This work is related mainly to methods aiming at detection of strabismus (cross-sightedness), eye alignment and amblyopia (“lazy eye”) in young test subjects, where patient cooperation is problematic and the detection of short-lasting episodes is important and even crucial in some cases. The robust application of the CWT makes it applicable to any type of optical systems acquiring and analyzing signals distinguishable in the joint time-frequency domain, especially in systems with abundant optical noise of origin different from that of the signals measured. A condition would be that the noise does not overlap in time and frequency simultaneously with the frequency and temporal location of the event. In addition to amblyopia, the technique described can be used for the detection of frequent and short-lasting losses of fixation, possibly indicative of nystagmus, attention-deficit-hyperactivity disorder (ADHD), autism, or other neuropsychologic disorders.
The CWT is not the only technique available for time-frequency expansion. The short-time Fourier transform, STFT, (also known as the windowed Fourier transform) localizes the signal in time- and frequency domain by modulating the time signal with a window function before performing the Fourier transform, to obtain the frequency content of the signal in the region of the window. As a rule, it is a compromise between time and frequency resolution; the wider the window, the higher the frequency resolution, at the cost of poorer time resolution, and vice versa . Any attempt to increase the frequency resolution causes a larger window size and therefore a reduction in time resolution, and vice-versa. Also, in order to be able to analyze transients, overlapping windows need to be used, which can slow down analysis considerably, and acts like a low-pass filter in the time domain.
Another alternative, most widely used two or three decades ago, is the Wigner-Ville distribution (WVD). Its definition for time-frequency analysis is:
where is the imaginary unit, and * denotes complex conjugation [19–22]. In essence, the WVD is the Fourier transform of the input signal’s autocorrelation function, i.e. the Fourier spectrum of the product between the signal and its delayed, time reversed copy, as a function of the delay. Unlike the short-time Fourier transform, the Wigner distribution function is not a linear transform. A cross term (“time beats”) occurs when there is more than one component in the input signal, analogous in time to frequency beats.
In order to reduce the cross term problem, many other transforms have been proposed, the most prominent one perhaps being the Cohen’s class distribution. The best known member of Cohen’s class distribution function is the Choi–Williams distribution function . This distribution function adopts an exponential kernel to suppress the cross-term:
and the kernel function is
However, the kernel gain does not decrease along the η and τ axes in the ambiguity domain, and, consequently, the kernel function of the Choi–Williams distribution function can only filter out the cross-terms resulting from components away from the η and τ axes and away from the origin.
In summary, the CWT appears to be the tool of choice when time-frequency distributions are needed in order to detect different frequency components appearing simultaneously, or in different moments in time. This is particularly true of high frequency events of short duration and low amplitude (small scales), or longer-lasting low-frequency oscillations of higher amplitude (large scales), as is the case here.
There are some limitations of the optical and opto-mechanical hardware involved in this technology. Mechanical vibrations, optical back-reflections (reflections from reflective surfaces back to the sensors before actually the light reaches the eyes), multiple internal reflections not captured by the light traps, some birefringence caused by the beam splitters, and others, can all cause instrumental noise which translates into parasitic frequency components. Some of them may overlap with the central fixation frequency of interest and thus mask it, making a differentiation in the joint time-frequency domain difficult. Such artifacts can be avoided or moved away from the region of interest, by careful choice of the mechanical parameters (like scanning speed), optical design (i.e. spinning wave plates), or proper design of the analog electronics (i.e. filters).
Other limitations arise from the measurement principle. Because only about 1/1000 of the polarized light used for measuring is returned by the fovea, the distance between the measurement system and the eye cannot be increased much beyond 40 cm. Further, the room illumination should be dimmed, to prevent pupil constriction and reduction of the near-infrared light going through it. This not only decreases the signsl-to-noise ratio, but can cause patient discomfort, and is one more reason to aim for fast and reliable tests.