Ih 2m vVxt Ccyt J x Hxt 2 dxVxt

where G(.) is a Gaussian kernel MAP introduced to nonlinearly modulate the spatial potential field that excites the dynamics of the quantum object. In fact, (G(.) = V(x,t), where V(x,t) is given in (9.24).

The nonlinear Schrodinger wave equation given by (9.27) is one-dimensional with cubic nonlinearity. Interestingly, the closed-form dynamics of the recurrent quantum neural network (equation (9.27)) closely resembles a nonlinear Schroodinger wave equation with cubic nonlinearity studied in quantum electrodynamics [20]:

where m is the electron mass, e the elementary charge and r the magnitude of | x |. Also, nonlinear Schroodinger wave equations with cubic nonlinearity of the form dA(t) = c\A + c3 | A |2 A, where ci and c3 are constants, frequently appear in nonlinear optics [12] and in the study of solitons [24, 11, 15, 33]. Application of the nonlinear Schrodinger wave equation for the study of quantum systems can also be found in [34].

In (9.27), the unknown parameters are weights Wi(x,t) associated with the Gaussian kernel, mass m, and Z, the scaling factor to actuate the spatial potential field. The weights are updated using the Hebbian learning algorithm r)W- (x t)

The idea behind the proposed quantum computing model is as follows. As an individual observes a moving target, the uncertian spatial position of the moving target triggers a wave packet within the quantum brain. The quantum brain is so hypothesized that this wave packet turns out to be a collective response of a classical neural lattice. As we combine (9.27) and (9.29), it is desired that there exist some parameters m, Z and ยก3 such that each specific spatial position x(t) triggers a unique wave packet, f(x,t) =| '^(x,t) |2, in the quantum brain. This brings us to the question of whether the closed form dynamics can exhibit soliton properties that are desirable for target tracking. As pointed out above, our equation has a form that is known to possess soliton properties for a certain range of parameters and we just have to find those parameters for each specific problem.

We would like to reiterate the importance of the soliton properties. According to our model, eye tracking means tracking of a wave packet in the domain of the quantum brain. The biological motor control aligns the eye pupil along the spatial position of the external target that the eye tracks. As the eye sensor receives data y from this position, the resulting error stimulates the quantum brain. In a noisy background, if the tracking is accurate, then this error-correcting signal v(t) has little effect on the movement of the wave packet. Precisely, it is the actual signal content in the input y(t) that moves the wave packet along the desired direction that, in effect, achieves the goal of the stochastic filtering part of the eye movement for tracking purposes.

9.4.3 Simulation Results II

In this section we present simulation results to test target tracking through eye movement where targets are either fixed or moving.

For fixed target tracking, we have simulated a stochastic-filtering problem of a dc signal embedded in Gaussian noise. As the eye tracks a fixed target, the corresponding dc signal is taken as ya(t) = 2.0, embedded in Gaussian noise with SNR (signal-to-noise ratio) values of 20 dB, 6dB and 0dB.

We next compared the results with the performance of a Kalman filter [19] designed for this purpose. It should be noted that the operation of the Kalman filter is based on a priori information that the embedded signal is a dc signal, whereas the RQNN is not provided with this information. The Kalman filter also makes use of the fact that the noise is Gaussian and estimates the variance of the noise based on this assumption. Thus it is expected that the performance of the Kalman filter will degrade as the noise becomes non-Gaussian. In contrast, the RQNN model does not make any assumption about the noise.

Notice that there are certain values of 3, m, Z and N for which the model performs optimally. A univariate marginal distribution algorithm was used to get near optimal parameters while fixing N = 400 and h = 1.0. The selected values of these parameters are as follows for all levels of SNR:

The comparative performance of eye tracking in terms of rms error for all the noise levels is shown in Table 9.1. It is easily seen from Table 9.1 that the rms tracking error of RQNN is much less than that of the Kalman filter. Moreover, RQNN performs equally well for all the three categories of noise levels, whereas the performance of the Kalman filter degrades with the increase in noise level. In this sense we can say that our model performs the tracking with a greater efficiency compared to the Kalman filter. The exact nature of trajectory tracking is shown for 0dB SNR in Fig. 9.9. In this figure, the noise envelope is shown, and obviously its size is large due to a high noise content in the signal. The figure shows the trajectory of the eye movement as the eye focuses on a fixed target.

To better appreciate the tracking performance, an error plot is shown in Fig. 9.10. Although Kalman-filter tracking is continuous, the RQNN model tracking consists of "jumps" and "fixations". As the alignment of the eye pupil becomes closer to the target position, the "fixation" time also increases. Similar tracking behavior was also observed for the SNR values of 20 and 6 dB.

These theoretical results are very interesting when compared to experimental results in the field of eye-tracking. In eye-tracking experiments, it is known that eye movements in static scenes are not performed continuously, but consist of "jumps" (saccades) and "rests" (fixations). Eye-tracking results are represented as lists of fixation data. Furthermore, if the information is simple or familiar, eye movement is comparatively smooth. If it is tricky or new, the eye might pause or even flip back and forth between images. Similar results are given by our simulations. Our model tracks the dc signal that can be thought of as equivalent to a static scene, in discrete steps rather than in

Table 9.1. Performance comparison between Kalman filter and RQNN for various levels of Gaussian noise

Noise level

RMS error

RMS error

in dB

for Kalman filter

for RQNN

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