Shape Moments for Region- Based Active Contours

Shape Moments for Region-
Based Active Contours
Peter Horvath, Avik Bhattacharya,
Ian Jermyn, Josiane Zerubia and
Zoltan Kato
SSIP 2005

Goal
o Introduce shape prior into the
Chan and Vese model
Improve performance in
the presence of:
•Occlusion
•Cluttered background
•Noise
SSIP 2005

Overview
o Region-based active contours
o The Mumford-Shah model
o The Chan and Vese model
o Level-set function
o Shape moments
o Geometric moments
o Legendre moments
o Chebyshev moments
o Segmentation with shape prior
o Experimental results
SSIP 2005

Mumford-Shah model
oD. Mumford, J. Shah in 1989
oGeneral segmentation model
o••R 2 , u 0 -given image, u-segmented
image, C-contour
E
MS
2
2
( u,
C)
= λ ( u − u)
dx + ∇u
dx + µ
∫
Ω
0
∫
Ω\
C
1 2 3
1. Region similarity
2. Smoothness
3. Minimizes the contour length
2
| C
|
SSIP 2005

Chan and Vese model I.
o Intensity based segmentation
o Piecewise constant Mumford-Shah
energy functional (cartoon model)
o Inside (c 1 ) and outside (c 2 ) regions
2
2
ECV ( c1 , c2,
C)
λ1
( u0
− c1
) dx + λ2
( u0
− c2)
dx +
= ∫ ∫
Ω
in
o Active contours without edges [Chan
and Vese, 1999]
o Level set formulation of the above model
o Energy minimization by gradient descent
Ω
out
C
SSIP 2005

Level-set method
o S. Osher and J. Sethian in 1988
o Embed the contour into a higher
dimensional space
o Automatically handles the topological
changes
o φ(., t) level set function
o Implicit contour (φ = 0)
o Contour is evolved
implicitly by moving
the surface φ
SSIP 2005

Chan and Vese model II.
o Level set segmentation model
o Inside φ>0; outside φ

Overview
o Region-based active contours
o The Mumford-Shah model
o The Chan and Vese model
o Level-set function
o Shape moments
o Geometric moments
o Legendre moments
o Chebyshev moments
o Segmentation with shape prior
o Experimental results
SSIP 2005

Geometric shape moments
o Introduced by M. K. Hu in 1962
o Normalized central moments (NCM)
η
o Translation and scale invariant
o (x c , y c ) is the centre of mass (translation
invariance)
x x
p
y y
q
c
c
pq = ( − ) ( − )
∫∫
M
( p+
q+
2) 2
00
Area of the object
(scale invariance)
dxdy
SSIP 2005

Legendre moments
λ pq
=
(2 p
+ 1)(2q
4
+ 1)
1
1
∫ ∫
−1
−1
P
( x)
P
( y)
f ( x,
y)
dxdy
o Provides a more detailed representation than
normalized central moments:
Shape
p
•Where P p (x) are the
Legendre polynomials
•Orthogonal basis functions
NCM (h) Legendre (l)
q
NCM is dominated
by few moments
while Legendere
values are evenly
distributed
SSIP 2005

Chebyshev moments
o Ideal choice because discrete
T
mn
=
N − 1 N −1
1
ρ(
m,
N)
ρ(
n,
M )
∑∑
i=
0
j=
0
( i)
T
o Where, ƒ(n, N) is the normalizing term,
T m (.) is the Chebyshev polynomial
o Can be expressed in term of
geometric moments
Chebyshev
polynomials:
T
m
n
( j)
f
( i,
j)
SSIP 2005

Overview
o Region-based active contours
o The Mumford-Shah model
o The Chan and Vese model
o Level-set function
o Shape moments
o Geometric moments
o Legendre moments
o Chebyshev moments
o Segmentation with shape prior
o Experimental results
SSIP 2005

New energy function
o We define our energy functional:
E ( c1,
c2,
φ,
π
ref
) = ECV
( c1,
c2,
φ)
+ E
prior
( π
ref
, φ)
o Where E prior defined as the distance
between the shape and the reference
moments
E prior
p,
q ≤ N
∑
pq
( π , φ)
= ( π −π
ref
p,
q
pq
ref
2
)
o π pq shape moments
SSIP 2005

Overview
o Region-based active contours
o The Mumford-Shah model
o The Chan and Vese model
o Level-set function
o Shape moments
o Geometric moments
o Legendre moments
o Chebyshev moments
o Segmentation with shape prior
o Experimental results
SSIP 2005

Geometric results
Legendre moments
Reference
object
p, q •12 p, q •16 p, q •20
Chebyshev moments
p, q •12 p, q •16 p, q •20
SSIP 2005

Result on real image
Original
image
Reference
object
Chan and Vese
Chan and Vese
with shape prior
SSIP 2005

Conclusions, future work
o Legendre is faster
o Chebyshev is slower but it’s
discrete nature gives better
representation
o Future work:
o Extend our model to Zernike
moments
o 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)
SSIP 2005