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