3.8 Shape approximation using DOPs
3.8.2 Experiments on shape representation
3.8.2.1 Characterizing shapes using curvature properties
The curvature properties of the shape boundary are among the important perceptual features describing the shape. The curvature can be defined as the rate of change of slope along the shape boundary and it can be expressed in terms of the first and the second order derivatives.
Let f denote a binary shape andBbe the corresponding shape boundary. Assume, B(u)=(x (u),y (u)) as a continuous vector valued function defined by the position vectors (x(u),y(u)). Given the boundary points (x,y)∈Bof length L, the curvature function is computed as [87]
κ(u)= ˙x (u) ¨y (u)− ¨x (u) ˙y (u)
˙x (u)2+˙y (u)23/2 (3.81)
where u is a real value such that 0 ≤ u ≤ (L−1). For the discrete case, the derivatives of x and y are approximated by the corresponding finite differences.
The zero-crossings of the curvature function are the inflection points on the shape boundary. The local absolute maximum in the curvature corresponds to a generic corner in the shape [167]. If the maximum value
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−0.75
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Boundary points -u
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− Zero−crossings
− Negative maxima
− Concave points
− Inflection points Concave segments
Concave segment Concave
segment
Figure 3.9: Illustration of finding the concave segments of a shape from the curvature function derived from the corre- sponding shape boundary. (a) Geometric shape used for illustration; (b) The curvature function derived from the boundary of the geometric shape and (c) Representing the inflection points and the concave segments on the shape boundary. The zero-crossings correspond to the inflection points. Similarly, the negative maxima correspond to the concave points.
Table 3.2: Types of concavities based on the width and the depth of the concave segments.
Width Depth Type of concavity
Small Small Narrow - Shallow concavity
Small High Narrow - Deep concavity
High Small Wide - Shallow concavity
High High Wide - Deep concavity
is positive, the corner is considered as the convex point and if the maximum is negative, the corner is considered as the concave point. The boundary section between the inflection points that contain the convex point is the convex segment and the section constituting the concave point comprises the concave segment.
Since, the shape boundary is a closed curve, the concave segments provide comparatively more detailed information and they constitute the transition between different parts of the shape. An example illustrating the inflection points and the concave segments of a geometric shape along with the corresponding curvature function is shown in Figure 3.9.
The structural variations in shape can be characterised by the variations in the width and the depth of the concave segments. The width of a concave segment can be computed as the length of the line connecting the corresponding two inflection points. The depth of the concave segment is the distance between the concave point and the line connecting the corresponding inflection points. Based on the width and the depth values, the concavities can be divided in to four types as listed in Table 3.2.
The characteristics of the concavities present in the shape can be represented using the curvature scale space (CSS) representation. The CSS representation is a map of the location of the zero-crossings in the curvature functionκ(u) obtained over successive smoothing of the shape boundary [87]. ConvolvingBwith a 1D Gaussian kernel g (u, σ) of standard deviationσresults in the smoothed curveBσ={(X,Y)}. The smoothed
3.8 Shape approximation using DOPs
boundary points are given by X (u, σ)= x (u)∗g (u, σ) Y (u, σ)=y (u)∗g (u, σ)
(3.82)
Accordingly, the curvature onBσis computed as κ(u, σ)= X (u, σ) ¨˙ Y (u, σ)−X (u, σ) ˙¨ Y (u, σ)
X (u, σ)˙ 2+Y (u, σ)˙ 23/2 (3.83)
The CSS descriptors that represent the location of the inflection points on the curve are extracted for varying values ofσand are used to obtain the CSS image that is defined as
ICS S (u, σ)={(u, σ)|κ(u, σ)=0} (3.84)
The CSS map consists of several arch-shaped contours, each related to the concave segments of the shape boundary. The height and the base width of the arch-shaped contours reflect the depth and the width of the concavities respectively [168, 169]. The height of the CSS contours is larger for wide-shallow, wide-deep and narrow-deep concavities.
