Wavefront sensing has become a useful tool to assess the image quality of the eye, with applications to both research and clinical evaluation. Ocular aberrometry has been used for studying ocular properties as a function of accommodation [1
], aging [2
], or refractive error [4
], as well as for the assessment of refractive correction techniques (refractive surgery [5
], cataract surgery [7
], and contact lenses [9
]), or the correction of ocular aberrations to visualize the eye fundus [12
]. The evaluation of the optical outcomes of refractive surgery has led to an increasing importance of aberrometry in recent years, and commercial aberrometers are now commonly used to assist in surgery [15
Most current aberrometry techniques measure the ray aberrations of the eye, i.e., the local slopes of the wavefront, by estimating the deviation of the light beams from a reference, either as the light goes into the eye (i.e., laser ray tracing (LRT) [17
] and spatially resolved refractometer (SRR) [18
]) or out of the eye (i.e., Hartmann-Shack [19
]). The wave aberration of the eye is then reconstructed from a discrete number of sampling points. This reconstruction can be local [20
], modal [21
], or a mixture of both. The most widely used method in ocular aberrometry is a modal reconstruction that is based on the expansion of the derivatives of wave aberration as a linear combination of a set of basis functions (most frequently a Zernike polynomial expansion) and a subsequent least-squares fit of the expansion coefficients to the measured gradients [22
The actual sampling pattern and density differ between aberrometers. The lenslets in a Hartmann-Shack (HS) wavefront sensor are typically arranged in either a fixed rectangular or a hexagonal configuration, and the number of samples range from around 50 to more than 15,000 (for instance, the aberrometer Haso3 128, by Imagine Eyes, Orsay, France) spots within the dilated pupil. Ray-tracing aberrometers (such as LRT or SRR), on the other hand, sample the pupil sequentially and can use a variable sampling configuration. However, given the sequential nature of these devices, high sampling densities are not typically used to reduce measuring times.
The optimal number of sampling points represents a trade-off. There has been a tendency to increase the number of lenslets of the HS sensor (i.e., increasing the sampling density) with the aim of improving resolution and the accuracy of the wavefront reconstruction. However, smaller lenslet diameters decrease the amount of light captured by each lenslet and increase the size of the diffraction-limited spots. Although it is possible to optimize the size of the CCD array and the focal length of the lenslets to gain accuracy (pixels per spot), an excessive number of spots can compromise the dynamic range of the device, as well as increase the processing time and potentially decrease the reproducibility, due to the lower signal intensity. In addition, increasing the number of samples may not decrease the variance of the estimates of the wavefront [21
] nor the aliasing error [23
The determination of a sampling pattern with the minimum sampling density that provides accurate results is of practical importance for sequential aberrometers, since it would decrease measurement time, and of general interest to better understand the trade-offs between aberrometers. It is also useful to determine whether there are sampling patterns that are better adapted to typical ocular aberrations, or particular sampling patterns optimized for measurement under specific conditions.
To our knowledge, there has not been a systematic experimental study investigating whether increasing the sampling density over a certain number of samples provides significantly better accuracy in ocular aberration measurements, or whether alternative sampling configurations would be more efficient. There have been theoretical investigations of sampling configuration, although the applicability to human eyes should be ultimately tested experimentally.
The first studies on wavefront estimates date from the 1970s. Cubalchini [21
] was the first to study the modal estimation of the wave aberration from derivative measurements using a least-squares method. He concluded that modal estimates of the wavefront obtained using this method were sensitive to the number of samples and their geometry. He advised minimizing the number of samples used to estimate a fixed number of terms and taking the measurements as far from the center of the aperture as possible in order to minimize the variance of higher-order Zernike terms.
In 1997, Rios et al
] found analytically for HS sensing that the spatial distribution of the nodes of the Albrecht cubatures [24
] made them excellent candidates for modal wavefront reconstruction in optical systems with a centrally obscured pupil. This sampling scheme could also be a good candidate for ocular aberrations, due to the circular geometry of the cubature scheme. In addition, as the Zernike order increases (i.e., higher-order aberrations), the area of the pupil more affected by aberrations tends to be more peripheral [21
], and therefore ocular wavefront estimates would potentially benefit from a denser sampling of the peripheral pupil.
