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cost of compression ratio. Hence, the choice of C should be made based on a
compromise between the quality of reconstructed image and the compression
ratio. Sections 2.4.2-2.4.4, provide details of approximation, along with a new
approach for the determination of polynomial order. In most of the cases,
order is seen to be 2 but it can go up to 3 or 4 depending on variations in the
segmented regions and the criterion, C.
Fig. 2.2.
Binary tree structure for hierarchical segmentation.
2.4.2 Approximation Problem
For approximation, one can first formulate the problem using Bezier-Bernstein
polynomial and then can consider the issue of the polynomial order determina-
tion. Choose the Bezier-Bernstein polynomial because the segmentation algo-
rithm we are considering is for image compression, for which Bezier-Bernstein
polynomial provides a number of merits during reconstruction. However, one
can use also other functions. The Bezier-Bernstein surface is a tensor product
surface and is given by
p
q
φ
rp
(
u
)
φ
zq
(
v
)
V
rz
,
s
pq
(
u, v
)=
r
=0
z
=0
(2.17)
p
q
B
rp
D
zq
u
r
u
)
p−r
v
z
v
)
q−z
=
(1
−
(1
−
V
rz
,
r
=0
z
=0
p
!
q
!
where
u, v
∈
[0
,
1] and
B
rp
=
(
p−r
)!
r
!
,
D
zq
=
(
q−z
)
z
!
.
p
and
q
define the order
of the Bezier-Bernstein surface.
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