**Purpose.**:
To examine the extent to which visual acuity (VA) for broadband optotypes is scale invariant by determining whether the same object frequencies mediate VA for individuals with different levels of VA.

**Methods.**:
LogMAR (minimum angle of resolution) VA for briefly presented tumbling E's was measured in 10 visually normal individuals and in five patients with VA loss. The E's were either unblurred or blurred with Gaussian low-pass filters that had cutoff frequencies spanning a 1.2-log unit range. The data were fit with a standard equivalent intrinsic blur model to estimate each subject's unblurred VA (MAR_{0} in minutes of arc) and equivalent intrinsic blur *(*σ_{int} in minutes of arc). From these estimates, the high-frequency cutoff of the band of retinal frequencies (cpd_{crit} in cycles per degree) and object frequencies (cpl_{crit} in cycles per letter) mediating VA were derived.

**Results.**:
LogMAR_{0} was related linearly to log σ_{int} with a slope of 1.47, which is steeper than that predicted by scale invariance. Log cpd_{crit} was related linearly to logMAR_{0} by a slope of −0.64, which is shallower than that predicted by scale invariance. This lack of scale invariance is due to a linear relationship between log cpl_{crit} and logMAR_{0} that had a slope of 0.36.

**Conclusions.**:
The overall pattern of results is not consistent with the expectation of scale invariance underlying the MAR scale. Optotypes that conform to the expectations of scale invariance are needed to improve vision assessment and to provide equivalency of VA defined in terms of MAR and cpd.

^{ 1 }A study of the Fourier components of the Landolt C and tumbling E concluded that, whereas VA for the tumbling E could be based on object frequencies near 2.5 cpl, VA for the Landolt C was most likely based on object frequencies in the range of 1.3 cpl.

^{ 2 }This proposal for the Landolt C was verified experimentally using band-limited targets in a study of the effect of crowding on VA.

^{ 3 }VA measurements using low- and high-pass–filtered tumbling E's showed that the VA for this target in the normal visual field periphery is based on object frequencies lower than the 2.5 cpl proposed previously.

^{ 4 }

^{ 5 –7 }Specifically, there is a linear relationship between log object frequency and log angular subtense of the letter that has a slope of approximately ⅓, such that contrast sensitivity is mediated by low object frequencies for letters of small angular subtense but higher object frequencies for larger letters. Given that individuals with decreased VA require larger-than-normal letters at the acuity limit, it is possible that they may use higher object frequencies than normally sighted individuals, and scale invariance would not hold for VA.

^{ 8 –10 }and have also been used in studies of the object frequencies mediating contrast sensitivity for letter optotypes.

^{ 11,12 }The amount of Gaussian blur necessary to reduce VA by a factor of √2 was used to estimate the high-frequency cutoff of the object frequencies mediating VA. This degree of Gaussian blur has been termed “equivalent intrinsic blur” and is assumed to be an estimate of the amount of blur within an individual's visual system.

^{ 8,9 }

Subject No. | Sex | Age (y) | Diagnosis | Refraction Sphere (D) | Refraction Cylinder (D × Angle) | Chart VA (logMAR) |
---|---|---|---|---|---|---|

