According to the standard linear model, perceived head-centered velocity
is a linear combination of retinal image velocity
R and eye velocity
E with gains
r and
e, respectively (
Equation 1). If
R and
E vary in time and the signals have different latencies, this will affect the perceived velocity
(
t) at time
t. As we used sinusoidal movements of one single frequency
f in our experiment, these latencies translate into phase shifts:
where
R and
E now represent movement amplitudes,
φ is the phase of the retinal image motion with respect to the pursuit target,
ρ is the phase shift of the retinal signal,
θ is the phase of the eye movement with respect to the pursuit target, and
ɛ represents the phase shift of the eye movement signal.
Figure 2 illustrates this equation in a phasor plot. Sinusoidal motion is represented as a vector, with the angle with respect to the positive
x-axis indicating phase and the distance to the origin representing amplitude. Adding sinusoids of the same frequency is equivalent to adding vectors in the phasor plot. By simple trigonometry, the amplitude
of the perceived motion
satisfies the equation:
In our experiment, observers judged the motion of a sinusoidally moving random dot pattern in two separate intervals. In the pursuit interval, they tracked a moving target with their eyes while making the judgment, whereas in the fixation interval the target was stationary. A staircase procedure was used to determine the point where the dot pattern appeared to have the same peak velocity in the two intervals (the point of subjective equality). At this point, the perceived motion amplitude
p in the pursuit interval equals the perceived amplitude during fixation
f. This gives:
where subscripts
f and
p refer to fixation interval and pursuit interval, respectively. Dividing both sides by
r 2 gives
According to the linear model, therefore, the velocity matches are determined by two free parameters: the gain ratio
e/
r and the phase difference
ɛ−
ρ. Note that the individual gains
e and
r and the individual phases
ɛ and
ρ cannot be resolved (see Freeman,
2001; Freeman & Banks,
1998; Souman et al.,
2006).