Sandia

A Topological Approach to Shape
Analysis and Alignment
David O’Gwynn
University of Alabama at Birmingham

Sandia National Laboratories – p. 1/5

The Problem: Shape Alignment

Importance of Part Structure

Part Structure from Topology

Topology → Parts → Alignment

Results


Problem Deﬁnition

Data

Problem

Existing solutions


Surface data from very different
sources
MRI

Stored in many different
databases

Range scans
CAT [Levoy 2000]


Predominantly stored as a mesh

MRI

Range scans
CAT [MeshLab] [Levoy 2000]


Information stored in a mesh...



Surface
sample points



Surface
Local connectivity of sample points
sample points


The Problem

Orientation of shapes rarely consistent across databases
... Or even in the same database


The Problem

How do we align two similar mesh shapes?


Further Complications

Variations in sampling density Variations in pose,
physiology


Given:
Input shapes P and Q
2-manifold triangle meshes
Same underlying shape class (hands, humans)
Transform class A
Isometry: Rigid or distance-preserving transforms

Find:
Transformation α ∈ A that minimizes

E(P, Q) = min d2 (α(P), Q)
α


Existing Solutions

Surface Registration
Local
Global

Orientation Normalization
Principle component analysis (PCA)
Function-space search



→ →

[Gelfand 2006]

[Levoy 2000] [Levoy 2000]


Local Registration
Iterative Corresponding Point (ICP)
[Besl and McKay 1992, Chen and Medioni 1991]
[Rusinkiewicz and Levoy 2001]

Given: initial guess α0
correspond α0 (P),Q → solve for α1 [Gelfand 2006]

correspond α1 (P),Q → solve for α2 Correspondence is based on a

... modiﬁed nearest neighbor

until convergence approach.

Process iterates to local minimum of E(P, Q).


Global Registration

Robust Global Registration
[Gelfand et al. 2005]
Select unique points on P and Q based on
volumetric feature space
Search correspondence space for optimal corre-
spondence

Four Points Congruent Sets (4PCS)
[Aiger et al. 2008]
Search 4-point set correspondence space

[Gelfand 2005]

The goal is to ﬁnd the global minimum of E(P, Q).



[Chaouch and Verroust-Blondet 2009]


Principal Component Analysis (PCA)

Each shape has a “correct” orientation characterized by its
principal components.

Continuous PCA (CPCA)
[Vranic et al. 2001]
Plane Reﬂection Symmetry PCA
[Chaouch and Verroust-Blondet 2009]

Find the principal axes of P and Q
(best ﬁtting ellipsoid) and align those axes


Function-space Alignment

Map P and Q to some functional space and solve for the
rotation in that space that best aligns them. Then map that
rotation back to R3 .
Axially Symmetric Alignment via
Spherical Harmonics
[Kazhdan 2007]

GPU-based Rotational Alignment
[Martinek and Grosso 2009]

[Kazhdan 2007]


Issues...
Predicated on “ground truth” model to which input shapes conform
Exceptions:
Anguelov et al. 2005
Chang and Zwicker 2008

Reliance on well-deﬁned axes/planes of symmetry
Poor performance on non-symmetric objects

Treat the shape as a geometric monolith.


Our Problem

How do we align these horses?
surface characteristics? principle axes?


Solution...

Helps to remember they’re horses
... and not a bag of surface samples, or planes of symmetry


Solution...

They have a part structure.
Aligning horses requires aligning their parts.


So we need part structure.

What do we mean by parts?

How are these parts related?

How do we get this part structure?

Segmentation
Subdividing a mesh into simpler, more manageable
sub-parts

Types of segmentations [Attene et al. 2006]: [Katz et al. 2005]
Geometric: clusters of similar geometry
Semantic: how would a human segment the mesh?

Salience [Hoffman 1997, Lee 2005, Golovinsky 2008]
Geodesic distance [Tierny 2007, Berretti 2009]
Medial/functional [de Goes 2008, Shapira 2008]
[Berretti et al. 2009]
Global part boundaries


Skeleton extraction

Simplifying a 3D surface to a 1D curve-skeleton
[Dey and Sun 2006]
[Tierny 2006]

Approaches [Cornea 2007]:
Volumetric thinning [Svensson 2002, Siddiqi 2008]
Geometric [Tierny 2006, Tagliasacchi 2008, Agathos
2010]
[Siddiqi 2008]
Global connectivity of parts


What do we want to do?
We want to correspond part junctions (i.e. boundaries) and
part caps (i.e. part end-points). Then we can align those
correspondences in closed form.

