2.1.

Stability

One of the more difficult aspects of measuring the in vivo tear film surface is that it is not a stationary surface that can be firmly mounted and aligned as is typical in most interferometry applications. In addition to the dynamic nature of the tear film, a normal healthy eye will have random, involuntarily motions.^33,34 Head motion is another significant source of random motion that will introduce additional motion.³⁵ The dynamic range for the TFI system using the 1 MP pixelated camera, as tested by the manufacturer and verified on the TFI, is $\sim 170$ waves/radius³¹ (corresponding to a 22.2 mrad slope error measured at the tear film surface). For a typical measurement, the predicted wavefront slope introduced by eye motion, head motion, and varying ocular geometry is 105 waves/radius²⁸ (13.7 mrad slope error). It should be noted that it was assumed that all motion was centered about a mean position to which the TFI could be aligned. Additionally, it does not include surface slope errors from the tear film surface itself. The initial estimate was that at least 70% of the captured data should be useable, where a useable frame of data is one that was successfully unwrapped (refer to Sec. 3) and contains $<10\%$ data dropouts. However, the first tests with the TFI yielded extreme difficulty in aligning to the test subject. Once aligned, only a few seconds of data could be collected, with $<30\%$ of the data being useable. The observed motion exceeded all previous predictions and severely limited the functionality of the TFI.

The excessive motion that was present during measurements was attributed to two sources: subject stability in the headrest and their inability to maintain sufficient fixation. Although both of these motions were accounted for in the initial design, the sensitivity of the design to these motions was underestimated. This required an iterative approach to redesigning both the headrest and fixation assemblies, which are discussed in the following sections.

After the redesign was implemented, a trained operator could align the TFI to a subject in less than a half minute, whereas previously the operator could struggle for minutes. An additional improvement was the quantity of valid data from a given collection increased from $<30\%$ to $>90\%$. Furthermore, there has been no determinable limit to the duration of a data collection, with acquisition times up to 120 s having been repeatedly demonstrated. Duration is limited by subject fatigue rather than the previous limitation where subject motion exceeded the dynamic range of the TFI.

2.1.2.

Fixation

Fixation of the eye is accomplished by directing the subject to lock their visual gaze on a stationary target. Despite the subject’s attempt to maintain fixation, their eyes will naturally make small, involuntary motions. These eye motions are composed of three movement types: saccades (quick, large flicks of the eye), drifts (slow, cyclic motions), and physiological nystagmus or tremors (small, random, high-frequency motions).³⁶ Previous studies have shown that color, luminance, contrast, or image quality of the fixation target have no effect on fixation except in the case where any of these factors render the target barely visible.^37,38 However, it was found that differences in fixation target sizes resulted in changes in saccades and drift.^38,39 Some studies have concluded that the optimal fixation target shape to minimize random eye movement is a set of cross-hairs surrounded by a circular target with a dot in the center.⁴⁰ External distractions, such as motion in the peripheral vision or auditory stimuli, were also found to increase the amount of microsaccades.^41,42 There is an important distinction between the TFI fixation and the systems described in the literature: the TFI uses monocular off-eye fixation, whereas the systems described in the literature used monocular on-eye fixation.

In the original implementation, the fixation target was an image of the Old Main building on the University of Arizona campus [Fig. 5(a)]. In an attempt to improve fixation stability, the experiments of Boyce³⁷ and Steinman et al.³⁸ were replicated to verify the optimal choice of fixation targets. The experiment was modified to account for the off-eye fixation used on the TFI, such that eye tracking system would monitor the eye under test while the subject performed monocular fixation with the opposite eye. A commercial eye tracker (Arrington Research, Scottsdale, Arizona) is attached to the side of the converger optics and images the eye at a 45-deg angle. Subjects were placed in front of the TFI in a normal test configuration, with the exception that the TFI laser was disabled. Subjects were then instructed to fixate on a series of different targets (Fig. 5) for up to 2 min. Additionally, subjects were tested with different permutations of illumination schemes: red, green, blue, and white illumination; low or high illumination; and constant or modulated illumination. A total of six subjects were tested with the eye tracking system.

Fig. 5

Fixation targets tested on the TFI.

The results were in good agreement with the previous tests of Boyce and Steinman; the largest contributor to eye motion was the target shape. Target performance is ordered in Fig. 5 with the worst starting on the left (image of the Old Main building) to the best performing on the right (cross-hairs with circle). The improvement gained by switching from the image of the building to the cross-hairs was a decrease in random eye motion by 50%. The distribution shown in Fig. 6 is from a single test subject but is representative of every subject tested. Although more difficult to quantify due to the limited data, it appears as though saccades were also reduced 50% by choice of the cross-hair target.

