SSIP 2005

Shape Moments for Region-Based Active Contours

Peter Horvath, Avik Bhattacharya, Ian Jermyn, Josiane Zerubia and

Zoltan Kato

SSIP 2005

Goalo Introduce shape prior into

the Chan and Vese model

Improve performance in the presence of:•Occlusion•Cluttered background•Noise

SSIP 2005

Overviewo Region-based active contours

o The Mumford-Shah modelo The Chan and Vese modelo Level-set function

o Shape momentso Geometric momentso Legendre momentso Chebyshev moments

o Segmentation with shape prior

o Experimental results

SSIP 2005

Mumford-Shah model

C

MS CdxudxuuCuE\

220

2 ||)(),(

1 2 3

1.Region similarity2.Smoothness3.Minimizes the contour length

oD. Mumford, J. Shah in 1989oGeneral segmentation model

oΩR2, u0-given image, u-segmented image, C-contour

SSIP 2005

Chan and Vese model I.o Intensity based segmentationo Piecewise constant Mumford-Shah

energy functional (cartoon model)o Inside (c1) and outside (c2) regions

o Active contours without edges [Chan and Vese, 1999]o Level set formulation of the above

modelo Energy minimization by gradient

descent

CdxcudxcuCccEoutin

CV

2202

210121 )()(),,(

SSIP 2005

Level-set methodo S. Osher and J. Sethian in 1988o Embed the contour into a higher

dimensional spaceo Automatically handles the

topological changes (., t) level set functiono Implicit contour ( = 0)o Contour is evolved

implicitly by moving the surface

SSIP 2005

Chan and Vese model II.o Level set segmentation model

o Inside >0; outside <0o H(.)-Heaviside step functiono It is proved in [Chan & Vese, ‘99]

that a minimizer of the problem exist

dxHdxHcudxHcuccECV |)(|))(1()()()(),,( 2202

210121

SSIP 2005

Overviewo Region-based active contours

o The Mumford-Shah modelo The Chan and Vese modelo Level-set function

o Shape momentso Geometric momentso Legendre momentso Chebyshev moments

o Segmentation with shape prior

o Experimental results

SSIP 2005

Geometric shape momentso Introduced by M. K. Hu in 1962

o Normalized central moments (NCM)o Translation and scale invariant

o (xc, yc) is the centre of mass (translation invariance)

dxdyM

yyxxqp

qc

pc

pq

2)2(00

)()(

Area of the object (scale invariance)

SSIP 2005

Legendre moments

o Provides a more detailed representation than normalized central moments:

1

1

1

1

),()()(4

)12)(12(dxdyyxfyPxP

qpqppq

Shape NCM () Legendre ()

•Where Pp(x) are the Legendre polynomials•Orthogonal basis functions

NCM is dominated by few moments while Legendere values are evenly distributed

SSIP 2005

Chebyshev momentso Ideal choice because discrete

o Where, ρ(n, N) is the normalizing term, Tm(.) is the Chebyshev polynomial

o Can be expressed in term of geometric moments

1

0

1

0

),()()(),(),(

1 N

i

N

jnmmn jifjTiT

MnNmT

Chebyshev polynomials:

SSIP 2005

Overviewo Region-based active contours

o The Mumford-Shah modelo The Chan and Vese modelo Level-set function

o Shape momentso Geometric momentso Legendre momentso Chebyshev moments

o Segmentation with shape prior

o Experimental results

SSIP 2005

New energy functiono We define our energy functional:

o Where Eprior defined as the distance between the shape and the reference moments

pq shape moments

),(),,(),,,( 2121 refpriorCVref EccEccE

Nqp

qp

pqref

pqrefpriorE

,

,

2)(),(

SSIP 2005

Overviewo Region-based active contours

o The Mumford-Shah modelo The Chan and Vese modelo Level-set function

o Shape momentso Geometric momentso Legendre momentso Chebyshev moments

o Segmentation with shape prior

o Experimental results

SSIP 2005

Geometric results

Reference object

Legendre moments

Chebyshev moments

p, q ≤12 p, q ≤16 p, q ≤20

p, q ≤12 p, q ≤16 p, q ≤20

SSIP 2005

Result on real imageOriginal image

Chan and Vese

Reference object

Chan and Vese with shape prior

SSIP 2005

Conclusions, future worko Legendre is fastero Chebyshev is slower but it’s

discrete nature gives better representation

o Future work:o Extend our model to Zernike

momentso Develop segmentation methods

using shape moments and Markov Random Fields

SSIP 2005

Thank you!

Acknowledgement:•IMAVIS EU project (IHP-MCHT99/5)•Balaton program•OTKA (T046805)