Image Processing Reference
In-Depth Information
2
2
max((
xx
- )
+ (
yy
- )
)
i
i
IS
() =
(7.72)
AS
()
where
(,
xy
represent the co-ordinates of the centre of mass of the region. Notice that the
numerator defines the area of the maximum circle enclosing the region. Thus, this measure
describes the density of the region. An alternative measure of dispersion can actually also
be expressed as the ratio of the maximum to the minimum radius. That is,
)
max
(
xx
- )
2
+ (
yy
- )
2
i
i
IR S
() =
(7.73)
2
2
min
(
xx
- )
+ (
yy
- )
i
i
This measure defines the ratio between the radius of the maximum circle enclosing the
region and the maximum circle that can be contained in the region. Thus, the measure will
increase as the region spreads. One
disadvantage
of the irregularity measures is that they
are insensitive to slight
discontinuity
in the shape, such as a thin crack in a disk. On the
other hand, these discontinuities will be registered by the earlier measures of compactness
since the perimeter will increase disproportionately with the area.
Code
7.4
shows the implementation for the region descriptors. The code is a straightforward
implementation of Equations 7.67, 7.69, 7.70, 7.72 and 7.73. A comparison of these measures
for the three regions shown in Figure
7.19
is shown in Figure
7.20
. Clearly, for the circle
the compactness and dispersion measures are close to unity. For the ellipse the compactness
decreases whilst the dispersion increases. The convoluted region has the lowest compactness
measure and the highest dispersion values. Clearly, these measurements can be used to
characterise and hence discriminate between areas of differing shape.
Other measures, rather than focus on the geometric properties, characterise the structure
of a region. This is the case of the
Poincarré measure
and
the
Euler number.
The Poincarré
measure concerns the number of holes within a region. Alternatively, the Euler number is
the difference between the number of connected regions and the number of holes in them.
There are many more potential measures for shape description in terms of structure and
geometry. We could evaluate global or local
curvature
(convexity and concavity) as a
further measure of geometry; we could investigate
proximity
and
disposition
as a further
measure of structure. However, these do not have the advantages of a unified structure. We
are simply suggesting measures with descriptive ability but this ability is reduced by the
correlation between different measures. We have already seen the link between the Poincarré
measure and the Euler number. There is a natural link between circularity and irregularity.
As such we shall now look at a unified basis for shape description which aims to reduce
this correlation and provides a unified theoretical basis for region description.
7.3.2
Moments
Moments
describe a shape's
layout
(the arrangement of its pixels), a bit like combining
area, compactness, irregularity and higher order descriptions together. Moments are a
global
description of a shape, accruing this same advantage as Fourier descriptors since
there is an in-built ability to discern, and filter, noise. Further, in image analysis, they are
statistical moments
, as opposed to
mechanical
ones, but the two are analogous. For example,
the mechanical moment of inertia describes the rate of change in momentum; the statistical
second-order moment describes the rate of change in a shape's area. In this way, statistical
