Effects of defocus and pupil size on human contrast sensitivity

Transcription

1 PII: S (99) Ophthal. Physiol. Opt. Vol. 19, No. 5, pp. 415±426, 1999 # 1999 The College of Optometrists. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain /99 $ Effects of defocus and pupil size on human contrast sensitivity Niall C. Strang, a David A. Atchison a and Russell L. Woods b a Centre for Eye Research, School of Optometry, Queensland University of Technology, Victoria Park Road, Kelvin Grove, QLD 4059, Australia and b Department of Vision Sciences, Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK Summary Defocus lowers the contrast sensitivity function (CSF), producing a complex function with local dips and peaks. Previously, we were able to predict the shape of the CSF with large pupils from measured transverse aberrations with hypermetropic defocus but not with myopic defocus (Atchison et al., 1998c, J. Opt. Soc. Am. A. 15, 2536). As there is no reason that myopic defocus should be more difficult to predict than hypermetropic defocus, we modified the procedure to try to improve CSF predictions with myopic defocus. Also, we extended the study to consider a range of pupil sizes. CSFs were measured for three subjects at three defocus levels (infocus, 2D and +2D) and three pupil sizes (2 mm, 4 mm and 6 mm). Using a diffraction optics model, transverse aberration measures and in-focus CSF measures, we predicted the defocused CSFs. The predicted defocused CSFs were lower than the in-focus CSF as expected, and had complex shapes that varied with defocus and pupil size and between subjects. While a few predictions were poor, generally, the overall magnitude and shape of the defocused CSFs were well predicted and similarly so for myopic and hypermetropic defocus. Some further improvements in technique are indicated. # 1999 The College of Optometrists. Published by Elsevier Science Ltd. All rights reserved, and Introduction Since the seminal work of Campbell and Green (Campbell and Green, 1965; Green and Campbell, 1965) other authors have examined the e ect of defocus on the contrast sensitivity function (e.g. Charman, 1979; Kay and Morrison, 1987). However, only recently have detailed measurements of the defocused contrast sensitivity function (CSF) revealed that, in addition to the well recognised reduction in contrast sensitivity, defocus dramatically alters the shape of the CSF measured with large pupils (Woods et al., 1996a; Bour and Apkarian, 1996). While the in-focus CSF is a monotonically decreasing function for spatial frequencies higher than peak contrast sensitivity, commonly the defocused CSF contains oscillations between the peak and the cut-o spatial frequency. As Correspondence and reprints requests to: Dr Niall C. Strang, Department of Optometry, University of Bradford, Richmond Road, Bradford, BD7 1DP, West Yorkshire, UK Received: 8 September 1998 Revised form: 29 January 1999 shown by Atchison et al. (1998c), the shape of the defocused CSF can be predicted from measured ocular transverse aberration functions using a di ractionbased optical model. However, this previous study provided accurate CSF predictions only for hypermetropic (negative) defocus, and was less reliable for myopic (positive) defocus. As there is no theoretical reason why the quality of prediction should vary with the direction of defocus, and despite the quality of most of those predictions, we had reason to doubt our procedures. In most techniques of determining aberration, such as the subjective vernier alignment technique we have used (Woods et al., 1996b), the aberrations are measured in object space. The path of any ray in the eye between the fovea and a point in the pupil must be the same, whatever the level of induced defocus. Hence we should expect that the terms of the transverse aberration function apart from the defocus term (eg spherical aberration, coma) would not vary with induced defocus. As these terms of the measured transverse 415

2 416 Ophthal. Physiol. Opt : No 5 aberration functions did vary with induced defocus (Atchison et al., 1998c), we suspected that there were limitations in our subjective vernier technique. For example, the ability of the subject to localise the image may be reduced in the presence of defocus (this task was particularly di cult in the peripheral pupil where the image was often very distorted). Consequently it should be possible to predict the defocused aberration function from the in-focus aberration function by the addition of a defocus term. Further, it should be possible to predict the e ect of defocus on the CSF using an aberration function derived from the in-focus aberration function with the addition of a defocus term. As the CSF with myopic defocus was predicted more successfully using this approach (Atchison et