3.1. Image processing and analysis
Segmentation of the OCT volume, performed on each A-line, permitted us to generate en face projections of the cone mosaic by co-adding the intensity values found at ISOS and PT for each A-line in the volume. (left, center) shows examples of these projections, from two volumes in a single acquisition (S1). Registration of volumes to one another allowed en face projections to be averaged. Such averaging improved signal to noise ratio (SNR) and, when an en face image without visible motion artifacts was used as a reference image, removed motion artifacts from the average, improving visibility of the cones. (right) shows a registered average of the 15 en face images, i.e. 15 images such as the one shown in (center), using (left) as a reference. Improved image quality in the average image provides qualitative evidence that the volumes were registered correctly–a key first step in the subsequent tracking of cones.
Fig. 2 En face projections of the cone mosaic. (Left and center) Single en face projections of the cone mosaic, produced by co-adding the segmented images of ISOS and PT from single volumes, acquired at 1.5°. A registered average (right) using the image (more ...)
Combining (1) A-line inclusion criteria, (2) spatial unwrapping, (3) registration, and (4) cone tracking permits θOS
to be monitored for all cones in a series of volumes. (
) shows a subset of the cones tracked, in intensity (upper left) and phase (upper right). Due to eye motion, not all cones are present in all frames of the video. Cones appearing in a large number of frames were selected for this display, and these are naturally clustered in a portion of the mosaic mostly present in all frames. For quantitative analysis, however, all cones in each frame were tracked. (bottom) shows stabilized views of the selected cones, arranged in two rows. The top box in each row contains the intensity image (labeled I
) and the bottom box in each row contains the θOS
image (labeled θ
). In the stabilized view it appears that θOS
is correlated among pixels inside the cone aperture in the intensity view.
Fig. 3 (
Media 1). Tracking intensity and phase of individual cones over time. (Upper left) An en face projection of the cone mosaic (S1), constructed by co-adding intensities of ISOS and PT pixels for each A-line in the volume. Video shows en face projections (more ...)
3.2. Experiment 1. Phase sensitivity in the cone outer segment
shows the phase distribution of all cones in a single volume (S2) before (left) and after (right) spatial unwrapping. In each case, the plot shows θOS
for all A-lines in all cones in the volume, grouped by cones such that each column of points represents the median-subtracted set of θOS
values from a single cone. In the wrapped distribution (, left) it is evident that in many cones θOS
is clustered near the edges of the ±2π
range, indicating the presence of phase wrap. These are not present in the spatially unwrapped case (, right). The log-scale bar graphs show the marginal phase distribution, and indicates a standard deviation of 1.13 rad in the wrapped case and 0.57 rad in the unwrapped case. Among all subjects and all volumes, the standard deviation of the wrapped phase distribution ranged from 1.1 rad to 1.8 rad. That is, without unwrapping phase, in the worst case the correlation of phase within cones is equal to that of a random uniform distribution (1.814 rad). In the best case, without unwrapping, phase measurements may be pooled from large numbers of cones. However, meaningful measurements cannot be made from a few cones without a robust unwrapping procedure. A clear demonstration of the unwrapping problem is given in in the appendix
. The range of unwrapped standard deviations was 0.48 rad to 1.5 rad, in all cases smaller (p<0.05) than that of a uniform random distribution. The RMS of unwrapped standard deviations, over all cones in all subjects (considering a total of 255,531 A-lines in 26,898 cones), was 1.0 rad. Applying Equation 1
this average σθ
suggests an OS length sensitivity, σL
, of 45 nm. In the best single volume σL
= 23 nm, and over all volumes acquired from the best subject (S1), σL
= 30 nm.
Fig. 4 Within-cone phase distributions (S2). (Left, scatter plot) The median-subtracted phase of each pixel in each cone. Each column of points depicts θOS within a cone. Spatial phase wrapping is evident in the cluster of points near the ±2 (more ...)
The distributions of θOS in are visibly nonuniform, and this was confirmed using a χ2 test for uniformity. The χ2 statistic was computed for the observed θOS distribution and an expected uniform distribution over the same range. The χ2 statistic was very large, ranging from 410 to 11000, with p < 10−6 (the probability that θOS represents a uniform, random distribution) for all volumes from all subjects. By contrast, the same test for uniformity applied to wrapped phase, yielded a χ2 statistic as low as 1.1 (p = 0.79), suggesting that the distribution of wrapped θOS, even when median-subtracted and averaged over cones, may be uniform.
We used ANOVA to test whether the differences in θOS among cones was significant in comparison to the variance of θOS within cones. The F-statistic was very large, ranging from 11.5 to 173, yielding a p < 10−6 for all volumes from all subjects. This suggests that the observed differences in θOS from cone to cone are meaningful, and not due to phase noise. It should also be noted that the F-statistic is highly conservative in this case, since between-group variance is artificially reduced by the [0,2π) wrapped range.
For confirmation that θOS was correlated within cones, we computed the autocorrelations of the intensity and phase images. These are shown in . The first two images (far left, near left) show the autocorrelations of en face projections of θISOS and θPT, respectively, which are the absolute phase at the corresponding reflections. It is clear from these images that very little phase correlation exists among A-lines, though a small amount can be seen at the pixels above and below the midpoint, parallel to the fast scan direction. This brief vertical phase correlation is likely a consequence of the vertical orientation of the fast scan. Neighboring pixels along the fast scan are acquired in quick succession, separated by only 6 – 8 μs. Axial eye motion–responsible for decorrelating phase across much of the image–must be sufficiently small over these brief time intervals to preserve correlation of phase between neighboring pixels. Two pixels away from the midpoint, the correlation disappears completely.
