Robust Global Registration

Natasha GelfandNiloy Mitra

Leonidas GuibasHelmut Pottmann

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.

Applications

• Shape analysis– similarity, symmetry detection, rigid decomposition

• Shape acquisition[Koller 05] [Pauly 05] [Anguelov 04]

Approaches I• Iterative minimization algorithms (ICP)

• Properties– Dense correspondence sets– Converges if starting positions

are “close”

2. Align corresponding points 3. Iterate1. Build a set of corresponding points

[Besl 92, Chen 92]

Approaches II• 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

[Stockman 87, Hecker 94, Barequet 97]

Approaches III• 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

•

• Relation to mean curvature

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

Feature Properties

• Sparse

• Robust to noise

• Non-canonical

• 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

Rigidity Constraint

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

PQ

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 distance matrix

Search Algorithm• Few features, each with few potential

correspondences– Minimize dRMS– Exhaustive search still too expensive

PQ

• 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

qi

Rc

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

…

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

…

…

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 Our alignment Refined by ICP

Alignment Results

Input: 10 scans

Our alignment Refined by ICP

Symmetry Detection• Symmetry detection

– Match a shape to itself

Rigid Decomposition

• Repeated application of partial matching

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

Future Work

• Linear features

Future Work

• Linear features

• Fast rejection tests

• More descriptors

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

Questions?