al., 1998c), the poor predictions of the e ects of myopic defocus may have been due to methodological problems. Various methodological problems can a ect the accuracy of the predictions, including problems with the subjective transverse aberration measurement technique. To investigate whether the poor CSF predictions with myopic defocus were due to such problems, we introduced a series of improvements in the experimental design over that in the previous reports. The quality of the predicted CSF was compared with equal levels of hypermetropic and myopic defocus with large pupils. The quality of the predicted CSF was compared using aberration functions measured with defocus and using aberration functions derived from infocus aberrational data with a defocus term. Furthermore, we extended our study to evaluate the robustness of the predictive model by measuring CSFs across a range of pupil sizes. Overall the complex shape of the defocused CSF was predicted well, though in a few cases predictions were poor. Generally, the shape of the defocused CSF was predicted as well for myopic as for hypermetropic defocus over the range of pupil sizes. Aberration functions measured with defocus and aberration functions using in-focus aberrational data with a defocus term predicted the defocused CSF equally well. The CSF measured with small pupils was not necessarily better than that with larger pupils, there being complex interactions between individual aberrations, defocus and pupil size. Methods Subjects Three subjects, aged between 29 and 42 years and all experienced in psychophysical experiments, participated in the study. Only right eyes were tested and prior to experiments a full cycloplegic subjective refraction (arti cial pupil size 4.5 mm) was conducted at the 4 m test distance. One drop of 1% cyclopentolate HCl was instilled every 2 h during the experimental trials. Refractive errors ranged from D to 2.00 D. Two subjects (RLW and DAA) participated in the previous study (Woods et al., 1996a; Atchison et al., 1998c). To minimise head movements, subjects were restrained with a bite bar during experimental procedures. In experiments, measurements were made for defocus levels, relative to the corrected state, of 0 D (in-focus), 2 D (hypermetropia) and +2 D (myopia), provided by placing appropriate trial lenses as close as possible to subjects' eyes. Transverse aberrations and predicted CSFs Transverse aberrations were measured along the horizontal meridian using the subjective vernier alignment technique described fully by Woods et al. (1996b). A polarised laser spot is seen projected into the gap between a pair of vertical lines displayed on a computer monitor. The lines are visible throughout a cross-polarizing lter placed close to the eye, but the spot is visible only through a round aperture in the lter. The lter is translated horizontally across the pupil in steps. At each step, the subject adjusts the location of the vertical lines until they are aligned subjectively with the spot. The visual axis (foveal achromatic axis) is used to de ne the centre of the pupil (Thibos et al., 1990). The following changes in technique were evaluated or included:. a smaller pupil-location selective aperture;. compensation for small eye rotation in the measurement;. matching the measurement wavelength to that used in CSF measurements;. use of in-focus aberrations with a defocus term to predict the aberration function with defocus;. alignment of the contrast sensitivity target with the visual axis. Aperture size. The subject viewed a pair of vertical lines presented on a monitor through the whole pupil while also viewing a spot through a small aperture. Measurements were made with both a 0.75 mm diameter (as in previous study) aperture and a 0.5 mm diameter aperture to sample the subject's pupil along the horizontal meridian. As the smaller aperture sampled from a smaller region of the pupil, the aberrations across the aperture which distort the image with defocus and in the periphery were expected to be reduced,

3 Effects of defocus and pupil size on human contrast sensitivity: N. C. Strang et al. 417 thereby a ording a more accurate measurement of the transverse aberrations. Ray height issue. When doing vernier alignments, the subjects had to rotate their eyes to align the two green lines seen through the whole pupil with the target (spot) as seen through the small aperture. This altered the small aperture position relative to the achromatic axis. This eye rotation was ignored previously (Woods et al., 1996a,b; Atchison et al., 1998c), but an allowance was made for it in this investigation. We corrected the height of a ray at the stop according to Equation (A14) in the Appendix. To use this equation, we measured the distance between the stop and the corneal vertex