Fig. 5 Autocorrelations of ISOS phase, PT phase, intensity, and θOS, from left to right; autocorrelations clipped at 0.5 to enhance visibility of correlation tails and zero crossings. The autocorrelations of absolute phase, at ISOS and PT (far left and (more ...)
(near right) shows the autocorrelation of an en face
intensity projection (S1). It has the stereotypical appearance of a uniformly packed mosaic. The distance between concentric peaks agrees with the predicted cone row spacing of approximately 5 μ
]. (far right) shows the autocorrelation of an en face
projection of θOS
. There is substantial agreement between the central peaks of the intensity and phase autocorrelations, but while the intensity autocorrelation has bright, concentric rings surrounding the central peak, the phase autocorrelation lacks any such features. The similarity between central peaks suggests that both intensity and phase are correlated among pixels within the cone, while the dissimilarity between the tails of the autocorrelation suggests that periodicity exists in the intensity image but not in the phase image, which is consistent with our predictions.
3.3. Experiment 2. Eccentricity dependence of phase sensitivity
We sought to determine if phase sensitivity in the OS varied with retinal eccentricity, and in particular at eccentricities greater than the 1.5° at which the data in Experiment 1 were collected. depicts this relationship. The black squares (left axis) show phase sensitivity as a function of retinal eccentricity, indicating a generally positive relationship that appears to plateau around 3.5°.
Fig. 6 Phase sensitivity as a function of retinal eccentricity, as measured in one subject (S3). Phase sensitivity (black squares) is better near the foveal center, which may be due to structural differences among cones at different eccentricities. It is known (more ...)
The dotted red line (right axis) shows predicted cone diameter [26
] as a function of retinal eccentricity. Up to 4° the two curves show similar eccentricity-dependence–strong near the fovea and weaker outside the fovea.
3.4. Experiment 3. Phase changes over hours
We imaged two subjects (S1 and S4) over several hours, calculating θOS for all cones in all volumes, as in Experiment 1. Automated cone identification and tracking permitted θOS to be monitored over time.
As a first pass, to determine whether statistically significant phase changes were present over time, we employed ANOVA for each cone in each subject. ANOVA, employed this way with respect to time, determines whether differences in average phase over time are significant (α = .01), in comparison to variance of phase within cones at single time points. We found that 97% (S1) and 88% (S4) cones underwent significant phase changes over time.
shows plots of θOS
over time for two cones from subject S1. For each cone, the temporally wrapped (top) and unwrapped (bottom) values are shown. Temporal unwrapping was performed using the linear fitting method described in the appendix
(§A.5). For these two cones, rates of elongation were 111 nm/hr and 84 nm/hr, reasonably consistent with previous findings [15
]. Measurements were absent when, in a given volume, the cone was out of view or suffered from a motion artifact. Also, it must be pointed out that at any given time as many as 21 points are shown in the plots (corresponding to the 21 pixels in the circular footprint of the OS); as a result, the precise average phase may be difficult to discern visually.
Fig. 7 Representative phase changes in two single cones (S1). For each cone, the temporally wrapped data are shown in the top plot (green markers) and temporally unwrapped data shown in the bottom plot (blue markers). In the unwrapped data, linear fits were (more ...)
In these two cones, instances of phase wrapping are apparent (1.9 h and 2.7 h, left, and 1.7 h, right). The linear fitting technique readily identified such instances of wrapping and corrected them accordingly. This was true for many cones in each data set, but examples of unwrapping error were common, and the bulk rate of renewal was consequently indeterminate by this method.
In order to assess the average rate of change over all cones, we performed Lomb-Scargle fitting of the sine of measured phase. In short, it involves computing sin(θOS
) for each cone, generating its frequency spectrum by sinusoidal fitting, and averaging these spectra. The spectra for S1 and S4 are shown in . The frequency (x) axis of the plot is expressed in terms of rate of length change, ΔLOS
. The spectra from both subjects have clear peaks at approximately 150 nm/hr. These rates are higher than what was typically observed using AOCFI [15
] but within the range of observations made over a larger number of subjects and broader range of retinal eccentricities [27
Fig. 8 Frequency spectra of sin(θOS), averaged over all cones in each of the two over-hours data sets. The frequency (x) axis has been expressed in terms of rate of OS length change, using Equation 3. Peaks at approximately 150 nm/hr can be seen in both (more ...)
3.5. Experiment 4. Phase sensitivity outside OS
We applied the same methods described in Experiment 1 to investigate phase sensitivity outside of the OS, in the inner segment (IS) between ELM and ISOS, and the extracellular matrix (SOS) between PT and RPE. We found the distribution of phase in these layers, confined to the same cone apertures used in Experiment 1, to be significantly nonuniform, with σθ values of 1.24 rad and 0.88 rad for IS and SOS, respectively, corresponding to σL of 56 nm and 40 nm in the best subject (S1). Over all subjects, σL was 72 nm and 58 nm for IS and SOS, respectively.
Additionally, we examined autocorrelations of the IS and SOS en face phase projections, which are shown in (top left and top right, respectively), in comparison to the autocorrelation of the OS en face phase projection (top center). As expected from the phase sensitivities above, the autocorrelations of these projections show local phase correlation. It is clear that SOS exhibits less local autocorrelation than OS, and IS still less, but in both cases the local autocorrelation of phase is better than random, i.e. correlated over more than 1 pixel. The profile of each image is plotted below (blue lines). Superimposed on each profile is the profile of the intensity projection (red, dashed). The important feature in these plots is the first zero. The first zero for θOS (center, blue) is at 5 μm. For θIS (left, blue) and θSOS (right, blue), the first zero occurs at 4 μm.
Fig. 9 Autocorrelations of θIS (left) and θSOS (right), as compared to that of θOS (center), and profile plots below (blue). Profiles of the intensity autocorrelation (red, dashed) are superimposed on each profile, for comparison. The (more ...)