Reflective Symmetry Detection in 3 Dimensions

Michael Kazhdan

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

GoalPresent a shape-descriptor for model analysis

Tasks:• Registration• Matching

Properties:• Parameterized over canonical domain• Insensitive to noise• Global

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

Related Work

Alignment:• Generalized Hough Transform Ballard (1981)• Geometric Hashing Lamdan (1989)• Iterative Closest Point Besl, McKay (1992)

Locally Parameterized Features:• Spin Images Johnson, Hebert (1999)• Harmonic Shape Images Zhang, Hebert (1999)

Related Work

Canonically Parameterized Features:• Extended Gaussian Images Horn (1984)• Spherical Attribute Images Dellinguette, Hebert,

Ikeuchi (1993)• Orientation Histograms Sun, Si (1999)• Moments Elad, Tal, Ar (2001)• Shape Distributions Osada, Funkhouser, Chazelle,

Dobkin (2001)

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

A function associating a measure of reflective symmetry to every plane through the origin

Need to address:– How do we measure symmetry?– How do we compute the measure efficiently?

Reflective Symmetry Descriptor

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

Measure of SymmetryQ: How close is a function f to be symmetric

w.r.t to a reflection r?

A: What is the distance to the nearest function g that is symmetric w.r.t. to r?

2)(|

min),( gfrfSymgrgg

fr

g = ?

Measure of SymmetryBecause the space of functions is a Hilbert space…

Because reflection preserves the inner product…

The closest symmetric function to f is the average of f with its reflection

2

+ =

Measure of SymmetrySo that the measure of symmetry of f w.r.t. the

reflection r is the (scaled) distance of f from its reflection:

22

)(),(

frfrfSym

),( rfSym-

22

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

Functions on a Circle

If f(t), t[0,2], is a function defined on a circle then the measure of symmetry of f with respect to reflection about the angle is:

2

)α2()(

2

)α2()(

)α,(

n termconvolutionormL

2

2

2

tftff

tftf

fSym

2-t

t

Functions on a Disk

...)α, ()α, ()α, ()α,( 2

factor scale

32

factor scale

22

factor scale

1 SymrSymrSymrfSym

f {fr1,fr2

,fr3,…}

Functions on a Sphere

projectedprojected f f

ff

Step 1: “North pole” symmetries by projection.

Step 2: All symmetries by walking a great circle.

Decompose the grid into concentric spheres, and apply the results for symmetry descriptors of spheres to the voxel grid.

Voxel Grids

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

Distinguishing Between Classes

Similarity Within Classes

Symmetry Within Classes

How Well is “Shape” Captured?

Evaluate how well models can be:

• Registered

• Aligned

by only using their symmetry descriptors

Registration Experiment (Ideal)

Given a collection of models that are classified into groups and aligned:

• For each pair of models within a group:– Find the rotation minimizing the L2-distance of

the symmetry descriptors– Evaluate how close the minimizing rotation is

to the registering rotation

Evaluating the Rotation

If M is the ideal registering rotation and N is the minimizing rotation found, how close is M to N?

What is the angle of the rotation of MN-1?

Registration Experiment (Practice)

Searching over the space of all rotations is computationally prohibitive:

• We know the axis about which the ideal aligning rotation occurs

• Search for best rotation about this axis

Registration Results

Symmetry Principal Axes

Model Database: Subset of Osada database that fully voxelized to 128x128x128

87 models, 24 Groups

Rotation Error

Rotation Error

% of Models % of Models

Problems With Covariance

Multi-Dimensional eigenspaces:

Registration Results

Matching Experiment (Ideal)

Given a collection of models classified into groups:

• For each pair of models:• Find rotation minimizing the L2-distance of the

symmetry descriptors

• Use the minimum L2- distance as a measure of the match quality.

Matching Experiment (Heuristic)

Searching over the space of all rotations is computationally prohibitive:

• Find the principal axis of symmetry of each of the models

• Search for minimizing rotation that maps principal axes of symmetry to each other

Matching Results

First Tier: If the query model belongs to a class with n models, how many of the top (n-1) matches are also in that class?

Nearest Neighbor: How often is the top match in the same class as the query model?

First Tier First Two Tiers

Nearest Neighbor

Time

Shape Distributions

47% 67% 62% 7.4 seconds

Symmetries 48% 61% 69% ~4,500 seconds

Model Database: Subset of Osada database that fully voxelized to 128x128x128

87 models, 24 Groups

Matching Results

Properties

• Parameterized over canonical domain:– Parameterized over the (projective) sphere

• Insensitive to noise:– Integration scales down high frequency Fourier

coefficients

• Global– For functions f and g, and any reflection r:

2),(),( gfrgSymrfSym

Overview

• Introduction

• Related Work

• Definitions and Computation– Definition– Computation

• Results

• Future Work

Future Work

• Consider applications of the L properties of the symmetry descriptor

• Determine what information about shape can be easily extracted from the descriptor

• Explore the potential orthogonality of different matching methods

• Apply other alignment methods to the symmetry descriptors