One obvious way to reduce noise is by weighted (e.g. 
Gaussian) or unweighted (i.e. box) moving averaging. Al- 
though the image noise is effectively reduced the apparent 
disadvantage is that also the edge strength is reduced. 
To reduce noise without affecting seriously edges, edge 
preserving (non-linear) smoothing filters are required. Three 
examples are: 
The Median Filter The grey value of the center pixel 
of a w x w window, w odd, is not replaced by the (weighted) 
mean, but by the median of the grey values in the win- 
dow. Median filters effectively reduce noise while preserv- 
ing edges. However, thin structures are removed; e.g., one 
pixel thick lines in 3 x 3 windows disappear. Additionally 
at corners anomalies are introduced. 
The Conditional Average Filter is a modification of 
the box filter. Only those values within the window, that 
don't deviate too much from the present value of the centre 
pixel are involved in the averaging process. A prescribed 
threshold is required which defines the allowed difference. 
The Kuwahara Filter is also a modification of the 
box filter. Four w x w windows are placed around the 
pixel such that it is located in one of the four corners. The 
variance of the grey values in each window is computed. 
Next the mean of the window with the lowest variance is 
assigned to the pixel. If the pixel is located at or nearby 
a ramp edge, the windows that cross the edge have a high 
variance, while the windows that occupy only one region 
have low variance. The unweighted averaging is performed 
over the low variance window, resulting in edge preserving 
properties. We have also implemented a modified version, 
in which a fifth w x w window is centred around the pixel. 
Now, on pixels within a homogeneous region, a standard 
box filtering is carried out. We call this filter the extended 
Kuwahara filter. 
4 Edge Based Region Growing 
4.1 Background 
Consider an image to be a two-dimensional curved surface 
in 3-D space. Consequently, we may look at it as a land- 
scape, much in the same way as a digital elevation model. 
Suppose that we have traced a homogeneous region in that 
landscape, e.g. a part of a plane. Now the idea is to pro- 
ceed from that initial region step-by-step, pixel-by-pixel, 
until one collapses against a mountain. 
Measures are required whether a mountain is reached. 
Our method predicts the grey value of each pixel, adjacent 
to a region of grouped homogeneous pixels. If the predicted 
grey value shows sufficient similarity with the observed one, 
the pixel is added to the region. If not and neither with one 
of the other surrounding regions, a new region is formed, 
consisting initially only of the concerning pixel. Rejection 
is due to the following properties of the concerning pixel: 
- The pixel belongs to a new region; 
- The pixel contains impulse noise; 
- The pixel is a mixed pixel, i.e. located at a boundary; 
- The pixel belongs to a previous region. 
If the pixel belongs to a previous region that is not 
connected with the present pixel, a new region is created. 
Where the ungrouped pixels encounter the grouped pixels 
of the same region, phantom edges are created. 
So, the above prediction procedure requires a second 
stage, to remove noisy clusters, elongated mixed pixels re- 
gions and phantom regions. In principle, this second stage 
is equivalent to the second stage in the split-and-merge 
paradigm. 
4.2 Stage I: Prediction 
Let us characterize each homogeneous image part, Ry, con- 
sisting of a collection of pixels g,(7,7) as a random field 
where the grey value of each individual pixel can be pre- 
dicted from the grey values of the pixels that have been 
previously examined to be part of R, 
9k(%,7) = 33,5) + nes, 7) (1) 
where gi (1, j) is an arbitrary prediction of g,(i, 7) and ny (i, 5) 
is another random field such that Eq.(1) realizes the co- 
variance properties of Rj (cf. Jain, 1989, p. 207). Let 
nk(¢,7) be a signal-independent, indentically-distributed, 
region-independent, Gaussian white noise field, that means 
n(i,j) = n,(3,7) ^ N(0,021). Consider the random field 
to be homogeneous or wide-sense stationary than the mean 
value: 
Elgx(i, j)] = Kk, 
and the covariance function: 
Cal, 3; 1,7) = Crlt— 1, — I) = Ci(p, 9) 
are spatial invariant. Suppose that the covariance function 
is region-indepen-dent, or in other words, each region has 
the same covariance function, i.e. C,(p, q) — C(p,q). A re- 
alistic image covariance function is the circularly symmetric 
or isotropic exponential function: 
pg 
C (p, q) ^ c? exp 2, (2) 
  
where c? represents the variance of the random field. Let 
c? — gl with o2 the variance of the additive noise field 
n(?,j). Than the prediction function to obtain g,(1, 7) such 
that the random field has covariance function according 
Eq.( 2) is: 
ptg 
207 
  
r(p,q) — ezp - (3) 
Fig. 1 illustrates the sampling of Eq.( 3) for use on a regular- 
spaced grid. The coefficients of the weight function should 
sum up to 1: 3°, cg, r(p,q) = 1. Consequently, the weight 
coefficients r(p,q) are divided by D, ger, r(p, q). Since it is 
our aim to test whether pixel (7,7) is part of region k, we 
have to check the hypothesis 
Ag; (7,7) = [dx (8, 7) = gx(5, 7)] =0 
than (7,7) is part of region k, against the hypothesis 
Agr(t, 3) = [gx (8,3) — gx(8,3)| # O 
Using the prediction equation Eq.( 3) under the restric- 
tion 3°, ger, "(P,4) = 1 the prediction g,(1,7) of g,(1, 7), 
becomes: 
A r(p, q)ax(t 4 p,J + p) (4) 
> pack, r(p, q) 
  
gu (7, J) = 
with prediction variance: 
) Yen; 72 (p, q) 
(Toren, r(p, q))? 
o2 (,J 
794 
S