1.

Introduction

Central visual field loss (CFL) is an increasing problem in the aging population and magnifying devices are currently the only help for affected persons to use their remaining visual function. However, we have recently shown that the remaining vision for these people can be improved by correcting peripheral defocus and astigmatism.^{1 2} The aim of the current work is to more thoroughly evaluate eccentric corrections. We have therefore developed a Hartmann–Shack (HS) technique to measure the peripheral wave front aberrations of individuals with CFL.

In an eye with dense or absolute CFL, the fovea is not working and the person has to rely on peripheral vision. Some people with CFL can utilize their remaining vision better by actively using eccentric viewing, i.e., using a certain part of the peripheral retina, a so called preferred retinal location. Peripheral vision can never fully replace direct or central vision, but with eccentric viewing people with CFL can, to a limited extent, perform visual tasks such as reading or watching television. CFL is the result of a variety of pathologies, like atrophy in the nervus opticus, but the most common cause of CFL is age-related macula degeneration and this problem is increasing as the population gets older. It is estimated today that about eight million people worldwide are severely visually impaired due to macula degeneration.³ In the USA age-related macula degeneration was found in 5% of the population age 65 and older.

Vision in eccentric viewing angles is limited both by the resolution capacity of the peripheral retina and by the large aberrations in the peripheral optics of the eye. It has commonly been assumed that the main limitation is poor retinal capacity and the only help prescribed to CFL patients has been magnifying devices. However, in a previous study we have shown that the remaining vision of subjects with CFL, using a well-defined preferred retinal location, can be improved by correcting the peripheral optical errors.^{1 4}

Measuring the peripheral optics and refractive errors with traditional methods in subjects with CFL is often difficult due to the large aberrations, reduced retinal function and poor fixation. In the earlier-mentioned study, off-axis refractive errors were measured by photorefraction with the PowerRefractor instrument^{5 6} and a special fixation target was developed to help the subject hold a stable fixation. Other methods used to investigated peripheral refraction and oblique astigmatism are: the Zeiss parallax optometer,^{7 8 9} the Hartinger optometer,¹⁰ retinoscopy,¹¹ and the double-pass method.^{12 13 14 15 16 17} However, all of these methods only give the spherical and cylindrical correction (even though the double-pass method can give an indirect measurement of the higher order aberrations via the point-spread function). In large eccentric viewing angles of 20°–30° off-axis, the oblique astigmatism and higher order aberrations are considerably larger.^{18 19} There is thus a need for methods that give more detailed information about the peripheral optics of the eye.

A way to assess more detailed information about the optical system of the eye is to measure the total wave front aberrations with a HS sensor. Atchison etal.¹⁹ have previously reported measurements of peripheral wave front aberrations in normal subjects with a HS sensor. In the present paper we describe a HS sensor adapted to measure the off-axis wave front of subjects with large CFL. The measurement method incorporates special fixation targets, an eye tracker, and analyzing software. The special software has been designed to handle the elliptic pupil and the large wave front aberrations, and to calculate how the retinal point-spread function can be improved with different corrections. This paper describes the setup and software and gives examples of measurements on subjects with CFL.

2.

Method

The HS sensor has been designed to measure the eccentric wave front aberrations of the eye. The principle of the optical setup resembles a traditional HS sensor for foveal measurements²⁰ with the pupil of the eye imaged onto a lenslet array (Sec. 2.1). The setup incorporates fixation targets to enable subjects with CFL to use eccentric viewing and align the measurement axis with the direction of the preferred retinal location. An eye tracker is also built into the system to follow the angle of gaze and the position of the eye. The raw data from the HS sensor is handled by software, specially developed for this purpose (Secs. 2.2. and 2.3). As a first step the eccentric refraction is calculated from the wave front data and compared to the refraction found with the PowerRefractor instrument.^{1 2 4 5 6} All measurements on subjects followed the Declaration of Helsinki and were approved by the local Research Ethics Committee. The subjects gave informed consent prior to participation.

2.1.

