And the variance of the estimate Agx(1,j) becomes since 
according Eq.( 3) r(0,0) — 1: 
t c fo S^ r? (p q) 
2 = a? ure e 
ó,(1, 7) On(t,3) (1 + py r(p, q))? 
We apply the standard z-score to test whether the hypoth- 
esis that pixel (i, j) is part of the region R,, which leads to 
the null-hypothesis Ag (?,j) = 0. 
z — Agr(i, 3) 
à, (1, J) 
(5) 
(6) 
where Agi(1, 3) — |gx(3, 7) — gx (5, 7)| and ôŸ(i,7) as defined 
in Eq.( 5). 
Remark: During our experimentation we got the ex- 
perience that the estimated prediction variances of larger 
regions is becoming rather small, and consequently the deci- 
sion that the pixel (7,7) under consideration doesn’t belong 
to region k is taken too easily. In order to avoid this disad- 
vantage, we do not involve the original number of pixels, but 
their square root. The rationale guiding this approach is 
based on the fact that the pixels are scattered over a plane, 
i.e have a two-dimensional extension. To bring them back 
to one-dimensional proportions, the square root is taken. 
The implementation of this rationale resulted in a consider- 
ably improvement of the performance of Stage I and, since 
the same considerations had to be applied to Stage II, a 
considerably improvement of the final performance. 
According the above considerations the estimation of 
the prediction variance becomes now: 
1/2 
2 [ 1+ Speer, (2,0) ]" 
62 1,3) — o; 
eme a Ey 
So, the final decision rule becomes: IF 
minkeu (Agk(i, 7)) < ZaO0n > 
rk 
(7) 
(8) 
THEN assign pixel (7, j) to region R, for which Ag;(1, 7) is 
minimum ELSE start a new region. 
U is the set of adjacent regions of pixel (7,7) a is the 
probability that pixel (7,7) is wrongly assigned to region 
Rı, and 
4 
> = 1 + PER. di (p, q) V 
rk (1 Xen, r(p, 4)? 
Before entering Stage I, the image may require noise reduc- 
tion by one of the smoothing scheme's of section 3. 
4.3 Stage II: Merging 
The result of Stage I is that homogeneous regions are cre- 
ated. Although Stage I tends to trace edges at the correct 
location, many phantom edges are introduced, due to de- 
pendency on the scan direction of the predictor as explained 
before. Further, also noisy image parts and mixed pixels 
boundary regions are traced as separate homogeneous re- 
gions. The aim of Stage II, is to remove: 
a) phantom regions; 
b) small regions, and 
c) mixed pixels boundary regions. 
4.3.1 Stage IIa: Removal of Phantom Regions 
Statistical formulation whether two adjacent regions R, and 
R, are homogeneous requires information about the means 
and variances of R, and R,. 
795 
As in Stage I we could use the variance of the image 
noise o7. However, in Stage I, the value of o? is not critical. 
Since in real images the noise is not isotropic distributed 
over the image, we need a better estimate for Stage II. We 
estimate the variance of the noise for each of the Stage I 
regions individually. Consequently, we have to replace the 
z-score by the t-score. 
Let E, and R, be two adjacent regions, each being ho- 
mogeneous. Let (u,,0;) and (u,,02) be the mean grey value 
and variance of E, and R,, respectively: 
1 1 
H)-— S^ gi et) => ST (ut) - gi)? (9) 
ny.) iER(.) ny. iER() 
Where (.) = k,£. Than the t-score becomes: 
_ Hike — Pkt(hyp) 
t ^ 
Okt 
(10) 
where: Ure = |Ur — ne] 
Vre(hyp) = 0 
01, — ln. + 62 /n, — ó?(n, t nj)/nyn, 
Where 82, the pooled variance, is obtained by a weighted 
averaging of the variance estimates of the two regions R; 
and R,, with the weights based on their respective degrees 
of freedom: 
(n — 1)6; t (n4 — 1)607 
(nk — 1) + (ne — 1) 
52 = 
p 
  
(11) 
~ 
If we may assume that n, and n, are large: n, 
and n, z n, — 1, than 
ny, —1 
6? 6? 
A2 k £ 
oO — — 
n, nk 
As in Stage I, it is our experience that larger regions are 
prevented from merging, due to the low joined variance es- 
timate 67, for large regions. Employing the same rationale 
as in Stage II we replace n, and n, by Vy and 4/ne, re- 
spectively. The final decision rule becomes now: IF 
ok 
+ 
Vn 
  
Iu m te < tay (12) 
THEN merge region R; and R,. 
Where a is the probability that two similar regions are 
wrongly not merged and v = J/n, 4- Vn, — 2 the degrees 
of freedom of the t-score. To avoid that similar regions are 
wrongly not merged a should be rather small. 
Remark: Since the mean and variance are affected 
by the tail parts of the noise and by textures, inevitably 
present in aerial and satellite images, but which we have 
not modelled within our present approach, we have in our 
implementation, the means of the regions replaced by their 
medians. (The median is an estimate for the average which 
is robust against heavy tailed noise). To avoid influence of 
the deviating grey values on the computation of the region 
variances, we preprocess the original image before entering 
Stage II by a 3 x 3 median filter, resulting in removal of 
small clusters of deviating grey values. Extensive experi- 
mentation have shown the feasibility of our approach. 
4.3.2 Stage IIb: Removal of Small Regions 
The aim is simply to remove regions due to small noisy