Professor Michael Leyton.
Center for Cognitive Science, Rutgers University
The purpose of this web-site is to describe a grammar that I published in the journal Artificial Intelligence in 1987. The grammar expresses any shape-evolution in terms of six types of phase-transitions. After the grammar was published, it was applied by scientists in many disciplines: Meteorology, chemical engineering, radiology, dendrite morphology, computer vision, forensic science, mathematics, Mayan languages, philosophy, etc.
Inferring the History of a Shape
Let us begin by understanding the purpose for which the grammar was developed: inferring history from shape. The shape of an object often seems to tell us something about the object's history; i.e. the causal-processes that formed it. For example, the shape shown below can be understood as the result of various processes of growth, pushing, pulling, resistance, squashing, etc. This type of understanding is, in fact, a remarkable phenomenon. A shape is simply a single state, a frozen moment, a step outside the flow of time; and yet we are able to use it as a window into the past. How is this possible?
The answer generally was the subject of my book Symmetry, Causality, Mind (MIT Press), which is essentially a 630-page rule-system for deducing the past history that formed any shape. The Process Grammar is part of that rule-system. We will concentrate here only on aspects related to the grammar.
The PISA Symmetry Analysis
An essential aspect of the grammar is the use of 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 curved axis. For example, a branch of a tree tends not to have a straight reflectional axis. Nevertheless, one understands the branch to have an axial core that runs along its center.
How can such a generalized axis be constructed? There are good mathematical reasons to proceed in the following way. Consider the next figure. It shows two curves c1 and c2 (the bold curves), which can be understood as two sides of an object. We will construct a symmetry axis between the two curves. In fact, the axis, we will produce, will be indicated by the sequence of dots shown. It is produced in the following way: Introduce a circle that is tangential simultaneously to the two curves, as shown. The two tangent points A and B are then defined as symmetrical to each other.
Then move the circle continuously along the two curves, c1 and c2, while always maintaining the property that it is simultaneously tangential to the two curves. One might need to expand or contract the circle to maintain the double-touching property. Now define the symmetry axis to be the trajectory of the midpoint Q of the arc AB, as the circle moves. For example, in the above figure, the trajectory of Q is represented by the locus of dots shown. This definition of symmetry was proposed by me in Leyton (1988), where I argued that it is more suited for the particular task of process-inference than are other symmetry analyses, such as those proposed by Blum and Brady. The intuitively satisfying results of our alternative analysis will be seen later. I called the above definition of symmetry axis, Process-Inferring Symmetry Axis, or simply PISA .
The Process 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:
SYMMETRY-CURVATURE DUALITY THEOREM (Leyton, 1987b): Any section of curve, that has one and only one curvature extremum, has one and only one symmetry axis. This axis is forced to terminate at the extremum itself.
To illustrate: Consider the shape shown in the following figure. The section of curve between the two letters m, has only one curvature extremum - that indicated by the letter M. The theorem says that this section of curve can have only one symmetry axis, and that the axis is forced to terminate at the extremum.
Symmetry Axes and Processes
The reason for involving symmetry axes is that it will be argued that they are closely related to process-histories. This proposed relationship is given by the following principle:
INTERACTION PRINCIPLE (Leyton, 1984): The symmetry axes of a perceptual organization are interpreted as the directions along which processes are most likely to act or have acted.
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.
The Inference of Processes
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 pushing out along its axis, and each indentation was the result of pushing in along its axis. In fact, each axis is the trace or record of boundary-movement! (The vertical arrow at the top is a squashing and the rules for inferring this are the same, as we shall see later.)
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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