The Hartmann–Shack Setup

Figure 1 and the photo in Fig. 2 show the optics of the HS setup, which also includes relay lenses imaging the pupil of the eye in the same conjugate plane as the lenslet array. The light source is a helium–neon laser (5 mW at 633 nm), the exposure time is typically around 200 ms and the collimated laser beam entering the eye has an intensity of 30 μW, which is about one tenth of the recommended safety limits.²¹ The setup incorporates a fast rotating mirror to avoid speckles and an adjustable defocus step to compensate for large amounts of defocus. To optimize the amount of reflected light from the retina to the detector and minimize the reflexes from the lenses a quarter-wave plate, a polarizing beamsplitter, and a polarizer are used.

Figure 1

The HS setup with descriptions. The inset shows the principle of the HS sensor. The lenslet array consists of lenslets 325×325 μm in size, with a focal length of 18 mm. Each lenslet covers about 50×50 pixels on the charge coupled device detector in the focal plane of the lenslets.

Figure 2

Photo of the HS setup. Eye tracker camera, ring fixation target, IR diodes, and headrest can also be seen.

When measuring the peripheral wave front aberrations of an eye with CFL, it is essential that the measurement is performed in the direction that the subject normally uses for eccentric viewing. That is, the measurement axis of the HS sensor has to be aligned with the direction of the subject’s preferred retinal location. It is also important that the eye does not move, so that the measurement can be repeated. Two special fixation targets have therefore been developed to enable the subject to look in the correct direction and keep a stable fixation. The first fixation target (F1 in Fig. 1) consists of multicolored shining rings around the axis of the HS sensor (see Fig. 3). The subject will only see the parts of the ring pattern outside his/her scotoma (the dysfunctional region of the retina) and can thus be helped by the rings to place the scotoma in a certain direction and hold it steady. The same principle was used during the measurements with the PowerRefractor both in this study and in a previous study.¹ Due to space limitations the target is placed very close to the eye, about 100 mm, which makes it unsuitable for normal, healthy eyes since it might stimulate accommodation. However, this is not the case for subjects with central visual field loss since they fixate eccentrically to the second fixation target (F2 in Fig. 1) and only use the rings as orientation help. The second target is a green diode aligned with the measurement axis of the HS sensor via a beamsplitter. The subject should try to direct his/her eye so that the green light is seen as clearly as possible, i.e., placed in the preferred retinal location. Since the peripheral retina has limited resolution capacity, this central fixation mark is large and is thus not suitable as a foveal fixation target.

Figure 3

The fixation target with ring patterns seen through the headrest.

Like most HS sensors this setup has a pupil camera, which helps the examiner to move the subject’s eye to the correct position for the HS sensor. The camera has an additional purpose as an eye tracker that measures the angle of gaze of the eye. The novel eye tracker software (Tobii Technology Inc.®) is specially developed to track an eccentric pupil in real time and to save the data of the images, the pupil shape, and the angle of gaze. The pupil camera uses the illumination from infrared (IR) diodes and finds the direction of the eye either from the elliptic shape of the pupil or from the specular reflexes from the diodes in the cornea. The angle of gaze data is needed to make sure that the wave front is measured in the same angle as the subject normally uses and that the fixation direction remains constant between the measurements.

2.2.

Wave Front Reconstruction

After a measurement the HS image has to be computer processed to find the displacements of the spots and the shape of the wave front. For this purpose we have developed a software package in Matlab® that solves a number of problems associated with measuring the wave front aberrations in eccentric angles. The software differs from that of conventional HS sensors especially in how the unwrapping is performed and how the pupil shape is taken into consideration.

First of all, the HS image is processed to find the location of the centroids of the spots.²² The resulting spot pattern must then be unwrapped, i.e., for each spot the corresponding lenslet has to be identified. This puts a limit on the measurable shape of a wave front; if a spot has moved outside the region of its lenslet and closer to another lenslet, it will be more difficult to assign the spot to the correct lenslet—the so called unwrapping problem. To be able to measure large aberrations, we have developed a software-based unwrapping algorithm.²³ The algorithm uses a two dimensional B-spline polynomial to predict the deviation between a HS spot and a lenslet. It first connects the central 3×3 HS spots with the closest lenslets, performs a least squares fit to the deviations, and then extrapolates to the closest not-yet-connected lenslets. The algorithm continues outwards in circles in an iterative manner and is able to work around difficult parts in the HS spot pattern and handle missing HS spots [an example of this is seen in Fig. 4(c)].