Fig. 6

Eye motion during target fixation. Histogram shows radial deviation from mean position. (a) Fixation on image of building and (b) fixation on cross-hairs with circle. Gaze direction was not calibrated for absolute positions; therefore, the axes are normalized from 0 to 1.

Although Steinman³⁸ reports no variation for illumination color, he did not test the color blue. Based on our test results, we saw an $\sim 100\%$ increase in eye motion for targets that were illuminated with a solid blue light. Otherwise, all other color schemes were in agreement with Steinman, producing negligible variation in eye motion. When given a choice, most subjects chose a constant green illumination. Therefore, the implemented illumination system uses a fixed, constant green illumination scheme with the cross-hair target.

Finally, baffling was added to the TFI to minimize peripheral distractions. The front panel of the TFI enclosure was originally a specular black plastic panel that subjects could see directly in front of them when testing. This panel was replaced by a diffuse black panel. Instrumentation displays, such as the power meter, were easily visible by the subject when sitting in front of the instrumentation. Black diffuse panels were placed on both sides of the subject. These structures can be seen in Fig. 4. Although no empirical data have been collected on the use of baffles, it is expected to provide a benefit based on previous studies.^41,42

2.2.

Interferometer Source Modulation

Source modulation was not implemented in the original design of the interferometer. It was thought that the 785 nm wavelength laser source was sufficiently below the human visual response that modulation would be unnecessary. However, every test subject has reported being able to see the 785 nm source, which is focused to be concentric with the cornea. Subjects report that the source appears as a large, faint, dark red spot that fills their vision and is filled with smaller blobs that were either static or moving across their field of vision. It is speculated that the static blobs were a combination of dust particles within the interferometer and floaters in the subject’s eye. Moving structures were reported to move downward, which would be consistent with the observed tear film dynamics. Upon blinking, the tear film would be pulled upward by surface tension and the upper lid. The perceived downward motion of the structures is a result of an inverted image due to the source coming to focus inside the eye. Because the laser source subtends 45 deg of the subject’s field, resulting in significant peripheral distraction, visual perception of the source must be minimized to guarantee stable fixation during testing.

The first attempt at correction was to reduce the laser power and increase camera gain. However, even at the lowest practical levels, $\sim 50\mu \mathrm{W}$ or $1.8\mu \mathrm{W}/{\mathrm{mm}}^2$ at the surface of the cornea, the source was still perceptible to most subjects. Therefore, an obvious next step was to modulate the laser source. The laser was modulated by placing an AOM directly after the laser source with the modulation synchronized to the camera’s electronic shutter. The nominal camera frame rate operates at 33 ms with camera integration times limited to $100\mu \mathrm{s}$, resulting in $<0.5\%$ duty cycle of laser source illumination. Once the laser source was modulated, subjects reported no longer being able to see the laser source.

Modulating the laser source provides an added benefit: a reduction in detector blooming and subsequently an increase in fringe modulation. The detector in the 4D pixelated camera (Truesense Imaging KAI-1010) specifies blooming suppression to be $>100\times$, which is sufficient for nominal operation. However, the large fringe densities that are measured at the detector, which are a result of eye motion, ocular variation, misalignment, and tear film dynamics, are significantly reduced in modulation by detector sampling.⁴³ By modulating the source and reducing blooming, fringe modulation can be improved by as much as 10%.

3.1.

Phase Unwrapping

The use of the pixelated phase mask camera allows four simultaneous frames of phase-shifted data to be captured. Standard phase recovery algorithms can be applied to the instantaneous frame to recover the phase.⁴⁵ A single frame extracted from a measurement made on the TFI is shown in Fig. 8(a) along with the recovered phase [Fig. 8(b)]. Because the recorded interferograms are a cyclical function of the measured phase, the recovered phase will be a modulo $2\pi$ of the true phase. The process of removing $2\pi$ discontinuities and recovering the surface is known as phase unwrapping and is a widely researched topic.⁴⁶

Fig. 8

Example fringe pattern from (a) TFI data and (b) its recovered, wrapped phase.