plane carefully. Achromatic axis errors in presence of defocus. The foveal achromatic axis is a ected by the level of defocus. This was fully explained by Atchison et al. (1998b), who found that the e ect when measured at the cornea is approximately mm per dioptre of defocus, which is small enough to be ignored in this investigation. Chromatic aberration correction. Previously, we had to make corrections for chromatic aberration. In the transverse aberration measurements, the two lines had a central moment of 605 nm and the spot was provided by a laser of wavelength 633 nm. A small correction was required to measure aberrations at 605 nm (Woods et al., 1996b). Furthermore, as the CSF measurements were taken at 550 nm, a 0.25 D correction was employed with this to equalise the defocus for aberration and CSF measurements (Woods et al., 1996a). The need for these corrections was eliminated in this study by the use of the green gun on the monitor (central moment 544 nm) and a green He±Ne laser for the spot (543 nm). Conversion of transverse aberrations to CSF predictions. We used the method of converting ocular transverse aberration data to CSF predictions described by Atchison et al. (1998c) as follows: (1) measure the in-focus and defocused ocular transverse aberration of the eye, (2) t a polynomial to the transverse aberrations, (3) convert transverse aberration to a wave aberration of the form W x; y ˆA 1 x 2 y 2 A 2 x 2 y 2 x 2 2x 3 A 3 x 2 y 2 A4 x 2 y 2 A5 x 2 y 2 1 where A 1 ±A 5 are coe cients and (x, y) are relative pupil co-ordinates (given in Table 1 for 6 mm diameter pupils and 2 D, 0 D and +2 D defocus), (4) determine the MTF by the autocorrelation of the pupil function, incorporating average Stiles- Crawford apodisation (Applegate and Lakshminarayanan, 1993), and (5) predict the defocused CSF by multiplying the infocus CSF by the ratio of defocused MTF to infocus MTF. In-focus transverse aberrations plus defocus term. In the previous study (Atchison et al., 1998c) we found that the predictions in one subject with myopic defocus could be improved by using the in-focus aberrations with a defocus term. We tried this approach also in this study. The additional defocus term was waves ( 2 D defocus) and waves (+2 D defocus) for 6 mm pupils. The additional defocus term varied proportionally with the square of the pupil size. CSF Measurement To improve the consistency of alignment with the aberration technique, we centred arti cial pupils on the foveal achromatic axis. Apart from this, procedures were unchanged from those employed by Woods et al. (1996a). Adaptative probit estimation (Watt and Andrews, 1981) was used to determine the 50% point on the psychometric function. Previously, this method has been shown to be an e cient method Table 1. Individual wave aberration co-efficients (wavelengths at the edge of the pupil) for the three focus levels ( 2 D, 0 D, +2 D) and 6 mm pupil sizes. Each coefficient has been scaled appropriately for the other pupil sizes e.g. for DAA at 2 D and 2 mm pupil size, A 3 is waves 6 mm Pupil size 2 D defocus In-focus +2 D defocus Subject RLW A Ð A Ð A A 4 Ð Ð 0.73 A 5 Ð 0.39 Ð Subject DAA A Ð A 2 Ð A A 4 Ð A 5 Ð 0.5 Ð Subject NCS A A 2 Ð A 3 Ð 0.13 Ð A 4 Ð Ð 1.16 A

4 418 Ophthal. Physiol. Opt : No 5 of contrast sensitivity measurement which is virtually free of problems associated with changes in the subjects' decision criteria (Woods and Thomson, 1993; Woods, 1996). Monochromatic CSFs were measured using a custom-built, monitor-based system with a circular eld subtending a visual angle of 2.5 degrees. All contrast and luminance calibrations were performed using a Topcon BM-7 luminance colorimeter. The central moment of the grating (P4 phosphor viewed through an Edmund Scienti c interference lter) was 550 nm (full width at half height = 10 nm) and the mean luminance was 2.8 cd/m 2 with the lter in place. Standard deviations of the contrast sensitivity measures were in the order of 20.1 log unit. Pupil sizes The study was extended by measuring CSFs with 2 mm and 4 mm arti cial pupil diameters, as well as 6 mm pupil diameters. Analysis of ts To assess the accuracy of our predictions we used the root-mean-square error (RMSE) method employed by Atchison et al. (1998c). The RMSE was calculated using the equation: qx 2 RMSE ˆ C m C p = n 1 2 where C m and C p are the measured and predicted log 10 (contrast sensitivity) at each spatial frequency, respectively, and n is the number of tested spatial frequencies. When C p < 0 it was given the value 0 as the C m values could never be less than 0 (i.e. 100% contrast). The number of test points in each condition ranged from 16 to 35. The repeatability of CSF measurement limits the ability to predict the CSF. Atchison et al. (1998c) reported that the RMSEs of two repeated