Figure 4

Example of a wave front measurement on subject D showing (a) the image from the eye tracker and (b) the original HS image. In (c) the spot pattern is unwrapped and each HS spot (star) is connected to its corresponding lenslet (circle). In (d) the wave front is reconstructed from the average of seven separate measurements (the height of the wave front is in micrometers).

The deviations of the HS spots divided by the focal length of the lenslets give the local tilts of the wave front. This algorithm uses modal reconstruction with a least squares fit of Zernike coefficients to reconstruct the wave front.²⁴ The Zernike polynomials, Z_n ^m (where n is the radial order and m is the azimuthal frequency)^{25 26} constitute a complete, orthogonal set of polynomials defined over a unit circle. Thus, any physical wave front ϕ can be expanded into Zernike polynomials

$$\Phi (x,y)=\sum _{n,m}{c}_n^m{Z}_n^m(x,y),$$

where c_n ^m are the Zernike coefficients (in micrometers) and (x,y) are normalized coordinates in the pupil plane. When eccentric wave front aberrations are reconstructed, a complication arises since the pupil and the wave front will be elliptic while the Zernike polynomials are defined over a circle. We have solved this by reconstructing the wave front over a circle with a radius equal to the major axis of the elliptic pupil. There will thus be an extrapolated part of the wave front outside the pupil that has no physical relevance. When the aberrations are evaluated, this part is removed. To verify that the calculated Zernike coefficients describe the measured HS spot pattern within the pupil, the algorithm calculates backwards from the wave front described by the coefficients to the spot pattern it would generate. This spot pattern is then compared to the originally measured spot pattern.

Multiple wave front measurements are made to minimize the influence of fluctuations in the higher order aberrations, due to the tear film, minor changes in fixation direction, etc. The Zernike coefficients are calculated for each measurement and are then averaged to describe the off-axis wave front of the eye. It is thus very important to measure in the same angle each time and the eye tracker gives a good indication of whether two measurements can be averaged or not.

3.

Results

The HS sensor and the software have been tested successfully in eccentric viewing angles on six subjects with CFL. The subjects have found the fixation targets to be a useful help to align their preferred retinal location with the measurement axis of the sensor and to keep their fixation stable. It was thus possible to make multiple measurements in the preferred retinal location of each subject. The quality of the HS images is good, without disturbing reflexes and stray light, and the wave fronts have been successfully unwrapped and reconstructed. The measured aberrations and the pupil sizes have been used to find optimal spherical and cylindrical corrections by optimizing the Strehl ratio.

In Table 1 the eccentric refractions found with the HS sensor and the refractions found with the PowerRefractor are listed for all subjects together with the direction of their preferred retinal location. A comparison of the refractions shows that the HS sensor generally has measured a more oblique axis of the astigmatism (larger J_45°) and that the PowerRefractor found a larger total cylinder for all subjects except subject D. The equivalent sphere is approximately unchanged for subject A, B, C, and F. As an example the HS measurements of subject D is presented more in detail. Subject D is a man born 1962 with a large CFL of radius of about 20° (stable the last 5 years). His right eye has a well trained preferred retinal location approximately 20° in the horizontal nasal visual field. Seven measurements in his preferred retinal location has been preformed in a dark room with a natural pupil (no cycloplegia was used). Only the ring pattern fixation target was in use during the measurements, since the central diode fixation target was not yet implemented. He commented that it was much easier to keep the eye stable with the fixation target than without. Table 2 shows that his fixation was stable during all measurements, which were all taken within half an hour. The results from the HS measurements on subject D are presented in Fig. 4. The first three images show the eye tracker image, the original HS image and the unwrapped HS spot pattern for the first measurement. The missing spots in the pupil are due to opacities in the eyes optical media. Figure 4(d) is the wave front reconstructed from the average Zernike coefficients of all seven measurements. The optimum spherical and cylindrical refraction is M=−4.25 D, J_0°=−0.80 D, and J_45°=0.96 D with the Strehl method calculated for a pupil size of 4 mm.

