Noël Bonnet

mathematical techniques. When the "objects" considered have specific properties, it is possible to envisage (and even to recommend) how to exploit these properties for performing dimensionality reduction. One example of this approach consists in replacing images of centro-symmetric particles by their rotational power spectrum (Crowther and Amos, 1971): the image is split into angular sectors, the summed signal intensity within the sectors is then Fourier transformed to give a one-dimensional signal containing the useful information related to the rotational symmetry of the particles. 3. Methods for checking the quality of a mapping and the optimal dimension of the reduced parameter space Checking the quality of a mapping for selecting one mapping method over others is not an easy task and depends on the criterion chosen to evaluate the quality. Suppose, for instance, that the mapping is performed through an iterative method aimed at minimizing a distortion measure e.g. as MDS or Sammon’s mappings do. If the quality criterion chosen is the same distortion measure, this method will be found to be good, but the same result may not be true if other quality criteria are chosen. Thus, one has sometimes to evaluate the quality of the mapping through the evaluation of a subsidiary task, such as classification of known objects after dimensionality reduction (see for instance De Baker et al. 1998). Checking the quality of the mapping may also be a way to estimate the intrinsic (or true) dimensionality of the data set, that is to say the optimum reduced dimension (D’) for the mapping, or in other words the smallest dimension of the reduced space for which most of the original information is preserved. One useful tool for doing this (and checking the different results visually) is to draw the scatterplot relating the new inter-distances (dij) to the original ones (Dij). While most information is preserved, the scatterplot display remains concentrated along the first diagonal (dij≈Dij ∀i ∀j). On the other hand, when some information is lost because of excessive dimensionality reduction, the scatterplot is no longer concentrated along the first diagonal, and distortion concerning either small distances or large distances (or both) becomes apparent. Besides visual assessment, the distortion can be quantified through several descriptors of the scatterplot, such as: contrast: ( ) ( ) ij ij i i j ij ij d D p d D D C , . ) ' ( 2 ∑∑