Images as a Set of Multivariate Data

We assume that a set comprising N images is given, {pz(r); i = 1,..., Ng. Each of the images is represented by a set of discrete measurements on a regular Cartesian grid:

where the image elements are lexicographically ordered as before (see section 2.2 in chapter 3).

We further assume that, by using one of the procedures outlined in chapter 3, the image set has already been aligned; that is, any given pixel j has the same "meaning" across the entire image set. In other words, if we were dealing with projections of "copies" of the same molecule lying in the same orientation, that pixel would refer to the same projection ray in a 3D coordinate system affixed to the molecule. In principle, the images of the set can vary in any pixels: they form a multivariate data set.

Variations among the images are in general due to a linear combination of variations affecting groups of pixels, rather than to variations of a single pixel. In the latter case, only a single pixel would change while all other pixels would

Figure 4.2 A specimen used to demonstrate the need for classification: although the micrograph shows the same view of the molecule (the crown view of the 50S ribosomal subunit from E. coli), there are differences due to variations in staining and in the position of flexible components such as the ''stalk'' (indicated by the arrow-head). From Frank et al. (1985), reproduced with permission of Van Nostrand-Reinhold.

Figure 4.2 A specimen used to demonstrate the need for classification: although the micrograph shows the same view of the molecule (the crown view of the 50S ribosomal subunit from E. coli), there are differences due to variations in staining and in the position of flexible components such as the ''stalk'' (indicated by the arrow-head). From Frank et al. (1985), reproduced with permission of Van Nostrand-Reinhold.

remain exactly the same. Such a behavior is extremely unlikely as the outcome of an experiment, for two reasons: for one, there is never precise constancy over any extended area of the image, because of the presence of noise in each experiment. Second, unless the images are undersampled or critically sampled—a condition normally avoided (see section 2.2 in chapter 3)—pixels will not vary in isolation but in concert with surrounding pixels. For example, a flexible protuberance may assume different positions among the molecules represented in the image set. This means that an entire set of pixels undergoes the same coordinate transformation; for example, a rotation around a point of flexure (see figure 4.2). Another obvious source of correlation among neighboring image elements is the instrument itself: each object point, formally described by a delta-function, is imaged as an extended disk (the point-spread function of the instrument; see section 3.3 in chapter 2). Because of the instrument's imperfections, image points within the area of that disk are correlated; that is, they will not vary independently.

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