Laura Keyes 
  
Given two sets of Fourier descriptors, how do we measure their degree of similarity? An appropriate classification is 
necessary if unknown shapes are to be compared to a library of known shapes. If two shapes, A and B, produce a set of 
values represented by a(i) and b(i) then the distance between them can be given as c(i) = a(i) — b(i). If a(1) and b(i) are 
identical then c(i) will be zero. If they are different then the magnitudes of the coefficients in c(i) will give a reasonable 
measure of the difference. It proves more convenient to have one value to represent this rather than the set of values that 
make up c(i). The easiest way is to treat c(i) as a vector in a multi-dimensional space, in which case its length, which 
represents the distance between the planes, is given by the square root of the sum of the squares of the elements of c(i). 
2.2 Moment Invariants 
Moment invariants have been frequently used as features for image processing, remote sensing, shape recognition and 
classification. Moments can provide characteristics of an object that uniquely represent its shape. Invariant shape 
recognition is performed by classification in the multidimensional moment invariant feature space. Several techniques 
have been developed that derive invariant features from moments for object recognition and representation. These 
techniques are distinguished by their moment definition, such as the type of data exploited and the method for deriving 
invariant values from the image moments. It was Hu ( Hu, 1962), that first set out the mathematical foundation for two- 
dimensional moment invariants and demonstrated their applications to shape recognition. They were first applied to 
aircraft shapes and were shown to be quick and reliable (Dudani, Breeding and McGhee, 1977). These moment 
invariant values are invariant with respect to translation, scale and rotation of the shape. 
Hu defines seven of these shape descriptor values computed from central moments through order three that are 
independent to object translation, scale and orientation. Translation invariance is achieved by computing moments that 
are normalised with respect to the centre of gravity so that the centre of mass of the distribution is at the origin (central 
moments). Size invariant moments are derived from algebraic invariants but these can be shown to be the result of a 
simple size normalisation. From the second and third order values of the normalised central moments a set of seven 
invariant moments can be computed which are independent of rotation. 
Traditionally, moment invariants are computed based on the information provided by both the shape boundary and its 
interior region (Hu 1962). However, for the purpose of this paper, the moment invariants are computed using the shape 
boundary only (Chaur-Chin Chen 1993). 
Using the same data sets as in the Fourier descriptor method described earlier, the moments technique is applied. 
However, for moments the points extracted from the map are stored not as complex numbers but represent the x and y 
co-ordinates of the polygonal shape. These points are processed by a moment transformation on the outline of the 
shape, which produces seven moment invariant values that are normalised with respect to translation, scale and rotation 
using the formulae above. The resulting set of values can be used to discriminate between the shapes. Classification is 
carried out using the same method described in section 2.1 for Fourier descriptors. 
2.3 Boundary chain coding and scalar descriptors 
Boundary chain-coding and scalar descriptors are also used as shape description techniques for this experiment. Chain- 
code treats the points of a curve and in particular a region boundary, directly and in a strictly local fashion. The basis of 
the technique is, essentially, to start with a point that is believed to be on the boundary (some local edge point) and to 
extend the boundary by adding a neighbouring point in the contour direction (that is, the direction which is normal to 
the gradient direction). This process is reiterated, starting at this new boundary pixel. This boundary chain code 
technique encodes piecewise linear curves as a sequence of directed straight-line segments called links. There are eight 
possible directions for a link between a point and its neighbour. These eight directions are numbered ‘0’ through “7° and 
move counter-clockwise, as shown in figure 1a. Each of these can be considered as an angular direction, in multiples of 
45 degrees, which are moved to go from one pixel to the next. The absolute co-ordinates (x, y) of the first boundary 
pixel together with the chain code, represent a complete description of the discrete region boundary. 
As tracing around the boundary continues it builds a boundary chain code representation of the contour by recording all 
the boundary pixels visited. Figure 2 shows a boundary representation of a simple shape, which is based upon the work 
of Freeman (Freeman 1961). It is assumed that the boundary pixels correspond to the pixels exhibiting the local 
  
482 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.