Robert Osada, Tom FunkhouserBernard Chazelle, and David

DobkinPrinceton University

Matching 3D Models With Shape Distributions

Shape Similarity Determine similarity between 3D

shapes

ComputerGraphics

ComputerVision

Computational Biology

[Caltech][Insulin, PDB]

Previous Work in 2D Shape representations

Fourier analysis [Arbter90] Turning function [Arkin91] Size function [Uras95]

Metrics for comparing curves Hausdorff Fréchet Bottleneck etc.

Previous Work in 3D High-level representations

Generalized cylinders [Binford71] Medial axis [Bardinet00] Skeletons [Bloomenthal99]

Statistical Moments [Reeves45, Prokop92] Crease angle [Besl94] Shells decomposition around centroid

[Ankerst99] Extended Gaussian Images [Horn84] etc.

Desired Properties Match global properties of shape Invariance

Rotation, translation, scale, mirror Robustness

Noise, cracks, insertions and deletions Practicality

Concise representation Efficient comparison Working with degenerate models

Our Approach Shape distributions

Concise shape descriptor Common parameterization Function of random points

3D Model ShapeDistribution

Parameterization

Random sampling

Our Approach

SimilarityMeasure

Parameterization

Parameterization

3DModel

ShapeDistribution

ShapeFunction

Issues Which shape function? How to compare shape distributions?

Parameterization

Parameterization

SimilarityMeasure

Issues Which shape function? How to compare shape distributions?

Parameterization

Parameterization

SimilarityMeasure

Which Shape Function? Computationally simple options (~

1s) Based on random points Angles, distances, areas, volumes

A3 D1

D2 D3

[Ankerst99]

D4

Shape Function – D2 Distance between

two random points on surface Line Segment Circle

Triangle Cube

Cylinder Sphere

Two adjacent spheres

Two spheres moving apart

Which Shape Function? Sneak preview

Shape Function – Key Questions Invariant?

Rotation, translation, mirror (not scale)

Robust? Noise, cracks, insertions and deletions

Descriptive?

Issues Which shape function? How to compare shape

distributions?

Parameterization

Parameterization

SimilarityMeasure

Comparison1. Normalize for scale2. Compare shape distributions

Parameterization Parameterization

Normalization for Scale

max mean search

Compare shape distributions Computationally simple options

(~ .1ms) Ln norms of densities (PDF) or

cumulative densities (CDF) More complex options

Earth mover’s distance, Bhattacharyaa distance.

PDF CDF

Experimental Results Goal is to address the following:

Is the method robust? How well does it classify?

Shape Function

Normalization Comparison

• • •A3 D1 D2D3 D4

Max Mean Search

PDF L1 L2 L

CDF L1 L2 L

Robustness Experiment 10 Models

Car Chair Human Missile Mug

Phone Plane Skateboard Sub Table

Robustness Experiment 6 Transforms

Rotate, scale, mirror, noise, delete, insert

Total of 70 models

1% Noise 5% Deletion

Robustness Results Resulting distributions stable

7 Mugs

Distance

Prob

abilit

y

7 Missiles

Classification Experiment 133 Models

categorized into25 Groups

Large variety within a group among groups

4 Mugs

6 Cars

3 Boats

Classification Results

4 Balls5 Animals 2 Belts 3 Blimps 3 Boats

6 Cars 8 Chairs 3 Claws 4 Helicopters11

Humans

3 Lamps 3 Lightnings6

Missiles 4 Mugs4

Openbooks

Classification Results

Distance

Prob

abilit

y

4 Balls5 Animals 2 Belts 3 Blimps 3 Boats

6 Cars 8 Chairs 3 Claws 4 Helicopters11

Humans

3 Lamps 3 Lightnings6

Missiles 4 Mugs4

Openbooks

Classification Results

Line Segment Circle

Triangle Cube

Cylinder Sphere

Two adjacent spheres

Two spheres moving apart

NearestNeighbor1st

Tier2nd

Tier

Classification Results Avoid bias due to varying group sizes

Query

…

Resu

lts

an ba be bl bo car chair cl he human la li mis mug op pen ph plane ri sk so sp sub ta tank

animal

ballbeltblimpboat

car

chair

claw

helicopter

human

lamplightning

missle

mug

openbook

pen

phone

plane

rifle

skateboard

sofa

spaceship

sub

table

tank

an ba be bl bo car chair cl he human la li mis mug op pen ph plane ri sk so sp sub ta tank

animal

ballbeltblimpboat

car

chair

claw

helicopter

human

lamplightning

missle

mug

openbook

pen

phone

plane

rifle

skateboard

sofa

spaceship

sub

table

tank

Similarity matrix Nearest Neighbor 1st Tier 2nd Tier

Blocks Tanks Mugs Humans Airplanes Boats

Classification Results

Classification ResultsShape

FunctionFirstTier

SecondTier

Nearest Neighbor

A3 38% 54% 55%D1 35% 48% 56%D2 49% 66% 66%D3 42% 58% 58%D4 32% 42% 47%

Comparison to Moments Method

Align 1st moments (translation) Align 2nd moments (rotation and scale) Compare using remaining moments (L2)

ShapeFunction

FirstTier

SecondTier

NearestNeighbor

D2 49% 66% 66%M3 35% 46% 63%M4 41% 52% 64%M5 28% 38% 55%M6 34% 44% 54%M7 27% 33% 51%

Conclusion – Properties Match global properties of shape Invariance

Rotation, translation, scale, mirror Robustness

Noise, cracks, insertions and deletions Practicality

Concise representation Efficient comparison Works for degenerate models

Conclusion – Key Ideas Sampling gives common

parameterization Simplifies comparison

Comparing distributions is fast and easy Avoids registration, correspondence, etc.

Simple shape functions are discriminating Method suitable as preclassifier

Future Work Use a larger and more controlled

database Combine shape distributions with

other classifiers into a working shape-based retrieval system

Thank you Sloan

Foundation NSF