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As we have seen, an essential aspect of the inference of history is symmetry. For a simple shape, a symmetry axis is usually defined to be a straight line along which a mirror will reflect one half of the figure onto the other. However, observe that, in complex natural objects, such as the branch of a tree, a straight axis might not exist. Nevertheless, one might still wish to regard the object, or part of it, as symmetrical about some How can such a Then move the circle continuously along the two curves, c
The Free-Form Grammar relies on two structural factors in a shape: symmetry and curvature. Mathematically, symmetry and curvature are two very different descriptors of shape. However, a theorem that I proposed and proved in Leyton (1987b) shows that there is an intimate relationship between these two descriptors. This relationship will be the basis of the entire paper:
To illustrate: Consider the shape shown in the following figure. The section of curve between the two letters
The reason for involving symmetry axes is that it will be argued that they are closely related to
The principle was advanced and extensively corroborated in Leyton (1984, 1985, 1986a, 1986b, 1986c, 1987a, 1987b, 1987c), in several areas of perception including motion perception as well as shape perception. The argument used in Leyton (1984, 1986b), to justify the principle, involves the following two steps: (1) A process that acts along a symmetry axis tends to preserve the symmetry; i.e. to be structure-preserving. (2) Structure-preserving processes are perceived as the most likely processes to occur or to have occurred.
We now have the tools required to understand how processes are recovered from shape. In fact, the system to be proposed consists of two inference rules that are applied successively to a shape. The rules can be illustrated considering the following figure: The first rule is the Symmetry-Curvature Duality Theorem which states that, to each curvature extremum, there is a unique symmetry axis terminating at that extremum. The second rule is the Interaction Principle, which states that each of the axes is a direction along which a process has acted. The implication is that the boundary was deformed along the axes; e.g. each protrusion was the result of Under this analysis, processes are understood as creating the curvature extrema; e.g. the processes introduce protrusions and indentations etc., into the shape boundary. This means that, if one were to go backwards in time, undoing all the inferred processes, one would eventually remove all the extrema. Observe that there is only one closed curve without extrema: the circle. Thus the implication is that the ultimate starting shape must have been a circle, and this was deformed under various processes each of which produced an extremum.
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