Table 1

Table 2

4.

Summary and Discussion

We have recently shown that the remaining vision of subjects with CFL can be improved by correcting the eccentric refractive errors in the fixation direction of the preferred retinal location.^{1 2 4} The peripheral optics of the eye often suffers from large aberrations, and this together with the low visual acuity makes it more difficult to assess the required correction with traditional subjective methods for foveal vision. The technique we have developed measures the wave front aberrations in the subject’s preferred retinal location with an objective HS sensor designed for peripheral measurements. The software package includes a special unwrapping algorithm and the wave front from the elliptic pupil is reconstructed with Zernike polynomials defined over a circle, which encloses the pupil. In the subsequent trial and error procedure the wave front is cut so that only the central elliptic wave front is used to find the refractive correction, which optimizes the Strehl ratio. The main advantage with this method is that it takes all aberrations within the elliptical pupil into consideration. The higher order aberrations contain components resembling defocus and astigmatism and will thus affect the refraction, especially when the aberrations are large. Another advantage with the Strehl method and the trial and error process is that it will not be mislead by, what optics designers call, local minima. A local minimum for a certain correction means that the retinal image quality is better here compared to what it is for other, slightly different corrections. But if the correction is changed more radically, another minimum can be found with even better image quality. In the peripheral optics of the eye, the higher order aberrations increase the risk of being trapped in a local minimum. The trial and error technique avoids this by objectively comparing a wide range of corrections. Maximizing the Strehl ratio is not the only metric that can be used. Other retinal image plane metrics, as for example the intensity variance of the point-spread function or the volume of the modulation transfer function in certain regions of spatial frequencies, may also be useful.²⁷ Mouroulis etal.³⁰ suggest that the maximum Strehl ratio might be misleading if the aberrations are very large and the Strehl ratio is very low.

The measurement technique differs from that of Atchison etal.¹⁹ in the design of the fixation targets, in the unwrapping algorithm and in how the elliptical pupil is handled. Atchison stretched the coordinates of the pupil to a circular shape before the wave front reconstruction; we leave parts of the circular Zernike pupil without sampling points and only use the sampled parts when evaluating the Strehl value. This means that the Zernike coefficients cannot be used directly to find either the rms of the wave front error or the refraction that minimizes the rms. However, the backwards calculation, from the coefficients to the HS spot pattern, confirms that the Zernike coefficients give a good description of the central elliptic wave front. A simulation has also been performed to verify that the extrapolated parts of the Zernike wave front do not affect the reconstruction of the central elliptic wave front. In the simulation the spot pattern from a circular pupil was reconstructed both with the correct pupil size and with a pupil diameter 50% larger. In the second case the wave front was therefore extrapolated, since there were only spots in the central part of the large pupil. The central part of this extrapolated wave front was then cut out by recalculating the Zernike coefficients to the correct size.³¹ The resulting coefficients were equal to the coefficients that were calculated from the spot pattern with the correct pupil size. These simulations were performed on real measurements with large aberrations and we therefore conclude that the extrapolation process does not influence the reconstructed wave front.

As a first step the CFL subjects have been prescribed spherical and cylindrical corrections based on peripheral wave front measurements. The eccentric refractive correction in the subjects preferred retinal location has also been measured with the PowerRefractor as is shown in Table 1. The correction found with the HS sensor is used instead of that from the PowerRefractor, because the PowerRefractor has proven to be more inaccurate in oblique angles and cannot be used at all in larger angles.^{5 6}

For subject A the PowerRefractor could not measure in the preferred retinal location since it was too far off-axis, instead a slightly smaller angle was used. The differences found in the comparison between the PowerRefractor and the HS sensor might be explained by the fact that the HS sensor measures all aberrations, while it is not well documented how the PowerRefractor handles large higher order aberrations. This might also explain why subject D, who has large amounts of coma-like aberrations, showed a very large difference in astigmatism between the two methods. Another advantage with the HS technique is that it can also be used to optimize aberration corrections. The largest higher order aberration in the periphery is coma^{17 19} and it would therefore be interesting to investigate if a coma correction could improve the visual function of subjects with CFL.