Image Processing Reference
In-Depth Information
function). 3D Fourier descriptors were introduced for analysis of simple shapes (Staib,
1992) and have recently been found to give good performance in application (Undrill,
1997). Fourier descriptors have been also used to model shapes in computer graphics
(Aguado, 2000). Naturally, Fourier descriptors cannot be used for occluded or mixed
shapes, relying on extraction techniques with known indifference to occlusion (the HT,
say). However, there have been approaches aimed to classify partial shapes using Fourier
descriptors (Lin, 1987).
7.3
Region descriptors
So far, we have concentrated on descriptions of the perimeter, or boundary. The natural
counterpart is to describe the
region
, or the
area
, by
regional shape descriptors
. Here,
there are two main contenders that differ in focus: basic regional descriptors characterise
the geometric properties of the region; moments concentrate on density of the region. First,
though, we shall look at the simpler descriptors.
7.3.1
Basic region descriptors
A region can be described by considering scalar measures based on its geometric properties.
The simplest property is given by its size or area. In general, the area of a region in the
plane is defined as
AS
() =
I x ydydx
(,
)
(7.66)
x
y
where
I
(
x
,
y
) = 1 if the pixel is within a shape, (
x
,
y
)
S
, and 0 otherwise. In practice,
integrals are approximated by summations. That is,
Σ Σ
AS
() =
(,
I x y
)
A
(7.67)
xy
where
A
= 1, then the area is measured in pixels. Area
changes with changes in scale. However, it is invariant to image rotation. Small errors in
the computation of the area will appear when applying a rotation transformation due to
discretisation of the image.
Another simple property is defined by the perimeter of the region. If
x
(
t
) and
y
(
t
) denote
the parametric co-ordinates of a curve enclosing a region
S
, then the perimeter of the region
is defined as
A
is the area of one pixel. Thus, if
2
2
(7.68)
PS
(
) =
x t
( ) +
y t dt
( )
t
This equation corresponds to the sums of all the infinitesimal arcs that define the curve. In
the discrete case,
x
(
t
) and
y
(
t
) are defined by a set of pixels in the image. Thus, Equation
7.68 is approximated by
2
2
PS
() =
( -
x
x
)+ ( -
y
y
)
(7.69)
i
i
-1
i
i
-1
i
where
x
i
and
y
i
represent the co-ordinates of the
i
th pixel forming the curve. Since pixels
are organised in a square grid, then the terms in the summation can only take two values.