CSF measurements with 2 D defocus of two subjects were 0.14 and 0.15 log units, values similar to reported repeatability with optical degradation (Woods, 1993). Results Aberrations Table 1 illustrates the di erences in the coma (A 2 and A 4 ) and spherical aberration (A 3 and A 5 ) wave aberration co-e cients that existed between subjects. Subjects DAA and RLW had third-order spherical aberrations which were similar to results taken two years earlier (compare Table 1 with Table 1 in Atchison et al., 1998c) and large compared with subject NCS. NCS exhibited a small level of myopic defocus for the in-focus condition (A 1 = waves, D) which suggests that he was not corrected fully. CSF with 6 mm pupil The measured and predicted CSFs for both myopic and hypermetropic defocus conditions are shown in Figure 1 for the three subjects. These show similar characteristics to those found previously (Woods et al., 1996a; Atchison et al., 1998c), with large oscillations of the CSF (multiple `notches') and large inter-subject di erences. The shape of the predicted CSF found in subject RLW when measured with hypermetropic defocus di ered markedly from subjects' DAA and NCS because of the greater amount of coma found in this subject. As shown in Table 2, the agreement (RMSE) between the predicted CSF (thin line) and the measured CSF (circles) varied between 0.12 log units (Figure 1a) and 0.44 log units (Figure 1f). On average the RMSE was greater than the limits of prediction suggested from repeated measurement (i.e log units). Some of the ts were extremely good with most of the complex nature of the measured CSF represented in the predicted CSF (e.g. Figure 1aand1c). Other predicted CSFs failed to represent the overall level of the measured contrast sensitivity while showing similar oscillations (e.g. Figure 1d), while other predicted CSFs represented the overall reduction in the measured contrast sensitivity well but predicted a di erent pattern of oscillations (e.g. Figure 1e). The ability to predict the CSF was equally good with hypermetropic and myopic defocus. Comparison of our results with the previous study (Atchison et al., 1998c) revealed marked improvement in the accuracy of the hypermetropic defocus CSF predictions for subject RLW (compare Figure 1a (RMSE = 0.12) with Figure 4b (RMSE = 0.57) from previous study). Subject DAA showed similar predictions to that found previously (compare Figure 1c (RMSE = 0.34) with Figure 4c from the previous study (RMSE = 0.26)). Prediction of the CSF with myopic defocus, which had been a problem in that previous study, was good for both subject RLW (Figure 1b, RMSE = 0.26) and subject DAA (Figure 1d, RMSE = 0.28). This prediction for subject DAA was markedly better than in the previous study (RMSE = 0.59 log units). Though several predicted notches were not found in both the hypermetropic and myopic measured CSFs of subject NCS, the accuracy of the predictions was better in hypermetropic (Figure 1e; RMSE = 0.31) than myopic (Figure 1f; RMSE = 0.44) defocus conditions. Predictions made by using in-focus aberrations with the defocus term added were no better (RMSE values

5 Effects of defocus and pupil size on human contrast sensitivity: N. C. Strang et al. 419 Figure 1. CSFs of three subjects with a 6 mm pupil: measured in-focus, measured defocus and predicted defocus. (a) subject RLW 2 D (hypermetropic) defocus; (b) subject RLW +2 D (myopic) defocus; (c) subject DAA 2 D (hypermetropic) defocus; (d) subject DAA +2 D (myopic) defocus; (e) subject NCS 2 D (hypermetropic) defocus; (f) subject NCS +2 D (myopic) defocus. Predictions were based on the measurements of the defocused aberrations.

6 420 Ophthal. Physiol. Opt : No 5 Table 2. RMSE values for the three subjects for 2 D and +2 D focus and for 2 mm, 4 mm and 6 mm pupils. The values without parentheses are derived from the aberration data in Table 1. Each value in the parentheses was obtained from the aberration data for the in-focus condition, combined with an additional defocus term as described in the text Subject Defocus RMSE values (6 mm pupil) RMSE values (4 mm pupil) RMSE values (2 mm pupil) RLW 2 D 0.12 (0.36) 0.18 (0.22) 0.16 (0.27) +2 D 0.26 (0.27) 0.32 (0.30) 0.34 (0.34) DAA 2 D 0.34 (0.24) 0.27 (0.19) 0.34 (0.29) +2 D 0.28 (0.26) 0.25 (0.27) 0.24 (0.33) NCS 2 D 0.31 (0.29) 0.36 (0.36) 0.30 (0.25) +2 D 0.44 (0.29) 0.27 (0.23) 0.22 (0.20) Figure 2. CSFs of two subjects with a 6 mm pupil: measured in-focus, measured defocus and predicted defocus. (a) subject DAA 2 D (hypermetropic) defocus; (b) subject NCS +2 D (myopic) defocus; (c) subject NCS 2 D (hypermetropic) defocus. The measurements are reproduced from Figure 1. Predictions were based on the aberration data for the in-focus condition combined with an additional defocus term as described in the text.