He et al
] used numerical simulations to test the robustness of the fitting technique they used for their SRR (least-square fit to Zernike coefficients) to the interaction between orders as well as the error due to the finite sampling aperture. They found that the error could be minimized by extracting the coefficients corresponding to the maximum complete order possible (considering the number of samples) and by using a relatively large sampling aperture, so that the whole pattern practically covered the measured extent of the pupil. Although this large sampling aperture introduced some error due to the use of the value of the derivatives at the center of the sampling apertures to perform the fitting, and their rectangular pattern did not provide an adequate sampling for radial basis functions, their simulation confirmed that the overall effect was relatively small.
In 2003, Burns et al
] studied computationally the effect of different sampling patterns on measurements of wavefront aberrations of the eye by implementing a complete model of the wavefront processing used with a “typical” HS sensor and modal reconstruction. They also analyzed the effect of using a point estimator for the derivative at the center of the aperture, versus using the average slope across the subaperture, and found that the latter decreased modal aliasing somewhat but made little practical difference for the eye models. Given that the higher-order aberrations tended to be small, their modal aliasing (leakage of a high order into a lower order) was subsequently small. Finally, they found that nonregular sampling schemes, such as cubatures, were more efficient than grid sampling when sampling noise was high. One year later [28
], we compared the aberrations obtained using different patterns to measure experimentally the same eyes, and we applied the previous computational model to test some additional patterns. We concluded that patterns with a very small number of samples failed at reproducing the wave aberration, but for human eyes, the differences across the rest of the patterns were of the order of the measurement error. Spatial distribution of the samples was found to be more relevant than the density.
Recently, Díaz-Santana et al
] and Soloviev et al
] developed analytical models to test different sampling patterns applied to ocular aberrometry and HS sensing in astronomy, respectively. Díaz-Santana et al
] developed an evaluation model based on matrices that included as input parameters the number of samples and their distribution (square, hexagonal, or polar lattice), the shape of the subpupil, and the size and irradiance across the pupil (uniform irradiance versus Gaussian apodization) regarding the sampling. The other input parameters were the statistics of the aberrations in the population, the sensor noise, and the estimator used to retrieve the aberrations from the aberrometer raw data. The model of Soloviev et al
] used a linear operator to describe the HS sensing, including the effects of the lenslets array geometry and the demodulation algorithm (modal wavefront reconstruction). When applying this to different sampling configurations, using the Kolmogorov statistics as a model of the incoming wavefront, they found that their pattern with 61 randomly spatially distributed samples gave better results than the regular hexagonal pattern with 91 samples of the same subaperture size (radius=1/11 times the exit pupil diameter), which completely covered the extent of the pupil in the case of the 91-sample pattern. In these theoretical models, an appropriate statistical input is crucial so that their predictions can be generalized in the population. It has been recently found [25
] that high-order aberration terms show particular relationships (i.e., positive interactions that increase the modulation transfer function over other potential combinations), suggesting that general statistical models should include these relationships in order to describe real aberrations.
In this study, we used a configurable wavefront sensor, LRT, to measure wave aberrations in human eyes, using different sampling patterns and densities. Hexagonal and rectangular configurations were chosen because they are the most commonly used. We also used different radially symmetric geometries to test whether these patterns were better suited for measuring ocular aberrations. These geometries included uniform polar sampling, arranged in a circular pattern, and three patterns corresponding to the zeros of the cubatures of the Albrecht, Jacobi, and Legendre equations. We also tested different densities for each pattern in order to evaluate the trade-off between accuracy and sampling density. To separate variability due to biological factors from instrumental issues arising from measurement and processing, we also made measurements on artificial eyes. Finally, we used noise estimates in human eyes as well as realistic wave aberrations in computer simulations to extend the conclusions to eyes other than normal eyes (referred to in this paper as healthy eyes with no pathological condition and that have not undergone any ocular surgery).