The "practical mathematics" of recording three-dimensional eye position using scleral coils

The "gold standard" for recording the three-dimensional rotation of the eye involves placing two coils of wire, embedded in a soft plastic ring, on the sclera of the eye, then placing the subject inside a set of orthogonal oscillating magnetic fields, and using the currents induced in the eye coils to deduce the position of the coil, and hence of the eye, in space. Eye movements are actually eye rotations, which can be described mathematically by a special class of matrices, rotation matrices, or, alternatively, by a rotation vector related to the axis of the rotation. This article deals with the mathematical tools needed to implement the signal processing from such a multifield, dual-coil system and compute the precise rotational movement of the eye. One reason for making such careful measurements is to study an interesting constraint on eye movements, called Listing's law, which expresses ocular torsion, or rotation of the eye about its line of sight, in terms of the direction of gaze. Techniques for experimentally quantitating these constraints are also presented. Following a treatment of the "ideal" case, with coils and eye in perfect alignment, the additional techniques for dealing with various departures from ideality that are almost always encountered experimentally are examined. A final section deals with developing a validation protocol for eye movement analysis techniques using mechanical and computer simulations of eye movements.