assignment

• Discard partial correspondences that fail thresholding test–

• Prune if partial correspondence exceeds bound– Spaced out features make incorrect correspondences fail quickly

• Since we explore the entire search space, we are guaranteed to find optimal alignment– Up to cluster size

Search Algorithm

pi

pjqj

qiRc


Search Algorithm

• Branch and bound– Initial bound using greedy assignment

• Discard partial correspondences that fail thresholding test–

• Prune if partial correspondence exceeds bound– Spaced out features make incorrect correspondences fail quickly

• Since we explore the entire search space, we are guaranteed to find optimal alignment– Up to cluster size


Greedy Initialization

• Good initial bound is essential• Build up correspondence set hierarchically

…




…

…




…


Partial Alignment

• Allow null correspondences, while maximizing the number of matches points


Alignment Results

Input: 2 scans Alignment Refined by ICP


Alignment Results

Input: 10 scans

Alignment Refined by ICP