Shape Based Image Retrieval Using Fourier Descriptors

Dengsheng Zhang and Guojun Lu

Gippsland School of Computing and Information Technology

Monash University

Churchill, Victoria 3842

Australia

dengsheng.zhang, guojun.lu@infotech.monash.edu.au

Outline

Introduction Shape Signatures Fourier Descriptors Retrieval Experiments Conclusions

Introduction-I--shape feature

What features can we get from a shapeshape?

perimeter, area, eccentricity, circularity, chaincode…

Introduction-II--Classification

Shape

Contour Region

Structural

SyntacticGraphTreeModel-drivenData-driven

PerimeterCompactnessEccentricityFourier DescriptorsWavelet DescriptorsCurvature Scale SpaceShape SignatureChain CodeHausdorff DistanceElastic Matching

Non-Structural AreaEuler NumberEccentricityGeometric MomentsZernike MomentsPseudo-Zernike MmtsLegendre MomentsGrid Method

Introduction-III--criteria

Criteria for shape representation Rotation, scale and translation Invariant Compact & easy to derive Perceptual similarity Robust to shape variations Application Independent

FD satisfies all these criteria Problem

Different shape signatures are used to derive FD, which is the best?

Shape Signatures

Complex Coordinates Central Distance Chordlength Curvature Cumulative Angles Area function Affine FD

Complex Coordinates

z(t) = [x(t) – xc] + i[y(t) - yc]

1

0

1

0

)(1

,)(1 N

tc

N

tc ty

Nytx

Nx

Central Distance

r(t) = ([x(t) – xc]2+ [y(t) - yc]

2)1/2

Chordlength

The chord length function r*(t) is derived from shape boundary without using any reference point

Cumulative Angular Function

(t) = [ (t) - (0)]mod(2)

L is the perimeter of the shape boundary

tLt

t )2

()(

Curvature Function

K(t) = (t) - (t-1)

w is the jumping step in selecting next pixel

)()(

)()(arctan)(

wtxtx

wtytyt

Area Function

|)()()()(|2

1)( 1221 tytxtytxtA

Fourier Descriptors

Fourier transform of the signature s(t)

un, n = 0, 1, …, N-1, are called FD denoted as FDn

Normalised FD

Where m=N/2 for central distance, curvature and angular functionm=N for complex coordinates

1

0

)2

exp()(1 N

tn N

ntjts

Nu

]||

||,...,

||

||,

||

||[

00

2

0

1

FD

FD

FD

FD

FD

FD mf

Affine Invariants

**

**

pppp

pkpkk

XYYX

XYYXQ

k = 1, 2, …

where Xk, Yk are the Fourier coefficients of x(t), y(t) respectively

Convergence Speed-I• Finite number of coefficients are used to approximate the signal. The partial Fourier sum of degree n of u(t) is given by

nk

jktn ekutuS

||

)(ˆ))((

)())((lim tutuSnn

• For piecewise smooth function u(t), there exists a one-to-one correspondence between u(t) and the limit of their Fourier series expansion

• For shape retrieval application, the number of coefficients to represent a shape should not be large, therefore, the convergence speed of the Fourier series derived from the signature function is crucial

Convergence Speed-II

r(t)

(t)

r*(t) z(t)

k(t)

(t)

Convergence Speed-III

• Ten very complex shapes are selected to simulate the worst convergence cases

 Signature functions

 Number of normalized spectra greater than 0.1

 Number of normalized spectra greater than 0.01 

r(t) 15 120

r*(t) 40 360

A(t) 20 210

z(t) 10 50

(t) 40 280

(t) k(t) 100 600

Qk 20 100

FD Indexing

Indexing each shape in the database with its Fourier Descriptors

Similarity between a query shape and a target shape in the database is

2/1

1

2 ))((

m

i

ti

qi ffd

lyrespectiveshapestwotheofvectorsfeature

thearefffandfffwhere mtttt

mqqqq ),...,,(),...,,( 2121 ff

Retrieval Experiments

A database consisted of 2700 shapes is created from the contour shape database used in the development of MPEG-7. MPEG-7 contour shape database is consisted of set A, B and C. Set A has 421 shapes, set B has 1400 shapes which are generated from set A through scaling, affine transform and arbitrary deformation and defection. Set C has 1300 shapes, it is a database of marine fishes.

Performance measurement: precision and recall Precision P is the ratio of the number of relevant retrieved shapes r to the total

number of retrieved shapes n. Recall R is the ration of the number of relevant retrieved shapes r to the total number m of relevant shapes in the whole database.

m

rR

n

rP

Results

0

1020

30

4050

60

70

8090

100

0 10 20 30 40 50 60 70 80 90 100 110

Recall

Pre

cisi

on

aff ine FDs

area function FDs

central distance FDs

chord length FDs

curvature function FDs

position function FDs

psi function FDs

Conclusions A comparison has been made between FDs derived

from different shape signatures, FDs with affine FDs In terms of overall performance, FDs derived from

central distance outperforms all the other FDs Curvature and angular function are not suitable for

shape signature to derive FDs due to slow convergence Affine FD is designed for polygon shape, it does not

perform well on generic shape Indexing data structure will be studied in the future

research Comparison with other shape descriptors