Conventional interferometric testing typically results in data with high signal-to-noise ratios and minimal defects in the image. This reduces the burden on the phase unwrapping process, which tend to be sensitive to noise and defects in the phase surface. However, the tear film surface is unlike most optical surfaces. First, the structure of the tear film contains a large amount of mid- to high-spatial frequency content. This becomes an issue when structures exceed the dynamic range of the instrument, generally where tear film breakup occurs, or when mucus globules or other particles appear in the tear film surface. Eye disease, such as dry eye syndrome, may increase the prevalence of mid- to high-spatial frequency structures. These structures have to be avoided or minimally processed during the phase unwrapping routine. Second, since the TFI data are processed on a frame-by-frame basis, eye motion will induce a significant amount of tilt and defocus, further increasing fringe density in a given frame of data. Finally, ocular variation will contribute additional low-order surface features, typically astigmatism and spherical aberration. The interaction of all these effects will generate dense fringe patterns on the detector that are reduced in modulation by detector sampling. Reduced modulation will result in reduced signal, increasing the difficulty of a successful phase unwrapping. An example breakdown during phase unwrapping is shown in Fig. 9(a). The discontinuities result from the phase unwrapping algorithm’s failure to properly resolve $2\pi$ discontinuities. Path following algorithms, where pixels are processed from a neighborhood queue, tend to propagate these errors across the rest of the image. A successful phase unwrapping of the same data is shown in Fig. 9(b).

Fig. 9

(a) Example of surface with phase unwrapping errors and (b) same surface processed with a more robust phase unwrapping algorithm. Note: the surface on the left was processed with a different application than on the right, resulting in different color maps.

Two algorithms have been successfully implemented in phase unwrapping TFI data. The first algorithm is more generally known as a quality-guided phase unwrapping algorithm.^47,48 Quality-guided unwrapping routines utilize a quality metric to determine the path of unwrapping in the recovered two-dimensional wrapped phase image. When generating the queue of neighboring pixels to be unwrapped, the ordering is determined by the quality map and higher-quality pixels are given priority. Phase-shifting interferometry allows for the recovery of a modulation map, which has been shown to be a useful quality metric.⁴⁸ However, the typical measured fringe densities in a TFI image result in low modulation across large areas, negating the advantage of a modulation map.

It was found that the modulation quality map could be supplemented with a local slope variance map, similar to previous methods.⁴⁷ The initial phase slope along the $x$- direction $({\tilde{\mathrm{\Delta }}}_x)$ is defined as the forward difference of the measured phase at a pixel location $(i,j)$:

The unwrapped phase will contain $2\pi$ phase discontinuities, which can be removed from the initial slope to determine a proper slope estimate $({\mathrm{\Delta }}_x)$:

A $k\times k$ neighborhood mean of phase slopes $(\overline{\mathrm{\Delta }})$, where $k=2N+1$, can be determined:

The directional phase derivative variance ${\sigma }_x^2(i,j)$ is then given by the expression

The same math can be applied to solve for the directional phase derivative variance along the $y$ direction [${\sigma }_y^2(i,j)$], and the total phase derivative variance can be given by

Predictions of slope variance can be improved by weighting the average using the modulation quality map. The quality metric for slope variance can then be derived as an inverse of local slope variance, for example. An example of a quality guided algorithm using both modulation and slope variance quality metrics to unwrap the phase is shown on the right in Fig. 9. Quality guided phase unwrapping methods using modulation and slope variance as quality metrics have been successful at handling most of the data collected with the TFI.

The remaining set of data that cannot be processed with the quality guided method can usually be ignored. This type of data usually occurs around a saccade or when the subject’s fixation drifts momentarily, and will only exist for one or two frames. However, there are cases where data sets may have extreme surface features that need to be recovered. In these instances, a more robust algorithm is used to unwrap the data sets. An example of a more extreme set of data is shown in Fig. 10. The surface data on the left was unwrapped using the quality guided method as described previously and could only handle a small region before breaking down in the exterior regions.

Fig. 10

An example extreme data set: (a) unwrapped using quality-guided method and (b) unwrapped using a preconditioned conjugate gradient method.

The second, more robust, phase unwrapping algorithm uses a preconditioned conjugate gradient (PCG) method to minimize the difference between the gradients of the wrapped and unwrapped phases.^46,49 The weighted least-squares solution to this minimization problem reduces to Poisson’s equation, which can be solved by a number of means.⁵⁰ The least-squares solution has a closed-form solution that can be solved using fast Fourier transforms, but will only be useful as an initial guess for the final surface solution. The same quality map from the quality guided routine can be applied as a weighting. The preconditioned conjugate gradient method is then used to iteratively solve for a surface solution. Figure 10 shows the same surface being unwrapped with the quality guided method [Fig. 10(a)] and the PCG method [Fig. 10(b)].

