The natural difficulty is the effect of noise. This can change the result, as shown in Figure 6.16.
This can certainly be ameliorated by using the earlier morphological operators (Section 3.6)
to clean the image, but this can obscure the shape when the noise is severe. The major point
is that this noise shows that the effect of a small change in the object can be quite severe on
the resulting distance transform. As such, it has little tolerance of occlusion or change to its
perimeter.
(a) Noisy rectangle (b) Distance transform
Figure 6.16 Distance transformation on noisy images
The natural extension from distance transforms is to the medial axis transform (Blum, 1967),
which determines the skeleton that consists of the locus of all the centres of maximum disks in the
analysed region/shape. This has found use in feature extraction and description, so approaches
have considered improvement in speed (Lee, 1982). One more recent study (Katz and Pizer,
2003) noted the practically difficulty experienced in noisy imagery: ‘It is well documented how
a tiny change to an object’s boundary can cause a large change in its Medial Axis Transform’.
To handle this, and hierarchical shape decomposition, the new approach ‘provides a natural
parts-hierarchy while eliminating instabilities due to small boundary changes’. An alternative is
to seek an approach that is designed explicitly to handle noise, say by averaging, and we shall
consider this type of approach next.
6.4.2 Symmetry
The discrete symmetry operator (Reisfeld et al., 1995) uses a totally different basis to find shapes,
is intuitively very appealing and has links with human perception. Rather than rely on finding
the border of a shape, or its shape, it locates features according to their symmetrical properties.
The operator essentially forms an accumulator of points that are measures of symmetry between
image points. Pairs of image points are attributed symmetry values that are derived from a
distance weighting function, a phase weighting function and the edge magnitude at each of the
pair of points. The distance weighting function controls the scope of the function, to control
whether points that are more distant contribute in a similar manner to those that are close
together. The phase weighting function shows when edge vectors at the pair of points point to
each other. The symmetry accumulation is at the centre of each pair of points. In this way, the
268 Feature Extraction and Image Processing
accumulator measures the degree of symmetry between image points, controlled by the edge
strength. The distance weighting function D is
Di j =
1
√
2
e
−
P
i
−P
j
2
(6.59)
where i and j are the indices to two image points P
i
and P
j
and the deviation controls the
scope of the function, by scaling the contribution of the distance between the points in the
exponential function. A small value for the deviation implies local operation and detection
of local symmetry. Larger values of imply that points that are further apart contribute
to the accumulation process, as well as ones that are close together. In, say, application to
the image of a face, large and small values of will aim for the whole face or the eyes,
respectively.
The effect of the value of on the scalar distance weighting function expressed as Equa-
tion 6.60 is illustrated in Figure 6.17.
Dij =
1
√
2
e
j
√
2
(6.60)
Di
(
j,
0.6)
Di
(
j,
5)
0 5 10
0.5
j
0 5 10
0.5
j
(a) Small
σ (b) Large σ
Figure 6.17 Effect of on distance weighting
Figure 6.17(a)shows the effect of a small value for the deviation, =06, and shows that the
weighting is greatest for closely spaced points and drops rapidly for points with larger spacing.
Larger values of imply that the distance weight drops less rapidly for points that are more
widely spaced, as in Figure 6.17(b) where = 5, allowing points that are spaced further apart
to contribute to the measured symmetry. The phase weighting function P is
Pi j = 1−cos
i
+
j
−2
ij
×1 −cos
i
−
j
(6.61)
where is the edge direction at the two points and
ij
measures the direction of a line joining
the two points:
ij
=tan
−1
yP
j
−yP
i
xP
j
−xP
i
(6.62)
where xP
i
and yP
i
are the x and y coordinates of the point P
i
, respectively. This function
is minimum when the edge direction at two points is in the same direction
j
=
i
, and is a
maximum when the edge direction is away from each other
i
=
j
+, along the line joining
the two points,
j
=
ij
.
Flexible shape extraction (snakes and other techniques) 269
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