7 Effects of defocus and pupil size on human contrast sensitivity: N. C. Strang et al. 421 shown in brackets in Table 2) than the predictions made using the transverse aberration function measured with defocus which are reported above. Again (Figure 2), the accuracy of predictions varied between subjects, with the RMSE ranging from 0.24 to 0.36 log units. In two cases the shapes of the predicted CSFs di ered qualitatively from those reported above (i.e. made using the transverse aberration function measured with defocus). For subject DAA with hypermetropic defocus (compare Figure 2a with Figure 1c) and subject NCS with myopic defocus (compare Figure 2b with Figure 1f), the shape of the CSF prediction made using in-focus aberrations and the defocus term was an improvement, but in all other cases this alternative analysis had only minor e ects (e.g. compare Figure 2c with Figure 1e). There was little di erence in the transverse aberration functions measured using the smaller (0.5 mm) aperture to sample the subject's pupil compared to that measured with the original (0.75 mm aperture) despite the reduction in the distortion of the image of the spot seen through the aperture. Not surprisingly the aberration functions were similar, resulting in a similar level of CSF prediction accuracy (e.g. subject DAA: hypermetropic defocus RMSE = 0.24 (0.5 mm) vs 0.34 (0.75 mm); myopic defocus RMSE = 0.42 (0.5 mm) vs 0.28 (0.75 mm)). No improvement was found in the predictions when the smaller (0.5 mm) aperture was used and as the laser spot was much Figure 3. Effects of pupil size on the measured CSF for (a) 2 D (hypermetropic) defocus (subject DAA), (b) +2 D (myopic) defocus (subject DAA), (c) 2 D (hypermetropic) defocus (subject NCS), and (d) +2 D (myopic) defocus (subject NCS). The 6 mm measurements are reproduced from Figure 1.

8 422 Ophthal. Physiol. Opt : No 5 easier for subjects to see with the larger aperture in conditions of defocus and peripheral pupil locations, the 0.75 mm aperture was used throughout the rest of the study. Pupil size and the CSF CSFs were measured with 2 and 4 mm pupils to evaluate the robustness of the optical model. As the 6 mm pupil results reported above did not provide convincing evidence for changing our initial protocol for determining the predicted CSF, we developed CSF predictions for these smaller (2 and 4 mm) pupils using the transverse aberration functions measured with defocus. For each subject, pupil size had only small e ects on the in-focus CSF, although subject NCS had a small notch with a 6 mm pupil size (Figure 1e and1f) which was the result of the residual level of defocus previously mentioned. Complexly oscillating CSFs were noted with both 4 mm and 2 mm pupils in the presence of defocus (Figure 3). There were large inter-subject di erences in the e ect of pupil size on the defocused CSF, being largest for subject NCS, intermediary for subject RLW and least for subject DAA. For subject DAA, while the shape of the CSF varied with the direction of defocus (compare Figure 3a and 3b), the shape and magnitude of the CSF did not vary markedly with pupil size for each defocus condition. For subject NCS, the shape of the defocused CSF varied considerably as pupil size varied, in particular the rst Figure 4. CSFs of subject RLW: measured in-focus, measured defocus and predicted defocus. (a) 2 mm pupil and 2 D (hypermetropic) defocus, (b) 4 mm pupil and 2 D (hypermetropic) defocus, (c) 2 mm pupil and +2 D (myopic) defocus, and (d) 4 mm pupil and +2 D (myopic) defocus.

9 Effects of defocus and pupil size on human contrast sensitivity: N. C. Strang et al. 423 notch (local minimum) occurred at a lower spatial frequency and oscillations occurred at smaller spatial frequency intervals as pupil size increased. The e ect of pupil size also varied with the direction of defocus (compare Figure 3c and 3d). For subject RLW, the shape and magnitude of the defocused CSF varied with pupil diameter, but not to the same extent as NCS. This inter-subject di erence appears to be due to the interaction between the individual ocular aberrations (Table 1), defocus and pupil size. Subject DAA, having the largest spherical aberration had the least variation in CSF with pupil size, while subject NCS, who had the least spherical aberration, had the greatest variation in CSF with pupil size. It is interesting to note that due to the complex interactions between pupil size and defocus, at some spatial frequencies, the contrast sensitivity with a larger (i.e. more aberrated) pupil was better than with a smaller pupil. For example, for subject NCS, in the presence of hypermetropic defocus the contrast sensitivity at 8 cyc/deg was greater with a 4 mm than with a 2 mm pupil (Figure 3c). This is contrary to the common perception that smaller pupils moderate the impact of defocus. The accuracy of CSF predictions with 2 mm and 4 mm pupils was