30 Logo Recognition: Theory and Practice
[125, 211, 39, 104, 78, 112] to ﬁnd a simple, less time consuming and highly
accurate method for shape recognition. Most researchers have judged their
proposals for shape recognition with the criteria described in Chapter 1. Gen-
erally speaking, to ﬁnish a recognition task, a shape is ﬁrst represented by a
particular method, and then the matching procedure is applied.
3.1.1 Shape representation
Shape is a concept which is widely understood yet diﬃcult to deﬁne. A shape
representation is a language for describing shape or some aspects of shape. It
includes a set of shape descriptors, and a mapping between shape descriptors
Shape descriptors may be classiﬁed as being either internal or external de-
scriptors depending on whether they code the boundary of a shape (external
descriptors) or the area within the boundary (internal descriptors). Descrip-
tors may also be classiﬁed as being either scalar transform or space domain.
Scalar transform methods generate shape descriptors which are mathematical
summations of the shape whereas space domain techniques comprise descrip-
tors which express the structural and relational properties of the shape. These
descriptors are reviewed with respect to scalar or space domain and internal
or external types in the following.
22.214.171.124 Internal scalar methods
Internal scalar methods use mathematical properties derived from the area
within a complete shape contour.
Moments of area. The standard 2D moment, m(u, v), of an image func-
tion, f(x, y), is deﬁned as
The use of moments for shape description was initiated by Hu . He proved
that moment-based shape description is information-preserving. Sardana et
al.  presented an eﬃcient shape recognition method based on moment. In
their scheme, a set of circles is drawn around the centroid to map the area of
the shape into a set of concentric rings. The shape is then described in terms
of the area occupied with each ring. These area values form the n numerical
features of a shape. These features are then normalized with respect to the size
of the shape to achieve size invariance. The major advantage of this method
is that it is independent of rotation and scaling of shapes. Moments [220, 178]
have been used successfully as a technique for describing simple shapes but
they tend to fail for more complicated, occluded or distorted objects.
Morphological methods. Mathematical morphology has evolved as a
useful tool for various image processing applications. It is suitable for shape-
related processing since morphological operations are directly related to the