1. INTRODUCTION

The optics of a human eye is usually considered in static terms, that is, when the whole eye and its elements stand still. It should be noted, however, that such conditions are unrealistic and that a real retinal image is always perceived within the inherent dynamics of the whole eye, as well as any of its individual components. As early as the late 19th century, Helmholtz [1] noted that the defects that result from the vision inexactness and the smaller number of cones in the greater part of the retina are compensated by the rapidity with which we can turn the eye to one point of the field of vision, and it is this rapidity of movement that constitutes the chief advantage of the eye over other optical instruments.

The role of fixational eye movements in preventing sensory adaptation in the visual path has been well established, but the saccadic acceleration exerted during a typical eye movement results also in significant deviations of the lens from the optical axis, causing dynamic retinal image distortions [2]. Other movements of the eye elements, such as those related to accommodation microfluctuations [3] and to changes in the retinal image plane attributed to the fundus pulsation [4], also produce refractive changes, hence distorting the retinal image quality. Those distortions, however, are not normally perceived as factors lowering the quality of vision. Also, worth noting is that the spectral contents of accommodation microfluctuations, as well as the fundus pulsation, are not limited to the low frequency range [3].

It has been over 50 years since the work of Lohmann and Paris [5], who investigated the effect of defocus vibrations in an ideal optical system. Interestingly, they found that the integrated quality of an optical system placed out of focus increases in the presence of defocus vibrations. By defocus vibrations, they understood dynamic changes in any of the elements of the optical system that causes temporal changes in defocus. The presence of such a mechanism in the eye has been speculated elsewhere [6]. Furthermore, it has been shown through simulations that even slight amounts of defocus [of amplitude equal or less than 0.25 of a diopter (D)] and equally small levels of vibrations could lead to a substantial improvement of integrated retinal image quality with respect to a system in which defocus vibrations are absent [7]. The same mechanism of improved integrated image quality has been observed, using simulations, for dynamic changes in spherical aberration [8].

It is not exactly known how the visual system copes with variations in the retinal image quality changes exerted by the eye dynamics, and whether it uses them in a destructive (suppressing) or constructive (enhancing) mode.

To provide some insight into this problem, we propose a preliminary experimental study with visual acuity (VA) measurements in a group of subjects to test the hypothesis whether defocus vibrations applied to a real defocused eye also increase the retinal image quality and result in a higher VA. Using real measured wavefront aberrations of the subjects, we also present new computer simulations in relation to the results of the experiments.

2. METHODOLOGY

A. Measurements and Instrumentation

Eight volunteers (four females and four males aged between 21 and 33 years) with no history of eye diseases took part in the experiment. Since the study was conceived as a proof of concept, no reliable estimation could be made regarding the group sample size.

Informed consent was obtained from each participant after the nature of the study was explained. The study was approved by the ethical committee of the University of Murcia, and it adhered to the tenets of the Declaration of Helsinki. For two subjects whose habitual astigmatism correction was greater than 0.25 D, the experiment was performed with the correction of astigmatism using cylindrical lenses.

A custom-made Adaptive Optics system (MurciAO, University of Murcia, Spain) was used for the study. A scheme of the system is shown in Fig. 1. The main components of the setup are a Shack–Hartmann wavefront sensor receiving light from a small source of light (830 nm), projected onto the retina. A Badal system was used to correct the eye’s defocus, and a deformable mirror was used to generate rapid defocus changes while the subject was looking at an $800\times 600\mathrm{pix}$ black-and-white microdisplay, running at a 60 Hz video rate and covering a field of view of 2.8° horizontally and 2.1° vertically. Further details about the system can be obtained elsewhere [9]. Internal aberrations of the system were corrected by the deformable mirror so that the residual root mean square was always below 0.1 μm for the maximum allowable pupil diameter of 8.75 mm.

 

Fig. 1. Schematic of the custom-made adaptive optics system (MurciAO). Lenses L1, L2, L3, L4, and L6 are achromatic doublets. Lenses L5 and L7 are singlets. M1, M2, M3, and M4 are flat silver mirrors. BS is a pellicle beam splitter. DM is a deformable mirror. Blue stars denote conjugated planes.