similar to that obtained with the 6 mm pupils, with the RMSE ranging between 0.16 and 0.36 log units (Table 2). For example, the predictions showed a good level of accuracy for hypermetropic defocus in subject RLW with a 2 mm pupil (Figure 4a, RMSE = 0.16) and with a 4 mm pupil (Figure 4b, RMSE = 0.18). However, for myopic defocus and a 2 mm pupil, the predicted notch at 5.5 cyc/ deg was larger than measured and the predicted notch at 12 cyc/deg was not measured (Figure 4c). This prediction inaccuracy at higher spatial frequencies was also found for myopic defocus and a 4 mm pupil (Figure 4d). For subject NCS, a similar level of accuracy was found between 2 mm, 4 mm and 6 mm pupil sizes in hypermetropic defocus, but in myopic defocus the accuracy of the prediction was better with 2 mm (RMSE = 0.22) and 4 mm pupils (RMSE = 0.27) than with the 6 mm pupil (RMSE = 0.44) (e.g. compare Figure 5 with Figure 1f). For subject DAA, the accuracy of predictions was similar for all three pupil sizes. There was no di erence between the two directions of defocus in the ability of the model to predict the CSF with these smaller pupils, suggesting that the model is reasonably robust. Discussion We have extended our work on the in uence of defocus on the contrast sensitivity function (Woods et Figure 5. CSFs of subject NCS with 2 mm pupil and +2 D (myopic) defocus: measured in-focus, measured defocus and predicted defocus. al., 1996a; Atchison et al., 1998c) showing that the predictive model is as e ective with myopic defocus as with hypermetropic defocus and is robust over a wide range of pupil sizes. That predictions of the CSF were slightly better than the previous study can be attributed to the improvements in the methodology. Aberration functions measured with defocus and aberration functions using in-focus aberrational data with a defocus term predicted the defocused CSF equally well. There were marked inter-subject variations in the shape and magnitude of the CSF. The CSF measured with small pupils was not necessarily better than that with larger pupils, there being complex interactions between individual aberrations, defocus and pupil size. In general, the spatial frequency interval between oscillations increased as pupil size decreased, but this was dependent on the interaction between individual aberrations and defocus. Despite the modi cations incorporated to both the transverse aberration measurement technique and optical modelling, we were unable to predict the shape of the CSF accurately in all experimental conditions. Two modi cationsðmatching the wavelengths of the vernier alignment task with the CSF measures and accounting for the in uence of eye rotation during the vernier alignment techniqueðhelped to improve the accuracy of the ts in one subject (RLW) compared with previous results (Atchison et al., 1998c). The other two modi cationsðusing of a smaller aperture in the aberration measurement technique and using the in-focus aberrations with an additional defocus term in the analysisðdid not improve the accuracy of the CSF predictions. Thus the poor predictions of the e ect of myopic defocus on CSF found previously (Atchison et al., 1998c) may have been a consequence of the meth-

10 424 Ophthal. Physiol. Opt : No 5 odological problems addressed in this study, or may have been a statistical anomaly since fewer measures were made of myopic defocus than of hypermetropic defocus in that study. We can consider some of the reasons for the less than perfect predictions, and point the way for further investigation. The aberration technique can be improved by having the spot (seen through the small aperture) move rather than having the vernier target move (seen through the whole pupil). This would eliminate the need to compensate for eye rotation. The lenses to correct or induce defocus created considerable problems by giving changes in image size (a ecting the CSF), and a ecting the location of light rays in the entrance pupil. This problem can be overcome by using a two-lens relay system, with the lens closer to the eye acting as a Badal lens, and with the eye and this lens moving in unison to correct or induce refractive errors. At present we are measuring aberrations along one meridian. Atchison et al. (1998c) argued that this was probably not too critical, provided it was used for determining MTFs only for objects oriented at right-angles to the meridian of aberration measurement, but a more sophisticated analysis should include two-dimensional measurements. This would prove to be time consuming using the method outlined in this study; however, measurement of the wavefront aberration using a fast psychophysical procedure (He et al., 1998) or an objective method such as the HS sensor (Liang and Williams, 1997) could be considered. A more sophisticated analysis could include allowing for individual Stiles±Crawford e ects in MTF determinations, although