Post on 23-Jan-2016
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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)