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All measurements were collected at scotopic room light levels under the monocular condition (microdisplay luminance level of approximately $100\mathrm{cd}/{\mathrm{m}}^2$) and evaluated later for a fixed 4 mm pupil diameter. The eye with less refractive error was always chosen. Neither mydriatic nor cycloplegic medications were used, but care was taken to control for the pupil size to ensure that it exceeds that taken for the analysis. Subject’s head was positioned with respect to the target and wavefront sensor and stabilized using individually prepared bite bars. Before commencing measurements, subjects were acquainted with the system and trained to find their far point. A Maltese cross (angular subtense of 1.6°) was used as a target in the microdisplay. The initial position of the target was optically set at infinity. Subjects controlled the Badal system with a mouse and moved the target toward the eye (the left mouse button) or away from it (the right mouse button). After finding their initial subjective refraction, subjects were asked to move the target away from the eye to the last position before the just noticeable blur was perceived [10]. After several training trials, the procedure was repeated three times, and the mean was taken as the subject’s far point.

In the main part of the experiment, the microdisplay was used to project the targets using the Freiburg vision test with a Landolt C letter [11,12] through an artificial pupil 4 mm in diameter on the corneal plane. The test was calibrated for the system and used to measure experimental high contrast VA in the logMAR scale in a control and five defocus-induced conditions. The control condition consisted of measuring the VA with the best-perceived subjective sphero-cylindrical correction. For the other five conditions, the optical system induced 1 D of a static myopic defocus. The deformable mirror generated trapezoidal defocus vibrations centered on the added 1 D of defocus at 50 Hz with five levels of amplitude from 0 to 1 D in steps of a quarter of a diopter.

The value of 1 D was chosen to be one order of magnitude greater than the natural microfluctuations of defocus occurring in the eye [3]. The number of Landolt C optotype directions (i.e., response choices) was selected as eight with the number of trials equal to 24 for every measurement. Each trial was presented to the subjects with a display and response timeouts both equal to 30 s. Subjects were indicating their choices using a numerical keyboard with additional tabs for tactile key identification. VA measurements were repeated five times in each condition presented to a subject in a random order to ensure unbiased responses. Additionally, for the control condition, average aberrations were recorded with the Shack–Hartmann wavefront sensor for the purpose of computational simulations. To show the visual effect that the volunteers were observing during the experiment, images of a Landolt C letter chart were captured with a digital camera. The camera exposure time was 1/6 s to match the temporal summation properties of the human eye [13].

B. Simulations

Computational simulations of a visual Strehl ratio based on optical transfer function (VSOTF) were performed using the recorded aberrations of the subjects and the procedure described in Ref.  [14]. The measured wavefronts were decomposed into a set of Zernike polynomials up to the 8th radial order and rescaled to a pupil 4 mm in diameter, that is, the size of the artificial pupil used in the experiment [15]. Through-focus VSOTF curves were obtained for each subject following the procedure as in Ref.  [16]. The effect of periodic defocus vibrations was also investigated under the assumption that their frequency was high enough to perform temporal averaging of the VSOTF. Two types of periodic functions were chosen to simulate the defocus vibrations: a sinusoidal function, as in Ref.  [5], and a trapezoidal function, which represented the behavior of the deformable mirror used in the experiment [9]. The amplitudes of the functions simulating vibrations and a static defocus offset were changed both with a step of 0.01 D to cover the range from 0 D to 1 D and from $-1.5\mathrm{D}$to 1.5 D, respectively. Figure 2 shows an example of the procedure described above, illustrating the concept behind the relationship between vibrations in defocus and the retinal image quality.

 

Fig. 2. Blue curves show a classical through-focus VSOTF curve. Exemplary simulations presented are (a) a trapezoidal function with an amplitude of 1 D simulating defocus vibrations without a defocus offset (orange) and with a 1 D defocus offset (green); (b) a trapezoidal function with an amplitude of 0.75 D is simulating defocus vibrations with a defocus offset of 1 D. Empty circular marks represent VSOTF values at an appropriate defocus level, when no defocus vibrations occur. Arrows indicate the change in VSOTF values at those points when defocus vibrations are applied. The increase in the VSOTF value in (b) is smaller than in the case of 1 D amplitude of defocus vibrations in (a).