this is unlikely to have a large in uence even at pupil sizes as large as 6 mm (Atchison et al., 1998a). As for other aberration measuring techniques except the aberroscope method, the aberrations measured are those for light travelling out of the eye, and are not completely those operating for visual performance measures, e.g. the CSF. To get some estimate of the error involved, we performed theoretical ray-tracing both into and out of a Gullstrand number 1 schematic eye, whose cornea had been aspherised to give D of longitudinal spherical aberration at the edge of a 6 mm stop placed at the front of the eye. The D value is for tracing out of the eye. Tracing into the eye for 2 D (hypermetropic) and +2 D (myopic) defocus gave D and D longitudinal spherical aberration, respectively. This variation is too small to have an important in uence on CSF measurements. Another possibility for some of our worse predictions is that adjacent pixel non-linearity in the monitor may have a ected our measured CSF values. This problem is known to modify contrast, especially at high contrast levels and high spatial frequencies (Klein et al., 1996), and could account for the higher than predicted contrast values found at higher spatial frequencies levels in some of our conditions. Non-linearity would also reduce the measured depth of local sensitivity minima (notches) at higher spatial frequencies. This would have the e ect of reducing the quality of the predictions of the e ects of myopic defocus more than the predictions of hypermetropic defocus since, as shown in all paired gures (e.g. Figure 1a and 1b), myopic defocus tended to reduce the CSF more quickly (i.e. higher contrasts required for CSF measurement at more spatial frequencies). However, as the measured contrast values were not higher than the predicted values in all myopic defocus conditions, it seems unlikely that adjacent pixel non-linearity is solely responsible for the poor predictions found in some experimental conditions at high spatial frequencies. In summary, following improvements in our technique, measured CSFs at three pupil sizes and two defocus levels showed complexly shaped CSFs that were predicted from aberration measurements and diffraction-based optical theory. Generally, the shapes of CSFs were well predicted from measured transverse aberrations for myopic as well as hypermetropic defocus for a range of pupil sizes, although there were a few cases where predictions were poor. We have discussed further improvements in technique that we will adopt. Acknowledgements Russell Woods was supported by a visiting researcher award from the Centre for Eye Research, QUT. Part of this work was presented at the 1997 Vision Science and its Applications topical meeting of the Optical Society of America, Santa Fe, NM (Strang et al., 1997). We thank George Smith for the use of his OTF program. This work was supported by an Australian Research Council Large Grant to D. Atchison and N. Strang. References Applegate, R. A. and Lakshminarayanan, V. (1993). Parametric representation of Stiles±Crawford functions: normal variation of peak location and directionality. J. Opt. Soc. Am. A 10, 1611±1623. Atchison, D. A., Joblin, A. and Smith, G. (1998a). In uence of Stiles±Crawford apodization on spatial visual performance. J. Opt. Soc. Am. A 15, 2545±2551. Atchison, D. A., Smith, G. and Charman, W. N. (1998b). Errors in determining the direction of the visual axis in the presence of defocus. Ophthal. Physiol. Opt. 18, 463±467.

11 Effects of defocus and pupil size on human contrast sensitivity: N. C. Strang et al. 425 Atchison, D. A., Woods, R. L. and Bradley, A. (1998c). Predicting the complex e ects of optical defocus on human contrast sensitivity. J. Opt. Soc. Am. A 15, 2536±2544. Bour, L. J. and Apkarian, P. (1996). Selective broad-band spatial frequency loss in contrast sensitivity functions. Invest. Ophthal. Vis. Sci. 37, 2475±2484. Campbell, F. W. and Green, D. G. (1965). Optical and retinal factors a ecting visual resolution. J. Physiol. (London) 181, 576±593. Charman, W. N. (1979). E ect of refractive error in visual tests with sinusoidal gratings. Br. J. Physiol. Opt. 33, 10± 20. Green, D. G. and Campbell, F. W. (1965). E ect of focus on the visual response to a sinusoidally modulated spatial stimulus. J. Opt. Soc. Am. 55, 1154±1157. He, J. C., Marcos, S., Webb, R. H. and Burns, S. A. (1998). Measurement of the wave-front aberration of the eye by a fast psychophysical procedure. J. Opt. Soc. Am. A 15, 2449±2456. Kay, C. D. and Morrison, J. D. (1987). A quantitative investigation into the e ects of pupil diameter and defocus on contrast sensitivity for an extended range of spatial frequencies in natural and homatropinized eyes. Ophthal. Physiol. Opt. 7, 21±30. Klein, S. A., Hu, Q. J. and Carney, T. (1996). The adjacent pixel nonlinearity: problems and solutions. Vision Res. 36, 