Shape Moments for Region-Based Active Contours
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Transcript of Shape Moments for Region-Based Active Contours
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SSIP 2005
Shape Moments for Region-Based Active Contours
Peter Horvath, Avik Bhattacharya, Ian Jermyn, Josiane Zerubia and
Zoltan Kato
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SSIP 2005
Goalo Introduce shape prior into
the Chan and Vese model
Improve performance in the presence of:•Occlusion•Cluttered background•Noise
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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
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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
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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 )()(),,(
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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
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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
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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
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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)
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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
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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:
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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
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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)(),(
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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
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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
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Result on real imageOriginal image
Chan and Vese
Reference object
Chan and Vese with shape prior
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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
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Thank you!
Acknowledgement:•IMAVIS EU project (IHP-MCHT99/5)•Balaton program•OTKA (T046805)