What do we need?
Simple, robust algorithm for decomposing shape into part graph
that captures the part structures shared between two shapes of
the same object class

Must be:
Invariant to isometry
As minimal as possible


Inspiration

Connectivity Shapes of Isenburg et al. [2001]


ˆ
Breath-ﬁrst graph G

Let the topology of the mesh do the talking,
and this is what it says


ˆ
Breath-ﬁrst graph G
Given:

Mesh M = {V, E, F }
Seed vertices VS ⊆ V

Traverse M in a breadth-ﬁrst manner

Each frontier of the traversal becomes a node in a new
graph


ˆ
Breadth-ﬁrst graph G


ˆ
Breadth-ﬁrst graph G

mesh graph structure → simpliﬁed metagraph structure


ˆ
Segmentation from G

Cap segments: Contain a cap vertex
Pipe segments: Connect 2 junctions
Junction segments: Connect > 2 other seg-
ments

ˆ
Skeleton from G

Use edges of BFG
Vertex of skeleton is the centroid of mesh
vertices it encodes


Automatic BFG generation
Issues:
Seed vertices
“Hairs”


Seed vertices
Breadth-ﬁrst traversal tends to terminate at appendage tips

Use a “priming” run to ﬁnd appendage tips.

Problem: too many tips...


“Hairs”

Distilling the BFG:
Trim cap segments with only one edge
Get standard deviation (σ) of number of edges in all cap segments
Iteratively trim segments of length < σ

Usually < 4 iterations


Distilling the BFG

Original BFG skeleton Trim segments of length 1 Iteratively trim


Robust to...
AutoBFG
Start with mesh vertex closest to mesh
ˆ
centroid (vcent ), generate G0
ˆ
Distill G0
ˆ ˆ
Use caps of G0 , minus Vcent , to build G1
ˆ
Distill G1 Sampling density
ˆ
Return G1

TopoBFG
Purely topological BFG. Same as above except that the
initial seed vertex is chosen randomly.
Shape variation and pose


Benchmark for Segmentation [Chen 2009]
400 different models, 20 classes, 20 meshes/class
Based on human-generated segmentations
Tested 7 other algorithms, plus two sanity checks
Uses four different metrics (lower is always better)
Cut discrepancy:
Disagreement between segment boundaries and baseline
Hamming distance:
Disagreement between segment regions and baseline
Consistency error:
Disagreement between segment regions in a way that does not penalize
differences in hierarchical granularity
Rand index:
Unlikelihood that a pair of faces will agree on segment identity


Cut Discrepancy


Hamming Distance


Consistency Error


Rand Index


Shape Alignment


→ →

P and Q Skeleton for P and Q

→

Correspondence
Alignment


Shape Alignment
Two aspects of alignment:

Correspondence
Limit correspondence space to most semantically dissimilar vertices
Junction (part boundaries)
Cap (part tips)
Greedy search

Alignment
Given correspondence, the transform α which minimizes E(P, Q) can be solved for in closed
form [Arun 1987]

Real problem is correspondence


Correspondence
Very similar in spirit to greedy search of Mitra et al. [2005].

Greedy Correspondence:
ˆ ˆ ˆ ˆ
Require: BFGs GP = {VP , EP } and GQ = {VQ , EQ }ˆ ˆ
ˆ
FP = junctions and caps of GP ˆ
ˆ
FQ = junctions and caps of GQ ˆ
ˆ ˆ
for all vP ∈ FP do
ˆ ˆ
Topologically sort FP w.r.t. graph distance from vP
ˆ ˆ
Find the vQ ∈ FQ (of the same type as vP ) whose topological sorted FQ most agrees
ˆ ˆ
ˆ
with that of vP
end for

Imagine two graphs as made of string. Grab both at a vertex, and if the
knots line up, then it’s likely that they are corresponding points.


Complications
We’re extracting part structure from mesh topology.

What if the topology doesn’t reﬂect the part structure?


Complications
Correspondence is based on similarity of internal graph distance, as well as
Euclidean distance.


Alignment Results
Using Chen database, selected 10 object classes (200 meshes)
Selected semantically relevant landmarks (tops of heads, tips of
wings, front and back of body, etc.)
For some mesh MP and MQ , whose landmark vertices are LP ⊂ VP
and LQ ⊂ VQ , respectively (N = |LP | = |LQ |), and some aligning
transform α, the error associated with (MP , MQ , α) is

N
1
E(MP , MQ , α) = ||α(vi ) − wi ||, vi ∈ LP , wi ∈ LQ
N i=1

Tested against Generalized-ICP of Segal et al. [2009]
Results show mean error over each class


Alignment Results

1.4
Baseline
1.2 AutoBFG
TopoBFG
1 Gen ICP

0.8

0.6

0.4

0.2

0
0 50 100 150 200

Humans


Alignment Results
1.4
Baseline
1.2 AutoBFG
TopoBFG
1 Gen ICP

0.8

0.6

0.4

0.2

0
0 50 100 150 200

Hands


Alignment Results
1.4
Baseline
1.2 AutoBFG
TopoBFG
1 Gen ICP

0.8

0.6

0.4

0.2

0
0 50 100 150 200

Four legged animals


Conclusions
Shape alignment better with part information
Startling amount of part information in topology
Part information from breadth-ﬁrst traversal
ˆ
Alignment using the breadth-ﬁrst graph G

Future Work
Shape reconstruction from BFG
Applications to graph visualization
Integration into Blender3D [www.blender.org]
1. Alignment
2. Auto-rigging


Questions?


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