A few examples of the shapes used in this experiment and the corresponding CSS representations are shown in Figure 3.10 and Figure 3.11. The CSS representations of three star-shaped polygons consisting of 16 concave segments of almost same width and varying depth are illustrated in Figure 3.10. It can be observed that the number of arch-shaped contours in the CSS map is equal to the number of concave segments in the shape boundary. The shape in Figure 3.10(a) consists of shallow concave segments in comparison to the shapes illustrated in Figure 3.10(b) and 3.10(c). The star-shaped polygon in Figure 3.10(c) exhibits deeper concave segments and hence, the height of the arch-shaped contours in the corresponding CSS map is comparatively higher. By comparing the CSS maps in Figure 3.10, it can easily inferred that the height of the contours in the CSS map increases with the depth of the concave segments.
Figure 3.11 illustrates the CSS representation of three different shapes composed of different number of concave segments of different width. The shape of the character ‘T’ in Figure 3.11(a) consists of two wide- deep concave segments and hence, the corresponding CSS map exhibits two arch-shaped contours representing the concavities. Similarly, the CSS representations of a cross-shaped and the fork-shaped polygons in Fig- ure 3.11(b) and Figure 3.11(c) respectively reflect the number of concavities in the shape boundary. The cross- shaped polygon is composed of concave segments of almost same width and depth. Hence, the arch-shaped contours in the corresponding CSS map are of approximately same height and base width. The fork-shaped
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(a) Polygon with shallow concavities and its CSS map
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(b) Polygon with moderately deep con- cavities and its CSS map
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(c) Polygon with very deep concavities and its CSS map
Figure 3.10: CSS representation of star-shaped polygons composed of 16 concave segments of varying depth. The polygon shape in (a) consists of shallow concave segments and (c) consists of deeper concave segments. The figure illustrates the variation in the height of the arch-shaped contours in the CSS map with respect to the variation in the depth of the concavities.
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(a) T-shape and its CSS map
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(b) Cross-shaped polygon and its CSS map
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(c) Fork-shaped polygon and its CSS map
Figure 3.11: CSS representation of four different geometric shapes with varying number of concave segments and width.
The figure illustrates the variation in the base width of the arch-shaped contours relative to the variation in the width of the concavities. Also, the number of arch-shape contours is proportional to the number of concavities. The shape of character
‘T’ has comparatively less number of concave segments and the concavities are more wide. The cross-shaped polygon has concave segments of similar width and depth. Conversely, the fork-shaped polygon is composed of concave segments of different widths and depth.
3.8 Shape approximation using DOPs
polygon consists of five deep concave segments. Based on the base width of the arch-shaped contours in the CSS map, it can be easily verified that the fork-shaped polygon is composed of two wide concavities and three comparatively narrow concavities. The base width of the CSS contours in Figure 3.11(b) and Figure 3.11(c) is less than that of the CSS contours illustrated in Figure 3.11(a). This implies that the T-shape is composed of comparatively wide concavities. By comparing the base widths of the arch-shaped contours in the CSS maps shown in Figure 3.10 and Figure 3.11, it can be inferred that the shapes presented in Figure 3.10 are composed of comparatively narrow concave segments.
Using the above curvature properties of the shape boundaries, it is also possible to infer the various spatial frequency structures of the shapes. It can be easily verified that the shapes composed of many number of deep concave segments consists of large number of transitions between the background and the object regions. Such shapes can be considered as complex shapes with higher spatial frequency regions and hence, exhibit large structural details. The shapes composed of shallow segments such as the star-shaped polygon in Figure 3.10(a) consist of less number of transitions between the background and the object regions. Hence, such shapes are composed of low spatial frequency regions with less structural details. On the other hand, the shapes presented in Figure 3.11 consist of less number of concave segments than the star-shaped polygon in Figure 3.10(a).
However, the concave segments of the shapes presented in Figure 3.11 are comparatively deeper exhibiting high spatial frequency regions. From this discussion on the curvature properties, it is understood that the complexity of the shapes in terms of the structural details increases with the number of concavities and the depth of the concavities.