Theory of Handwriting

Handwriting production is viewed as a constrained modulation of an underlying oscillatory process. Coupled oscillations in horizontal and vertical directions produce letter forms, and when superimposed on a rightward constant velocity horizontal sweep result in spatially separated letters. Modulation of the vertical oscillation is responsible for control of letter height, either through altering the frequency or altering the acceleration amplitude. Modulation of the horizontal oscillation is responsible for control of corner shape through altering phase or amplitude. The vertical velocity zero crossing in the velocity space diagram is important from the standpoint of control. Changing the horizontal velocity value at this zero crossing controls corner shape, and such changes can be effected through modifying the horizontal oscillation amplitude and phase. Changing the slope at this zero crossing controls writing slant; this slope depends on the horizontal and vertical velocity zero amplitudes and on the relative phase difference. Letter height modulation is also best applied at the vertical velocity zero crossing to preserve an even baseline. The corner shape and slant constraints completely determine the amplitude and phase relations between the two oscillations. Under these constraints interletter separation is not an independent parameter. This theory applies generally to a number of acceleration oscillation patterns such as sinusoidal, rectangular and trapezoidal oscillations. The oscillation theory also provides an explanation for how handwriting might degenerate with speed. An implementation of the theory in the context of the spring muscle model is developed. Here sinusoidal oscillations arise from a purely mechanical sources; orthogonal antagonistic spring pairs generate particular cycloids depending on the initial conditions. Modulating between cycloids can be achieved by changing the spring zero settings at the appropriate times. Frequency can be modulated either by shifting between coactivation and alternating activation of the antagonistic springs or by presuming variable spring constant springs. An acceleration and position measuring apparatus was developed for measurements of human handwriting. Measurements of human writing are consistent with the oscillation theory. It is shown that the minimum energy movement for the spring muscle is bang-coast-bang. For certain parameter values a singular arc solution can be shown to be minimizing. Experimental measurements however indicate that handwriting is not a minimum energy movement.

ADEPT: A Heuristic Program for Proving Theorems of Group Theory

A computer program, named ADEPT (A Distinctly Empirical Prover of Theorems), has been written which proves theorems taken from the abstract theory of groups. Its operation is basically heuristic, incorporating many of the techniques of the human mathematician in a "natural" way. This program has proved almost 100 theorems, as well as serving as a vehicle for testing and evaluating special-purpose heuristics. A detailed description of the program is supplemented by accounts of its performance on a number of theorems, thus providing many insights into the particular problems inherent in the design of a procedure capable of proving a variety of theorems from this domain. Suggestions have been formulated for further efforts along these lines, and comparisons with related work previously reported in the literature have been made.

Slow and Smooth: A Bayesian Theory for the Combination of Local Motion Signals in Human Vision

In order to estimate the motion of an object, the visual system needs to combine multiple local measurements, each of which carries some degree of ambiguity. We present a model of motion perception whereby measurements from different image regions are combined according to a Bayesian estimator --- the estimated motion maximizes the posterior probability assuming a prior favoring slow and smooth velocities. In reviewing a large number of previously published phenomena we find that the Bayesian estimator predicts a wide range of psychophysical results. This suggests that the seemingly complex set of illusions arise from a single computational strategy that is optimal under reasonable assumptions.