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Niloy  J.  Mitra

University  College  London

Tutorial

Symmetry in Shapes

Theory and Practice

(2)

What we do not cover?

Symmetry detection on images

Global symmetry detection on 3D geometry

Intrinsic symmetry detection

(3)

Regular Structure

+

(4)

Difficult

• Which parts are symmetric objects are not pre-segmented

• Space of transforms: rotation, translation, scaling, etc.

• Brute force search is not feasible

Easy

• Proposed symmetries easy to validate

Problem Characteristics

(5)

Symmetry Detection

M

(6)

M 1 ⇡ T (M 2 )

Geometric Matching

M 1 M 2

(7)

Types of Symmetry

• Reflection

• Rotation + translation

• Uniform scaling

(8)

Typical Stages

• Feature selection

• Aggregation

• Extraction

F (M ) = F (T (M ))

(9)

Geometric Hashing

Features: quadratic patch parameters Aggregation: geometric hashing

Extraction: pre-segmentation

[Gal et al. 2006]

(10)

Symmetry Transform

Features:

Aggregation: FFT in transform domain

Extraction: clustering, region growing

(11)

Goal

A computational representation that describes all planar symmetries of a shape

?

(12)

Symmetry Transform

A computational representation that describes all planar symmetries of a shape

?

(13)

Symmetry Transform

A computational representation that describes all planar symmetries of a shape

?

Symmetry = 1.0 Perfect Symmetry

(14)

Symmetry Transform

A computational representation that describes all planar symmetries of a shape

?

Symmetry = 0.3 Local Symmetry

(15)

Symmetry Transform

A computational representation that describes all planar symmetries of a shape

?

Symmetry = 0.2 Partial Symmetry

(16)

Symmetry Transform

A computational representation that describes all planar symmetries of a shape

?

Symmetry = 0.2

d(M, T ) = M T (M )

2

(17)

Transform Domain Analysis

Features: curvatures

Aggregation: transform domain analysis Extraction: region growing

[Mitra et al. 2006]

(18)

Reflective Symmetry

(19)

Reflective Symmetry: A Pair Votes

(20)

Reflective Symmetry: Voting Continues

(21)

Reflective Symmetry: Voting Continues

(22)

Height of cluster size of patch

Spread of cluster level of approximation

Reflective Symmetry: Largest Cluster

(23)

Pipeline

(24)

Pipeline

(25)

Rigid Transformations

(26)

Mean-Shift Clustering

(27)

correction field detected symmetries

Detection Results: Dragon

(28)

Insight: Global to Local Problem

(Euclidean) symmetry in spatial domain

cluster(s) in transform domain

(29)

2D Example: Symmetrization

[Mitra et al. 2007]

(30)

2D Example: Symmetrization

(31)

2D Example: Symmetrization

(32)

2D Example: Symmetrization

(33)

2D Example: Symmetrization

(34)

2D Example: Symmetrization

(35)

2D Example: Symmetrization

(36)

Symmetrization: Bunny

(37)

Graph-based Symmetries

Features: slippage analysis

Aggregation: locally coherent line arrangements Extraction: simultaneous refinement

[Bokeloh et al. 2009]

(38)

Algorithm Pipeline

(39)

Symmetry of Symmetries

Features: curvatures

Aggregation: transform domain model extraction Extraction: simultaneous refinement

[Pauly et al. 2008]

(40)

Structure Discovery

Input Model

Structure Discovery

(41)

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Regular Structures Structure Discovery

(42)

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Aggregation Structure

Discovery

(43)

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Transform Generators Regular Structures

Aggregation Structure

Discovery

spatial

domain transform

domain

(44)

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Aggregation Structure

Discovery

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Aggregation Structure

Discovery

(45)

Transformations

(46)

Transformations

(47)

Transformations

(48)

Transformations

(49)

Transformations

spatial domain transformation space

(50)

Transformations

spatial domain transformation space

(51)

Transformations

spatial domain transformation space

(52)

Model Estimation

density plot of

pair-wise transformations

origin

(53)

Optimization in Transform Domain

(54)

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Aggregation Structure

Discovery

(55)

Structure Discovery

Input Model

Transform Analysis

Transform Clusters

Model Estimation

Transform Generators Regular Structures

Aggregation Structure

Discovery

Aggregation Aggregation Aggregation

Aggregation

(56)

Chambord Castle

(57)

Chambord Castle

(58)

Symmetry Factored Encoding

a ! b

b ! c

c ! a

a ! c

finding cliques amounts to spectral analysis

Referanser

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