Robust Global Registration

1
Robust Global Registration

• Natasha Gelfand
• Niloy Mitra
• Leonidas Guibas
• Helmut Pottmann

2
Registration Problem
Given
Two shapes P and Q which partially overlap.
Goal
Using only rigid transforms, register Q against P
by minimizing the squared distance between them.
3
Applications

• Shape analysis
• similarity, symmetry detection, rigid
  decomposition
• Shape acquisition

Anguelov 04
Koller 05
Pauly 05
4
Approaches

• Iterative minimization algorithms
• Voting methods
• Geometric descriptors

5
Approaches I
Besl 92, Chen 92

• Iterative minimization algorithms (ICP)
• Properties
• Dense correspondence sets
• Converges if starting positions are close

3. Iterate
2. Align corresponding points
1. Build a set of corresponding points
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Approaches II
Stockman 87, Hecker 94, Barequet 97

• Voting methods
• Geometric hashing, Hough transform, alignment
  method
• Rigid transform can be specified with small
  number of points
• Try all possible transform bases
• Find the one that aligns the most points
• Guaranteed to find the correct transform
• But can be costly

7
Approaches III
Mokh 01, Huber 03, Li 05

• Use neighborhood geometric information
• Descriptors should be
• Invariant under transform
• Local
• Cheap
• Improves correspondence search or used for
  feature extraction

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Registration Problem

• Why its hard
• Unknown areas of overlap
• Have to solve the correspondence problem
• Why its 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.
9
Method Overview

• Use descriptors to identify features
• Integral volume descriptor
• Build correspondence search space
• Few correspondences for each feature
• Efficiently explore search space
• Distance error metric
• Pruning algorithms

10
Integral Descriptors
Manay 04

• Multiscale
• Inherent smoothing

f(x)
Br(p)
p
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Integral Volume Descriptor

• Relation to mean curvature

12
Descriptor Properties

• Relation to curvature
• Influence of noise

13
Descriptor Computation

• Approximate using a voxel grid
• Convolution of occupancy grid with ball

14
Feature Identification

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

Features
15
Multiscale Algorithm

• Features should be persistent over scale change

16
Feature Properties

• Sparse
• Robust to noise
• Non-canonical

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Correspondence Space

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

Q
P
18
Evaluating Correspondences

• Coordinate root mean squared distance
• Requires best aligning transform
• Looks at correspondences individually

19
Rigidity Constraint

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

Q
P
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Rigidity Constraint

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

Q
P
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Rigidity Constraint

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

Q
P
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Evaluating Correspondences

• Distance root mean squared distance
• Depends only on internal distance matrix

23
Search Algorithm

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

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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

Rc
qi
pi
pj
qj
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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

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Greedy Initialization

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

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Greedy Initialization

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

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Greedy Initialization

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

29
Partial Alignment

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

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Alignment Results
Input 2 scans
Our alignment
Refined by ICP
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Alignment Results
Our alignment
Refined by ICP
Input 10 scans
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Symmetry Detection

• Symmetry detection
• Match a shape to itself

33
Rigid Decomposition

• Repeated application of partial matching

34
Conclusions

• Simple feature identification algorithm
• Used integral volume descriptor
• Applicable for any low-dimensional descriptor
• Correspondence evaluation using dRMS error
• Look at pairs of correspondences, instead of
  individually
• Efficient branch and bound correspondence search
• Finds globally best alignment

35
Future Work

• Linear features

36
Future Work

• Linear features
• Fast rejection tests
• More descriptors

37
Acknowledgements

• Daniel Russel
• An Nguyen
• Doo Young Kwon
• Digital Michelangelo Project
• NSF CARGO-0138456, FRG-0454543, ARO DAAD
  19-03-1-033, Max Plank Fellowship, Stanford
  Graduate Fellowship, Austrian Science Fund
  P16002-N05

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Questions?