Symmetry occluded shapes

The symmetry of occluded shapes, such that appear in microscopy studies of materials. 

Here we address a problem which is commonly encountered in microscopy studies of particulate materials Given a collection of similar objects, such that the individual shapes occlude each other, how can the symmetry of these objects be extracted using methods of automated image analysis. The method we apply to solve the problem is to evaluate the symmetry of the occluded shapes, locate the center of symmetry, and reconstruct the symmetric shape most similar to the unoccluded original. 

The closest symmetric shape obtained by angular selection about the center of symmetry (Figure 14c) is visually more similar to the original (Figure 14a) than that obtained by angular selection about the centroid of the occluded shape (Figure 14b). We consider the former a reconstruction of the unoccluded shape. 

The process of reconstructing the occluded shape can be improved by altering the method of evaluating the symmetry of a set of points. As described in Section III the symmetry of a set of points is evaluated by folding, averaging, and unfolding about the centroid of the points. We alter the method as follows ... 

Figure 14. (a) Original occluded shape, its centroid (+) and its center of symmetry ( ). (b,c) The closest Cs-symmetric shapes following angular selection about the centroid (b) and about the center of symmetry (c). 

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