Graphics Reference
In-Depth Information
3.4.4 Algorithm 2
Here each row (column) of pixels has been viewed as a space curve and is
segmented depending on the homogeneity among the pixels. Each segment is
then approximated by the modified approximation scheme. Here, we consider
n
1
n
v
1
i
,
v
1
=
(3.16)
i
=1
where
v
1
s are computed using equation (3.6).
Since the segments are all homogeneous, approximation for coding depends
on the homogeneity parameter and not on any external approximation param-
eter as required in the case of Algorithm 1. The appr
ox
imation is faster. Since
for each homogeneous segment
v
1
s are averaged for
v
1
, every approximation
has its own
max
that varies from segment to segment.
Small deformation space curve and the concept of homogeneity
An image may be considered as an intensity surface with surface contours
representing the space curves along the rows and columns of the image. Note
that for any curve
Γ
, the amount of information contained in it can be rep-
resented by its curvature vector
k
v
or by any other related quantity. The
curvature vector
k
v
is defined as
k
v
=
dt
ds
,
t
being the tangent vector and
s
being the arc length. For a curve
Γ
, with
given end points, its bending energy
B
e
can be written as
B
e
=
Γ
k
v
2
ds.
Here the deformation of the curve is in the direction normal to the axis of the
equilibrium position. Therefore, when the x-axis is along the axis of equilib-
rium position, the deformation may be represented by
z
(
x
) and consequently
we have
B
e
=
Γ
k
v
2
dx
=
Γ
(3.17)
[
z
(
x
)]
2
[1 + (
z
(
x
))
2
]
3
dx.
0and
B
e
≈
Γ
[
z
(
x
)]
2
dx.
Since
B
e
represents
the total energy of the curve,
k
v
2
or (
z
)
2
represents the energy of the curve
at an arbitrary point. Therefore, in an image plane,
k
v
2
For small deformation,
z
(
x
)
≈
will represent the
energy of the image space curve at a pixel position.
With the above principle, a curve (a set of pixels along a row or a column)
can be considered to be perfectly homogeneous if the bending energy is zero at
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