Multidimensional Shape Characterization

In order to analyze differences in shape precisely, more than one shape aspect must be monitored. Depending on the number of aspects considered using mutually independent shape parameters, two-, three-, or multidimensional representation of shape can be useful for comparison, and cluster analysis may be appropriate to quantify the significance of differences. Figure 26 shows a two-dimensional shape characterization of idealized particles in a ruggedness/elongation diagram.

Fig. 26 Characterization of the shape of twelve typical particle projections (or sections) in a two-dimensional shape space

These procedures require a computer with a large capacity, but which is modest compared to Fourier analysis, for example. The assessment of the individual parameters involves straightforward arithmetic calculation, and cluster analysis is performed on a few numbers for each feature measured.

Simple shape characterization, normally accomplished using Eq 6, prevails in practice, because the multidimensional procedures referred to previously are not yet implemented in commercial image analyzers. At present, the best approach to the difficult problem of quantitatively characterizing the shape of a system of particles in a powder mass is to use one or several shape parameters carefully adjusted to the problem under investigation.

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