Example wavefront from the set of 1000 randomly generated wavefronts. This wavefront was generated with the maximum amplitude considered here. Large circles indicate spots that were correctly located according to the internal error checking of each algorithm. We can see that in this specific example, the spiral algorithm was able to correctly sort the spots; the other algorithms show varying amounts of error. 

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... The range for the B-spline algorithm increased by 22 to 42%, depending on the aberration type, confirming that for the isolated aberrations considered here this algorithm surpassed the practical limit set by the overlap- ping of spots. We can therefore confidently state that any future algorithm would not provide a greater range. For reference, the Zernike algorithm showed minimal improvement in range under the same conditions, and the spiral algorithm showed an improvement only for defocus. This means that as far as a pure sorting task is concerned the B-spline algorithm is superior to the other two algorithms, but that this advantage is diminished under practical conditions due to spot overlap. It is also important to consider the range results in the con- text of the maximum levels of each type of aberration that the algorithm must be able to measure. For ocular aberrometry it is desirable to be able to measure as many eyes in the population as possible, both on- and off-axis. For astigmatism, the ranges given here are equivalent to ∼ 30 D for the improved algorithms (the conversion factor between Zernike coefficient and ophthalmic prescription is ∼ 0.77 for defocus and ∼ 1.09 for astigmatism at a pupil size of 6.0 mm). This is far in ex- cess of any case of astigmatism ever reported to our knowledge, even in eyes with severe corneal distortion 17 or far away from the optical axis. 18 The defocus results for the improved algorithms correspond to ∼ 13.3 to 16.6 D of refractive error. Even allowing for a 2-D variation over the central 40 deg of the visual field, 18 this is still sufficient to account for > 99.9% of the normal human eye population. Since we have already shown that the improved algorithms operate at or close to the hard limit set by the physical overlap of spots, a larger range would require a lenslet array with a shorter focal length (or some other hardware alteration as discussed in Sec. 1). The range for defocus for each algorithm can be adjusted as a linear function of the focal length of the lenslet array used. For example, if a lenslet array with half the focal length were used instead, a roughly equivalent spot pattern would result only when twice as much defocus is present. This is because the wavefront slope, and therefore the spot displacement, for pure defocus varies linearly in x and y across the pupil. Spherical aberration is very rarely greater than ∼ 0.4 μ m over a 6.0-mm pupil in the normal population, 19 and changes little off-axis, 18 although it may reach ∼ 1.0 μ m in eyes suffering from corneal pathology. 17 This is well within range of the 2.4 to 2.9 μ m shown for the improved algorithms here. Coma is very rarely greater than ∼ 0.5 μ m in the normal population 19 and can change by as much as ∼ 0.6 μ m over the central 40 deg of the visual field, 18 which is similarly well within the improved ranges seen here of ∼ 4.6 to 4.8 μ m. However, keratoconic eyes have been reported to feature up to ∼ 3.6 μ m of vertical coma, 17 which may give any of the algorithms considered here difficulty when combined with other aberration types, due to potential spot overlap. If it were desirable to accommodate a keratoconic population of eyes, a lenslet array with a shorter focal length may be desirable. The preceding results quantify the performance of each algorithm for particular aberration types in isolation. We next used a real adaptive optics system with a model eye to simulate a worse-case scenario—randomized, irregular wavefronts com- posed of many different types of aberration. The amplitude of the random wavefronts was varied systematically by altering the allowed amplitude of the commands applied to the deformable mirror, with 100 different wavefronts generated for each of the 10 amplitude conditions. Figure 3 shows an example Hartmann-Shack image that was generated using the maximum amplitude, and illustrates the attempts of each algorithm to an- alyze it. Closed symbols denote spot positions located by the image processing routine. Circles indicate spots that a given algorithm judged were unambiguously assigned to a lenslet. In this particular example, the spiral algorithm did not make any errors, the B-spline made a small number of errors, and the conventional and Zernike algorithms made a large number of errors. Figure 4 plots the average rate of failure for each algorithm as a function of the maximum allowed actuator amplitude. The failure criterion was met if 5 or more spots (of the ∼ 280 in the pupil) were missed. The error bars indicate standard error of the mean. The spiral and B-spline algorithms achieved similar results over much of the amplitude range tested; the spiral algorithm was superior at the highest end of the scale. This is probably because the spiral algorithm is far more local in nature and so is somewhat better suited to handle large wavefront deviations that are not well correlated with deviations in other parts of the pupil (the wavefront measured in Fig. 3 is a good example of this). There is a large difference evident between the Zernike algorithm and the spiral and B-spline algorithms. This is likely because the randomly generated mirror shapes would not of- ten have been well represented by the Zernike expansion. Note that this algorithm still performed significantly better than the conventional algorithm, and that the high-amplitude, irregular aberrations produced here are unlikely to be seen in eyes without significant disease affecting the ocular media. Finally, we report on the processing time for each algorithm. Absolute processing times, of course, depend highly on the computer hardware used. However, even the relative processing times given here should be taken as a rough approximation, since