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C. Statistical Analysis

To analyze the trend of VA improvement with the increase of the vibration amplitude of defocus (experimental viewing conditions), linear regression was applied for all subjects, and the overall slope was calculated. Spearman’s rank correlation between the experimental VA and simulated VSOTF was determined. Fisher’s test for correlation coefficients was used to ascertain whether sinusoidal and trapezoidal types of vibrations applied in simulations result in significantly different correlations. A two-way repeated-measure nonparametric analysis of variance (Friedman test) was used to ascertain the differences in VA for different experimental viewing conditions. A significance level of 0.05 was used throughout the study. Bonferroni correction was applied when appropriate to compensate for multiple comparisons.

3. RESULTS

A. Measurements in Subjects

The experimental VA measurements are presented in Fig. 3. The box plots show collectively an increase in the image quality with the increase of the defocus vibrations amplitude. The blue box plot summarizes the distribution of VA at the far point with no vibrations present (control condition). Green box plots summarize the distributions of VA measured at the 1 D static defocus with the increasing amplitude of defocus vibrations. Images of the Landolt C chart shown in Fig. 3 were acquired with a digital camera for illustrative purpose only. They show how the image of the target changed in each condition. Statistically significant differences were found after Bonferroni correction in VA among experimental viewing conditions (Friedman test, ${\chi }^2=173.81$, $p<0.001$). A linear regression analysis of the experimental VA versus the amplitude of defocus vibrations showed an overall negative slope of $-0.3\mathrm{logMAR}$ units per D of amplitude, which was statistically significant (Spearman’s $r=-0.581$, $p=0$; negative value due to logMAR notation).

 

Fig. 3. Left panel: box plots of VA measurements for all subjects in each condition A–F. Red horizontal bars show the median value for each condition, and red dots show outliers. The blue box plot shows the distribution of VA at the far point, whereas the green box plots show the distributions of VA at 1 D beyond the subjects’ far points and five different amplitudes of vibrations. Right panel: photographs of a Landolt C chart taken with a digital camera for illustrative purposes. Degradation of the image is visible when a static defocus of 1 D is introduced (A and B). The image gradually improves with the increase of the amplitude of vibrations (B to F).

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B. Simulations

Figure 4 shows baseline static higher-order aberrations for all the subjects, where the single-indexing standard Zernike coefficient notation scheme is used [17]. The results are within those reported in the literature [18]. These wavefront measurements were used for the VSOTF simulations.

 

Fig. 4. Baseline higher-order aberrations of all the subjects. The error bars show $\pm 1$ standard deviation.

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A three-dimensional surface plot of VSOTF as a function of defocus and defocus vibrations shown for one randomly chosen subject in Fig. 5 can serve as a typical example of subject’s VSOTF behavior in the simulated conditions. With no defocus vibrations present (the case of static defocus), the peak value of VSOTF occurs close to zero. This peak transforms into two ridges when a simultaneous increase of defocus vibrations and increase in the static defocus occur. Performing analogous simulations using a sinusoidal function, we found a slight increase in the VSOTF values in the valley area, formed between the two ridges, and a decrease in the peak values at the ridges. However, the general behavior of the through-focus VSOTF described above was maintained for all the subjects. The blue curve overlapped on the surface in Fig. 5 is a classical through-focus VSOTF when no defocus vibrations are present. The red curve is a through-focus VSOTF when the amplitude of defocus vibrations reaches 1 D. The green curve corresponds to a situation where 1 D of a static defocus was added, which was the value introduced by a Badal system in the experimental VA testing. Green triangles represent the points where the VA tests were taken.

