Object description - Feature Extraction and Image Processing - page 288

restriction on normalisation to the unit circle, as well as complex moments (Abu-Mostafa,
1985), again aimed to provide a simpler moment description with invariance properties.
Finally, there are affine invariant moments which do not change with position, rotation and
different scales along the co-ordinate axes, as a result, say, of a camera not being normal
to the object plane. Here, the earliest approach appears to be by Flusser and Suk (Flusser,
1993). One of the reviews (Teh, 1988) concentrates on information content (redundancy),
noise sensitivity and on representation ability, comparing the performance of several of the
more popular moments in these respects.

7.4 Further reading

This chapter has essentially been based on unified techniques for border and region description.
There is actually much more to contour and region analysis than indicated at the start of the
chapter, for this is one the start points of morphological analysis. The neighbourhood can
be extended to be larger (Marchand, 1997) and there is consideration of appropriate distance
metrics for this (Das, 1988). A much more detailed study of boundary-based representation
and application can be found in Van Otterloo's fine text (Van Otterloo, 1991). Naturally,
there are many other ways to describe features, though few have the unique attributes of
moments and Fourier descriptors. Naturally, there is an inter-relation between boundary
and region description: curvature can be computed from a chain code (Rosenfeld, 1974);
Fourier descriptors can also be used to calculate region descriptions (Kiryati, 1989). There
have been many approaches to boundary approximation by fitting curves to the data. Some
of these use polynomial approximation, or there are many spline-based techniques. A
spline is a local function used to model a feature in sections. There are quadratic and cubic
forms (for a good review of spline theory, try Ahlberg et al. (1967) or Dierckx (1995)), of
interest, snakes are actually energy minimising splines. There are many methods for polygonal
approximations to curves, and recently a new measure has been applied to compare
performance on a suitable curve of techniques based on dominant point analysis (Rosin,
1997). To go with the earlier-mentioned review (Prokop and Reeves, 1992) there is a book
available on moment theory (Mukundan and Ramakrishnan, 1998) showing the whole
moment picture and even how to calculate moments from Fourier and Hartley transforms.
The skeleton of a shape can be derived by the medial axis transform (Blum, 1967) and then
used for recognition. This is a natural target for thinning techniques that have not been
covered here. An excellent survey of these techniques, as used in character description
following extraction, can be found in Trier et al. (1996) - describing use of moments and
Fourier descriptors.

7.5 References

Abu-Mostafa, Y. S. and Psaltis, D., Image Normalisation by Complex Moments, IEEE
Trans. on PAMI , 7 , pp. 46-55, 1985
Aguado, A. S., Nixon, M. S. and Montiel, E., Parameterising Arbitrary Shapes via Fourier
Descriptors for Evidence-Gathering Extraction, CVIU: Computer Vision and Image
Understanding , 69 (2), pp. 202-221, 1998
Aguado, A. S., Montiel, E. and Zaluska, E., Modelling Generalised Cylinders via Fourier
Morphing, in press ACM Transactions on Graphics , 2000