the MATLAB code for each algorithm has not necessarily been optimized for speed. With that in mind, with 10 trials for the well-centered case on an Intel Core i7 2.66 GHz processor with 2 GB RAM and using a 17 × 17 lenslet array (i.e., detector size of 6.8 mm), we obtained processing times of 2.3, 4.2, 28, and 41 ms for the conventional, spiral, Zernike, and B-spline algorithms, respectively. This excluded components of each algorithm that involved the initialization of variables, which would optimally be performed only once in a session. The spiral algorithm was faster than the other improved algorithms, as predicted due to the maximally simplistic calculations involved in predicting each spot location. However, the magnitude of temporal variations in ocular aberrations for a given retinal point are generally small and well within the range of the conventional algorithm. A fast frame rate could therefore be achieved by employing a single iteration of any improved algorithm and feeding these results into the conventional algorithm to guide its search. This approach offers a clear benefit to frame rate because of the unique ability of the conventional algorithm to take advantage of parallel processing and so achieve even greater speed than reported here. The range of aberrometry was similar for the improved algorithms considered here, despite the differences in processing complexity. These algorithms approached the physical limits to the achievable range imposed by the overlap of Hartmann-Shack spots. The results presented here show that each algorithm is more than sufficient to accurately characterize the vast majority of human eye aberration, both on- and off-axis, with our chosen lenslet array parameters. The B-spline algorithm produced a high range and was highly robust to error, but was an order of magnitude slower than simpler algorithms. The Zernike algorithm was somewhat faster, but care must be taken to accurately estimate decentration of the pupil and of the input beam, and performance is noticeably reduced for wavefronts that are not well approximated by the Zernike expansion. The spiral algorithm was highly robust to error and to irregular aberration types. It was also 7 to 10 times faster than the other improved algorithms, affording the potential for real-time analysis. Real-time analysis for ocular aberrometry could alternatively be achieved by employing only a single initial iteration of any improved algorithm to inform subsequent iterations of the conventional algorithm, since the vast majority of human eye aberration is static. This work was supported by the Australian Research Council (Discovery Project Grant DP0984649), and the Rowden White Benevolent ...

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... errors for poorly sensed spots. This is by virtue of the large number of spots considered when making each prediction for the B-spline and Zernike algorithms, and by the use of the “dummy” grid for the spiral algorithm. In fact, the B-spline and spiral algorithms were also highly robust in the presence of the other error types considered; range decrease was no more than 6 to 10% across all of the error conditions for these two algorithms. The Zernike algorithm was not robust against decentrations in the estimated position of either the pupil or the input beam, making it advisable when using this method to use a calibrated scale when decentering the input beam, 10 and possibly a separate pupil monitor if maximal range is desired. It is of interest to know whether the range limits shown here could be improved on with some other as yet undiscov- ered sorting algorithm, or whether the obtained limits were a result of physical spot overlap (i.e., limited by image processing difficulties rather than the sorting algorithm). To test this, we repeated the range measurements by using the predicted spot locations from geometric optics rather than relying on the MATLAB centroiding functions. The range for the B-spline algorithm increased by 22 to 42%, depending on the aberration type, confirming that for the isolated aberrations considered here this algorithm surpassed the practical limit set by the overlap- ping of spots. We can therefore confidently state that any future algorithm would not provide a greater range. For reference, the Zernike algorithm showed minimal improvement in range under the same conditions, and the spiral algorithm showed an improvement only for defocus. This means that as far as a pure sorting task is concerned the B-spline algorithm is superior to the other two algorithms, but that this advantage is diminished under practical conditions due to spot overlap. It is also important to consider the range results in the con- text of the maximum levels of each type of aberration that the algorithm must be able to measure. For ocular aberrometry it is desirable to be able to measure as many eyes in the population as possible, both on- and off-axis. For astigmatism, the ranges given here are equivalent to ∼ 30 D for the improved algorithms (the conversion factor between Zernike coefficient and ophthalmic prescription is ∼ 0.77 for defocus and ∼ 1.09 for astigmatism at a pupil size of 6.0 mm). This is far in ex- cess of any case of astigmatism ever reported to our knowledge, even in eyes with severe corneal distortion 17 or far away from the optical axis. 18 The defocus results for the improved algorithms correspond to ∼ 13.3 to 16.6 D of refractive error. Even allowing for a 2-D variation over the central 40 deg of the visual field, 18 this is still sufficient to account for > 99.9% of the normal human eye population. Since we have already shown that the improved algorithms operate at or close to the hard limit set by the physical overlap of spots, a larger range would require a lenslet array with a shorter focal length (or some other hardware alteration as discussed in Sec. 1). The range for defocus for each algorithm can be adjusted as a linear function of the focal length of the lenslet array used. For example, if a lenslet array with half the focal length were used instead, a roughly equivalent spot pattern would result only when twice as much defocus is present. This is because the wavefront slope, and therefore the spot displacement, for pure defocus varies linearly in x and y across the pupil. Spherical aberration is very rarely greater than ∼ 0.4 μ m over a 6.0-mm pupil in the normal population, 19 and changes little off-axis, 18 although it may reach ∼ 1.0 μ m in eyes suffering from corneal pathology. 