 

Fig. 5. Simulation of a VSOTF in the presence of defocus vibrations. In this example, a trapezoidal function was used to simulate defocus vibrations. (a) A three-dimensional plot of a VSOTF as a function of defocus and amplitude of defocus vibrations. The blue continuous curve is a classical through-focus VSOTF when there are no defocus vibrations. The red curve is a through-focus VSOTF when the amplitude of vibrations is equal to 1 D. The green curve shows how the VSOTF improves with the increase of amplitude of vibrations when 1 D of a static defocus is added. (b) A side view showing VSOTF as a function of defocus. (c) A side view showing VSOTF as a function of amplitude of defocus vibrations. Green triangles represent points of the experimental VA measurements, and their results are shown in Fig. 3.

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To compare the simulations and real measurements results, we computed the correlation between experimental VA and VSOTF. Spearman’s rank correlation for simulations with sinusoidal vibrations of defocus was $r=-0.7782$ with $p=0$. For simulations with the trapezoidal function, the correlation was $r=-0.777$ with $p=0$. Fisher’s test revealed no statistically significant differences when using sinusoidal or trapezoidal defocus vibrations ($p=0.388$). Figure 6 shows the relationship between experimental VA and VSOTF simulated using the trapezoidal vibrations mimicking the experimental conditions.

 

Fig. 6. Experimental VA measurements versus VSOTF simulations using the trapezoidal function employed by the deformable mirror. Blue empty circles show VA for the control condition, green empty circles show VA for the condition with 1 D without vibrations, and the solid circles show VA for the rest of experimental conditions.

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4. DISCUSSION

In principle, for a certain level of a static defocus present in the optical system, the overall temporarily integrated optical transfer function (and in the case of this study, its visual counterpart, VSOTF) improves when sinusoidal or trapezoidal defocus vibrations are added. The VSOTF is a clearly nonlinear function of defocus (see Fig. 2). Consider the case of Fig. 2(a), where vibrations of 1 D in amplitude are added to a system with 1 D of defocus. Examining the slope of the through-focus VSOTF, it is apparent that the extent of the image quality enhancement is greater, when moving the defocus from 1 D toward 0 D, than the extent of the image quality diminishment, when the defocus is moved from 1 D towards 2 D. For such a system, the “glimpses” of a better image will be evident, while the “glimpses” of a worse image will be barely noticeable. This on average results in a better image quality. An interesting observation is that the average defocus in such a system with vibrating defocus does not provide means for intuitive assessment of the overall image quality, which raises the question of whether averaging higher-order aberrations—for example to estimate the retinal image quality—is the best approach to take into account their constant fluctuations [3,19]. An estimation based, for example, on integrating the area under the VSOTF through-focus curve could possibly result in a better evaluation of the retinal image quality for such systems.

Figure 7 shows the comparison between sinusoidal and trapezoidal functions. When the system has a defocus offset (green lines), the cumulative time, for which the system is at 0 D, is shorter in case of the sinusoidal vibration function (Fig. 7(a)) than in the case of the trapezoidal vibration function (Fig. 7(b)). This explains the slightly better performance of a defocused system with trapezoidal vibrations.

 

Fig. 7. Optical system with periodically vibrating defocus. The frequency of vibrations in this example is 50 Hz. Two cases are considered: the orange curve and lines represent a system that has no defocus offset; the green curve and lines represent a system that has a defocus offset of 1 D. Larger time, on average, is spent in the range of 0–0.25 D assumed as the in-focus range (blue zone) in the presence of a 1 D defocus in the sinusoidal (a) of trapezoidal (b) vibration functions. A larger number of dots on the green line also indicate that the increasing of the in-focus time effect is more evident in the trapezoidal defocus vibration (b) than in the sinusoidal one (a).

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The idea of adding defocus vibrations to a defocused system for improving the image quality can also be seen from an inverse perspective: if there are some inherent image-degrading vibrations of defocus present in the optical system that is in focus, adding a certain amount of defocus offset can effectively improve the image quality. Taking into account a certain depth of focus corresponding for instance to a quarter of a diopter (see the blue zone in Fig. 7), it is clear that the average time spent within the blue zone is greater when a defocus offset is added (green curve and lines) in comparison to the case without a defocus offset (orange curve and lines).

