A Segmentation Algorithm Using
Dyadic Wavelet Transform and
the Discrete Dynamic Contour
Bernard Chiu
University of Waterloo
Agenda





Introduction and Problem Definition
Proposed Solution
Dyadic Wavelet Transform
Discrete Dynamic Contour (DDC) Model
Conclusion
Introduction and Problem Definition



This segmentation algorithm is developed for
detecting the prostate boundary in 2D
ultrasound images.
Prostate segmentation is a required step in
determining the volume of a prostate.
The radiologists manually segment at least a
hundred cross-sectional ultrasound images
before they can obtain an accurate estimate
of the prostate volume  Too timeconsuming!
Introduction and Problem Definition


To reduce the time required for prostate
segmentation, a computerized method
has to be developed.
The quality of the ultrasound images
make it a very difficult task, because of
the presence of false edges.
Proposed Solution
Fast Dyadic
Wavelet
Transform
a0
M 2j
aj
Original Ultrasound Image
Four
user-defined
initial
points
Initialization
Four
Initial Points
Initial
Contour
DDC Model
Initial
Contour
Energy
Field
Resulting
Contour
Final Contour
The Dyadic Wavelet Transform
§ What is the relationship between edge
detection and wavelet analysis?
o It is always possible to find wavelet that have one
vanishing moment such that
 x, y 
  x , y 
2



x
,
y


 x , y   
and
y
x
1
where x, y  is a two-dimensional smoothing function
that converges to 0 at infinity and integrates to a
positive value.
The Dyadic Wavelet Transform
o Suppose f Î L2(R2), the wavelet transforms of
2
1
respect to  and  are defined by
W2kj f u, v    f x, y , 2kj x  u, y  v    f  2kj u, v 
for
f with
k  1,2
o The wavelet transform components are proportional to the
corresponding components of the gradient vector of
f  2 j :








f


j u , v 
2


f u ,v 
j u

2 
 2 j  f   2 j u ,v 

W22j f u ,v 
  f   j u ,v 
2
 v

W21j


The Dyadic Wavelet Transform
o In the previous equation, it is defined that, for any  Î
1  x y



x
,
y

 j , j 
j
2 2
 x , y    2 j  x , y  .
j
2
L (R ),
2  2 2  and 2 j
o The modulus of this gradient vector is proportional to the
wavelet transform modulus:
M 2 j f u ,v   W f u ,v   W f u ,v 
1
2j
2
2
2j
2
The Fast Discrete Dyadic Wavelet
Transform
§ How can the idea of the dyadic wavelet transform be
applied to a digital image?
o The original image can be modelled according to
a0 m,n  f x , y ,x  m, y  n 
o For any j  0 , we define the larger scale approximation a j by
a j m ,n  f x , y , 2 j x  m , y  n  
o Denote
the integer samples of wavelet transforms at scale 2j by
d 1j and d 2j :
d kj m,n  W2kj f m,n  f x, y ,2kj x  m, y  n  for k  1,2 .
The Fast Discrete Dyadic Wavelet
Transform
§ If  and  are separable, a j , d j and d j can be
obtained iteratively using a filter bank structure.
1
2
§ h and g, called the scaling sequence and the
wavelet sequence respectively, are defined by the
equations
 x    hk  22 x  k  and x    g k  22 x  k 
kÎZ
kÎZ
§ h j n is denoted to be to be the filter obtained by
inserting 2 j  1 zeros between each sample of hn .
The Fast Discrete Dyadic Wavelet
Transform
row by row
operation



 hj
aj
column by column
operation



 hj
a j 1
d 1j 1
gj
gj
d 2j 1
Example: The scale-24 gradient modulus calculated for a
typical ultrasound prostate image
The DDC Model
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Motivation for introducing the DDC
model
Terminology
Initialization
Internal and External Forces
Dynamics
Resampling
Motivation for introducing the DDC Model



After obtaining the gradient modulus using
the fast dyadic wavelet transform, one needs
to estimate the location of the edge.
In Mallat’s multiscale edge detection
algorithm, edge are defined by joining the
local maxima points of the gradient.
Unfortunately, it is almost impossible to get a
closed contour using this method.
The DDC model can be used to solve this
problem since it is always closed.
Terminology
Vi-1
Vi
di-1
pi-1
pi
di
Vi+1
pi+1
di+1
Terminology
dˆi 1
tˆi
Vi
dˆi
di-1
r̂i
Vi+1
di
ci
Vi-1
Define curvature, tangent and radial vector
ci  dˆi  dˆi 1
dˆi  dˆi 1
tˆi 
dˆi  dˆi 1
 0 1
rˆi  
tˆi

  1 0
Initialization

The spline interpolation techniques are used
to define the initial contour.
Internal and External Forces
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
The purpose of internal forces is to minimize the
curvature of the contour, so that the general shape of
the segment is not distorted by small/irregular
features.
User defines an energy function, Eim, that relates to
some kind of image feature.
f im  Eim


f im ,ri  f im ,Vi  r̂i r̂i
Contour Dynamics


Total force
f i  wex f ex ,ri  win f in,i  wdampvi
Calculate position, velocity and
acceleration
pi t  t   pi t   vi t t
vi t  t   vi t   ai t t
1
ai t  
f i t 
mi
Resampling
tion
resolu
ti o n
resolu
ti o
resolu
n
residual
ion
t
u
l
so
re
Vi+1
Vi+1
Vi
Vi
new point on the contour
after resampling
Vi-1
(a)
new point on the contour
after resampling
residual
Vi-1
(b)
Evaluation Criteria – Distance-Based Metric
d i
i
Evaluation Criteria – Distance-Based Metric
§ d is measured for 100 uniformly sampled angles in the
interval 0, 2 .
§ Three quantities are computed:
o The absolute mean deviation (MAD) defined by
100
MAD   d i  100
i 1
o The maximum deviation (MAXD) defined by
MAXD  max  d i  
i
o Out of the 100 values in the set  d 1  , d 2  ,  , d 100  , the
number of values that are smaller than 4 pixels.
Results
No.
1
2
3
4
5
6
7
8
9
10
Mean
MAD
MAXD % of points where
(in pixel) (in pixel) d  i   4 pixels
4.0435
4.4742
2.3161
3.9666
4.4374
2.0757
3.9595
3.5645
5.7222
3.2352
3.7795
27.1969
21.4126
7.2617
14.6658
28.3313
7.1438
18.9977
18.6715
29.7247
17.1046
19.0511
71
66
76
58
54
87
65
72
51
79
67.9
Conclusion

Proposed a new algorithm that


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

uses the gradient modulus of the image obtained using dyadic
wavelet transform, and
defines the edge using the DDC model, which is driven by an
external energy field proportional to the gradient modulus.
This algorithm always gives a closed segment, which
cannot be obtained by using Mallat’s multiscale edge
detection algorithm.
The proposed algorithm requires a radiologist to enter
four initial points, rather than 30-40 points required to
define the prostate boundary.
Our algorithm is obviously not perfect. In particular, the
performance of the DDC model is sensitive to the initial
contour. Fortunately, in most cases, the initialization
method introduced approximates the actual boundary with
reasonable accuracy.