Affine invariant descriptors have been widely used for recognition of
objects regardless of their position, size and orientation in
space. Examples of color, texture and shape descriptors abound in
the literature. However, many tasks in computer vision
require looking not only at single objects or regions in images but
also at their spatial relationships. In an earlier work, we
showed that the relative position of two objects can be quantitatively
described by a histogram of forces. Here, we study how affine
transformations affect this descriptor. The position of an object
with respect to another changes when the objects are affine
transformed. We analyze the link between (i) the applied
affinity, (ii) the relative position before transformation (described
through a force histogram), and (iii) the relative position after
transformation. We show that any two of these elements allow the
third one to be recovered. Moreover, it is possible to determine
whether (or how well) two relative positions are actually related
through an affine transformation. If they are not, the affinity
that best approximates the unknown transformation can be
retrieved, and the quality of the approximation assessed.