If we examine the results of simulations in Fig. 5, we can see that if the system has no defocus offset, the vibrations would only decrease the image quality. Naturally, any defocus degrades the image quality in a system without defocus vibrations. Yet if one of those degrading components exists in the system, it can be balanced by the other. Such balancing enhances the system performance. This has been proved experimentally here only in one direction, that is, by adding the defocus vibrations to a system at defocus. To prove experimentally the inversed hypothesis mentioned above, the subjects’ accommodation would have to be paralyzed to avoid any major defocus changes in the presence of defocus vibrations when there is no defocus offset. Following theoretical [5] and computational simulations, increasing the defocus offset then should lead to an improved VA. Such experiments are planned for the future.

Furthermore, the existence of an accommodative lag can be explained in terms of a lack of need for a perfect accommodation to perform a certain task, for example, reading or identifying a letter [20–22], or the use of the eye’s depth of focus during accommodation [23]. It has been shown that a certain defocus offset (lag) can improve the retinal image quality in the presence of higher-order aberrations [24]. Our results give another potential source for explaining the origin of that tolerance to defocus, since it is well known that the microfluctuations existing in the eye increase during accommodation [3] and that the accommodative lag increases as well [21,22]. The amplitude of the microfluctuations is typically much smaller (0.1–0.5 D [3] ) than the accommodative lag, which can be larger than 1 D for targets closer than 4 D [25]. However, our results show that the defocus vibrations of lower amplitudes (e.g., 0.25–0.50 D) increase VA, allowing targets with logMAR of approximately 0.3 to be seen (Fig. 3). This fact together with the natural depth of field of the eye without microfluctuations of approximately 0.9 D [23] can explain the presence of a lag even larger than 1 D. Nevertheless, the question of how small the vibrations could be to potentially trigger a lag response and how fast they should be for the neural processes to integrate them still needs to be answered.

VA experiments proved that the image quality in terms of sharpness perceived by the subjects improved with the increase in the amplitude of defocus vibrations introduced by the deformable mirror. This improvement can be appreciated by examining the photographs presented in Fig. 3, where the exposure time was set long enough for the image integration effect to take place. One should note, however, that the pictures were taken by a photographic camera that did not contain the higher-order aberrations present in the real human eyes.

There are several limitations to this study. Because subjects were not cyclopleged, the subjective assessment of the far point in the experiment could have led to residual accommodation. Further, when computing VSOTF in the presence of defocus vibrations, the assumption was made that the through-focus VSOTF was not temporally changing, which is not true in the presence of higher-order aberrations dynamics existing in the human eye [19,26]. This dynamics was not stopped when the subjects’ VA was measured, and it may contribute to lowering the correlation between the VA and VSOTF. Additionally, VA involves neural processes that are ignored in the optical simulations. It is also not clear how the presence of defocus vibrations influences those processes and, as stated in the Introduction, whether they unambiguously have a constructive or a destructive character.

The shape of the VSOTF surface plot can be modified by the type of a periodic function used for the defocus vibrations implementation and also by the higher-order aberrations that will affect the shape of the through-focus VSOTF of a non-vibrating system. Thus, despite defocus being the largest aberration and its vibrations having the strongest effect on VA, by enlarging the depth of focus, the dynamics and structure of higher-order aberrations, such as the spherical aberration [8], will in fact also have their influence on the image quality. Interestingly it has been found recently that the correction of lower-order aberrations induces changes in the dynamics of higher-order aberrations, and those changes are statistically different between the myopes and emmetropes [27]. This raises the question about the origin of those differences.

5. CONCLUSIONS

The paper explores the topic of defocus vibrations in optical systems, introduced for the first time by Lohman and Paris, but in the context of human vision. The theoretical approach given in Ref.  [5] was expanded here to computational simulations using real human eye wavefront aberrations rather than an ideal optical system. Experimental VA was measured, and it confirmed the patterns that can be predicted with in-silico simulations. This work has proved that defocus vibrations applied to a real defocused eye increased the quality of the retinal image, consequently leading to a better VA. More studies on this topic are needed to further explore this phenomenon and answer the question whether similar mechanisms could be naturally present and utilized by the human vision.