17 This is well within range of the 2.4 to 2.9 μ m shown for the improved algorithms here. Coma is very rarely greater than ∼ 0.5 μ m in the normal population 19 and can change by as much as ∼ 0.6 μ m over the central 40 deg of the visual field, 18 which is similarly well within the improved ranges seen here of ∼ 4.6 to 4.8 μ m. However, keratoconic eyes have been reported to feature up to ∼ 3.6 μ m of vertical coma, 17 which may give any of the algorithms considered here difficulty when combined with other aberration types, due to potential spot overlap. If it were desirable to accommodate a keratoconic population of eyes, a lenslet array with a shorter focal length may be desirable. The preceding results quantify the performance of each algorithm for particular aberration types in isolation. We next used a real adaptive optics system with a model eye to simulate a worse-case scenario—randomized, irregular wavefronts com- posed of many different types of aberration. The amplitude of the random wavefronts was varied systematically by altering the allowed amplitude of the commands applied to the deformable mirror, with 100 different wavefronts generated for each of the 10 amplitude conditions. Figure 3 shows an example Hartmann-Shack image that was generated using the maximum amplitude, and illustrates the attempts of each algorithm to an- alyze it. Closed symbols denote spot positions located by the image processing routine. Circles indicate spots that a given algorithm judged were unambiguously assigned to a lenslet. In this particular example, the spiral algorithm did not make any errors, the B-spline made a small number of errors, and the conventional and Zernike algorithms made a large number of errors. Figure 4 plots the average rate of failure for each algorithm as a function of the maximum allowed actuator amplitude. The failure criterion was met if 5 or more spots (of the ∼ 280 in the pupil) were missed. The error bars indicate standard error of the mean. The spiral and B-spline algorithms achieved similar results over much of the amplitude range tested; the spiral algorithm was superior at the highest end of the scale. This is probably because the spiral algorithm is far more local in nature and so is somewhat better suited to handle large wavefront deviations that are not well correlated with deviations in other parts of the pupil (the wavefront measured in Fig. 3 is a good example of this). There is a large difference evident between the Zernike algorithm and the spiral and B-spline algorithms. This is likely because the randomly generated mirror shapes would not of- ten have been well represented by the Zernike expansion. Note that this algorithm still performed significantly better than the conventional algorithm, and that the high-amplitude, irregular aberrations produced here are unlikely to be seen in eyes without significant disease affecting the ocular media. Finally, we report on the processing time for each algorithm. Absolute processing times, of course, depend highly on the computer hardware used. However, even the relative processing times given here should be taken as a rough approximation, since the MATLAB code for each algorithm has not necessarily been optimized for speed. With that in mind, with 10 trials for the well-centered case on an Intel Core i7 2.66 GHz processor with 2 GB RAM and using a 17 × 17 lenslet array (i.e., detector size of 6.8 mm), we obtained processing times of 2.3, 4.2, 28, and 41 ms for the conventional, spiral, Zernike, and B-spline algorithms, respectively. This excluded components of each algorithm that involved the initialization of variables, which would optimally be performed only once in a session. The spiral algorithm was faster than the other improved algorithms, as predicted due to the maximally simplistic calculations involved in predicting each spot location. However, the magnitude of temporal variations in ocular aberrations for a given retinal point are generally small and well within the range of the conventional algorithm. A fast frame rate could therefore be achieved by employing a single iteration of any improved algorithm and feeding these results into the conventional algorithm to guide its search. This approach offers a clear benefit to frame rate because of the unique ability of the conventional algorithm to take advantage of parallel processing and so achieve even greater speed than reported here. The range of aberrometry was similar for the improved algorithms considered here, despite the differences in processing complexity. These algorithms approached the physical limits to the achievable range imposed by the overlap of Hartmann-Shack spots. The results presented here show that each algorithm is more than sufficient to accurately characterize the vast majority of human eye aberration, both on- and off-axis, with our chosen lenslet array parameters. The B-spline algorithm produced a high range and was highly robust to error, but was an order of magnitude slower than simpler algorithms. The Zernike algorithm was somewhat faster, but care must be taken to accurately estimate decentration of the pupil and of the input beam, and performance is noticeably reduced for wavefronts that are not well approximated by the Zernike expansion. The spiral algorithm was highly robust to error and to irregular aberration types. It was also 7 to 10 times faster than the other improved algorithms, affording the potential for real-time analysis. Real-time analysis for ocular aberrometry could alternatively be achieved by employing only a single initial iteration of any improved algorithm to inform subsequent iterations of the conventional algorithm, since the vast majority of human eye aberration is static. This work was supported by the Australian Research Council (Discovery Project Grant DP0984649), and the Rowden White Benevolent ...