3167±3181. Liang, J. and Williams, D. R. (1997). Aberrations and retinal image quality of the normal human eye. J. Opt. Soc. Amer. A 14, 2873±2883. Strang, N. C., Atchison, D. A. and Woods, R. L. (1997). Predicting variations in visual performance caused by optical defectsð2. Vision Science and its Applications, topical meeting of the Optical Society of America, 60±63. Thibos, L. N., Bradley, A., Still, D. L., Zhang, X. and Howarth, P. A. (1990). Theory and measurement of ocular chromatic aberration. Vision Res. 30, 33±49. Watt, R. J. and Andrews, D. P. (1981). APE: Adaptive Probit Estimation of psychometric functions. Curr. Psych. Rev. 1, 205±214. Woods, R. L. (1993). Reliability of visual performance measurement under optical degradation. Ophthal. Physiol. Opt. 13, 143±150. Woods, R. L. (1996). Spatial frequency dependent observer bias in the measurement of contrast sensitivity. Ophthal. Physiol. Opt. 16, 513±519. Woods, R. L. and Thomson, W. D. (1993). A comparison of psychometric methods for measuring the contrast sensitivity of experienced observers. Clin. Vis. Sci. 8, 401± 415. Woods, R. L., Bradley, A. and Atchison, D. A. (1996a). Consequences of monocular diplopia for the contrast sensitivity function. Vision Res. 36, 3587±3596. Woods, R. L., Bradley, A. and Atchison, D. A. (1996b). Monocular diplopia caused by ocular aberrations and hypermetropic defocus. Vision Res. 36, 3597±3606. Appendix Correction for ray height due to eye rotation In the development that follows, we assume that paraxial and thin lens optics hold as far as ray-tracing is concerned through the trial lenses. This assumption is warranted because of the low power of the lenses and because the angle involved was small. The small distance between the lenses and the stop was ignored. The accompanying Figure A1 is not to scale; in particular the angles are grossly exaggerated. Height of the foveal achromatic axis at the stop (lens) and the entrance pupil We start our ray-trace at the green vernier target and nish at the centre of rotation of the eye at C. The ray component between the centre of rotation and the stop represents the foveal achromatic axis of the rotated eye. The transverse displacement of the vernier target necessary to achieve apparent alignment with the laser spot is h, the ray has heights h s and h e at the lens and entrance pupil respectively, the distance between the screen and the stop is r (4.0 m), the distance between the stop and the centre of rotation is z (13 mm plus measured vertex distance), the distance between the stop and the entrance pupil at E is v (3 mm plus measured vertex distance), the ray angle between the vernier target and the stop is u, the ray angle between the stop and the centre of rotation is u', and the lens power is F. Using the paraxial transfer equation, h s ˆ h ur which can be rearranged to give u ˆ h h s =r: Using the paraxial refraction equation, u 0 u ˆ h s F: A1 A2 Substituting the right-hand-side of Equation (A1) for u into Equation (A2) gives u 0 h h s =r ˆ h s F which can be rearranged to give u 0 ˆ h s Fr 1 h =r: A3 In the gure u 0 ˆ h s =z: A4 Substituting the right-hand-side of Equation (A4) for u' in Equation (A3) and rearranging gives h s ˆ hz= r z Frz : A5 In the gure h e = z v ˆh s =z from which

12 426 Ophthal. Physiol. Opt : No 5 Figure A1. Determination of the height of the light beam, passing through the small aperture, relative to the foveal achromatic axis. Dimensions are not shown to scale. See text for other details. h s ˆ h e z= z v : A6 h e ˆ h s u0 v: A11 Substituting the right-hand-side of Equation (A6) for h s in Equation (A5) and rearranging gives h e ˆ h z v = r z Frz : A7 Substituting the right-hand-side of Equation (A10) into Equation (A11) gives h e ˆ h s r v Fvr =r: A12 The height of the ray in the stop relative to the foveal achromatic axis is given by Laser ray height at the stop (lens) and entrance pupil relative to the foveal achromatic axis at the stop (lens) and the entrance pupil The laser ray has height h* = 0 at the screen, h * s at the stop and height h * e at the entrance pupil. Its angles before and after refraction are u* and u*', respectively. Other parameters are as given above. For transfer from the screen to the stop h s ˆ h u r ˆ u r from which On refraction at the stop u ˆ h s =r: A8 h rels ˆ h s h s: A13 Substituting the right-hand-side of Equation (A5) for h s into Equation (A13) gives h rels ˆ h s hz= z Frz r : A14 The height of the ray in the pupil relative to the foveal achromatic axis is given by h rele ˆ h e h e: A15 Substituting the right-hand-sides of Equations (A7) and (A12) for h e and h * e, respectively, into Equation (A15) gives h rele ˆ h s r v Fvr =r h z v = r z Frz : A16 u 0 u ˆ h s F: A9 Substituting the right-hand-side of Equation (A8) into Equation (A9) and rearranging gives u 0 ˆ h s Fr 1 =r: A10 Transferring from the stop to the entrance pupil gives