The terms "homogeneous" and "heterogeneous" are obviously relative. For instance, a certain range of orientation or variation is acceptable as long as the effect of these variations is much smaller than the resolution distance.
To consider the magnitude of this tolerance, we might take the 50S ribosomal subunit, whose base is roughly 250 A wide. A tilt of the subunit by 1° would lead to a foreshortening of the particle base to 250 x cos(1°) = 249.96 A, which leads to a very small change of shape compared to the size of the smallest resolved features (10 A). From this consideration, we could infer that orientation changes by 1° will not change the appearance of the projection significantly. In reality, however, the ability to differentiate among different patterns is a small function of the amount of noise. In the absence of noise, even a change that is a fraction of the resolution distance will be detectable. A general criterion for homogeneity could be formulated as follows (Frank, 1990): "a group of images is considered homogeneous if the intra-group variations are small compared with the smallest resolved structural detail.'' Boisset et al. (1990b) used such a test to determine homogeneous subgroups in a continuously varying molecule set.
Two ways of scrutinizing the image set based on a statistical analysis have already been introduced in chapter 3: the variance map and the rank sum analysis. Neither of these methods, however, is capable of dealing with a completely heterogeneous data set, that is, a data set consisting of two or more different groups of images. For such data sets, new tools are required, which are able to group similar images with one another, and discern clusters. Various classification techniques exist, but these are most efficient, given the low signal-to-noise ratio (SNR), when a reduced representation of the data is available. Much of the following is concerned with the introduction of such reduced representations. Once these are established, we then move on to the subject of classification techniques.
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