This method has two main advantages over the quality guided method. The first is that it is a global phase unwrapping process; noise defects that appear in the middle of an image will not be propagated throughout the image as they would be in a path-following method. The second is that it accepts a quality map, which ensures convergence to a proper surface solution. However, the major disadvantage of the PCG method is that it requires approximately four times the amount of processing time over the quality guided method. Therefore, the PCG method is only used sparingly.

3.2.

Surface Correction

When interferometrically testing optical components, it may be common for residual wavefront errors to show up in the measured wavefront due to a misalignment of the test surface with respect to the interferometer. For example, testing a decentered spherical part will show up as a tilt in the measured wavefront. Knowing that tilt is an artifact of an alignment error and not actually representative of the test surface, it is considered acceptable to subtract the tilt in postprocessing of the wavefront. As a result, a first approximation to correct the TFI data for eye motions between frames would be to simply subtract the average tilt from each frame. Since the goal of the TFI is to measure the mid- and higher-spatial frequency variations of the tear film, the measured power can also be subtracted. However, postprocessing of tear film surface measurements will become more complicated by the extreme ranges of data collected.

When removing measurement terms, such as tilt or power due to alignment, it is assumed that the wavefront was near null and the measured fringe densities were minimal. Larger displacements of the test surface and greater surface deviations will increase fringe densities and result in a non-null condition. When the null condition is violated, the wavefront measured at the detector is no longer representative of the test surface.⁵¹ In other words, subtraction of tilt and power is no longer sufficient to recover the actual test surface. The error that is accumulated on the measured wavefront is a result of the test wavefront passing through different portions of the interferometer optics than what the null wavefront would have propagated through. This systematic error is commonly referred to as retrace error. In ray tracing terms, it is the difference in path that a non-null test ray takes through the interferometer optics than that of a null ray.⁵²

The dominant component of retrace error is referred to in literature as a shearing or phase error.^53,54 When considering the fact that the interferometer is imaging the test object’s surface onto the detector, the systematic errors introduced by the interferometer can be described in terms of imaging aberrations. Because the source is spatially filtered, the phase across any detector pixel can be accurately represented by a single ray that maps to the test surface. Therefore, the phase error $\mathrm{\Delta }(H)$ is equivalently the wavefront error $W$ for the given field height $H$ and pupil coordinate $\stackrel{\rightharpoonup }{\rho }$.

Because this is a single ray that results from a reflection off of the test surface, the pupil location can be completely described by the surface departure.

As an example, we will consider the effect of eye motion in the TFI. The largest imaging aberration present in the TFI is distortion and the wavefront aberration can be defined by standard aberration theory.

The phase error is

The phase error will show up as coma in the surface that is scaled by the system parameters and surface decenter. For the TFI system, $A\approx 0.031\mathrm{\Delta }x$ $\mathrm{wv}/\mu \mathrm{m}$. In other words, every $27\mu \mathrm{m}$ of decenter of the eye will generate an additional wave of coma that will show up in the measured surface. With the expected range of eye motion of $\pm 120\mu \mathrm{m}$, tear film surface data will contain coma that can vary over a range of $\pm 4.5$ waves.

Based on the aforemetioned result, an initial thought would be to remove tilt and an amount of coma that is scaled by the magnitude of tilt present. However, this would be incorrect. The previous example only considered distortion in the imaging system and lateral displacement of the eye. A full analysis including all of the TFI image system aberrations, eye motions, and ocular variations would yield a number of cross-terms that would not allow for a straightforward correction. Furthermore, it would be difficult, if not impossible, to separate surface features from introduced phase error.

When the full analysis was applied to the TFI to determine the parameters that are most sensitive to eye motion, it was found that the first eight Zernike polynomial terms [Tilt (x2), Power, Astigmatism (x2), Coma (x2), and Spherical) are sufficient. In other words, to view the dynamic surface topography of the tear film, which does not contain random surface fluctuations that are a result of eye motion and retrace errors, the first eight Zernike polynomial terms should be removed. Trefoil and secondary astigmatism, the next four Zernike terms, are weakly affected by retrace errors and do not necessarily have to be removed. However, additional Zernike terms may be removed to enhance the display of higher spatial frequency structure in the tear film. The number of Zernike terms removed would have to be determined by the individual subject’s unique corneal topography. Data collected from most of the subjects tested to date have topographies that necessitated the removal of the first 12 Zernike terms, with the largest requiring the first 19 Zernike terms to be removed. By removing these terms, the measured corneal surface is effectively projected onto a plane that allows for high-resolution inspection of the dynamic tear film structure over the desired range of spatial frequencies.