1 | M | 46 | Normal | −1.50 | 0.00 | −0.08 |

2 | F | 48 | Normal | −1.00 | +0.50 × 90° | −0.08 |

3 | M | 53 | Normal | −7.75 | +0.50 × 0° | −0.01 |

4 | F | 53 | Normal | −7.00 | +0.75 × 90° | 0.03 |

5 | F | 54 | Normal | −5.75 | +0.25 × 100° | 0.06 |

6 | F | 56 | Normal | −0.50 | 0.00 | −0.10 |

7 | M | 56 | Normal | 0.00 | 0.00 | −0.07 |

8 | F | 57 | Normal | −3.25 | +1.00 × 100° | −0.01 |

9 | M | 58 | Normal | 0.00 | 0.00 | −0.07 |

10 | M | 63 | Normal | −4.00 | 0.00 | −0.16 |

11 | M | 44 | CRVO; CSDME | 0.00 | 0.00 | 0.66 |

12 | F | 58 | PDR; CSDME | 0.00 | 0.00 | 0.22 |

13 | F | 64 | NPDR; CSDME | +2.00 | 0.00 | 0.28 |

14 | F | 68 | NPDR; CSDME | −0.75 | +0.25 × 100° | 0.58 |

15 | M | 71 | BRVO | −0.50 | 0.00 | 0.34 |

^{ 10 }Briefly, stimuli were generated by computer (Macintosh G4; Apple Computer, Cupertino, CA) with commercial software (MatLab; The MathWorks, Natick, MA) and the Psychophysics Toolbox extensions.

^{ 13 }The stimuli were displayed on a 22-in. cathode ray tube (CRT) monitor (FE2111SB; NEC, Irving, TX) with a screen resolution of 1024 × 768 and an 85-Hz refresh rate, driven by a video card with 10-bit DAC resolution (ATI Radeon 9000 Pro; AMD, Sunnyvale, CA).

^{ 14 }such that the stroke width was one fifth of the overall optotype size and the three bars were of equal length. The stroke width ranged from 0.6 to 20 arcmin in 16 steps spaced approximately 0.1 log unit apart. The E at each size was blurred with a set of three 2-D Gaussian filters with standard deviations (σ

_{stim}) of 2, 8, and 32 pixels, corresponding to 0.4, 1.6, and 6.4 arcmin at the test distance of 4.5 m. Stimuli were presented for approximately 60 ms (five video frames) in the center of an adapting field that subtended 3.4° horizontally and 2.6° vertically. The luminance of the adapting field was 106 cd/m

^{2}and the luminance of the unblurred test stimulus was 1.4 cd/m

^{2}, yielding a Weber contrast of −99%. The blurred stimuli were presented without rescaling the contrast. The stimulus luminances were measured with a photometer (LS 110; Minolta Osaka, Japan), and the temporal characteristics of the display were confirmed with an oscilloscope and photocell.

^{ 15 }The subject's task was to judge the orientation of the tumbling E, which was randomly facing either to the right or up on each trial. A brief warning tone signaled the start of each stimulus presentation, and the subject verbally reported the orientation, which was recorded by the examiner. The subjects were given practice trials to become familiar with the task.

_{t}) for each value of σ

_{stim}(0, 0.4, 1.6, 6.4 arcmin) was determined using a two-alternative, forced-choice staircase procedure. An initial estimate of logMAR

_{t}was obtained by presenting the optotype at a suprathreshold size and then decreasing the size by 0.1 log unit until an incorrect response was recorded. After this initial search, logMAR

_{t}was determined using a two-down, one-up decision rule, which provides an estimate of the 71% correct point on a psychometric function.

^{ 16 }Each staircase continued until 10 reversals had occurred, and the mean of the last 6 reversals was taken as logMAR

_{t}. The staircase length was typically 40 to 50 trials, which produced stable measurements (the SEM of the last eight reversals was typically less than 0.05 log unit). One staircase measurement of logMAR

_{t}was obtained from each subject for each value of σ

_{stim}. The conditions were presented in order of increasing σ

_{stim}, but for a given staircase, E's of different sizes were convolved with a constant filter width (σ

_{stim}) so that the degree of blur varied across trials within the staircase.

_{t}for each subject was plotted as a function of log σ

_{stim}, and the data were fit with the log form of the following equation

^{ 8 }: where MAR

_{0}represents VA in the absence of blur (i.e., σ

_{s}_{tim}= 0), and σ

_{int}(corresponding to equivalent intrinsic blur) is the value of σ

_{s}_{tim}at which MAR = MAR

_{0}· √2. This value of MAR was defined as MAR

_{crit}. MAR

_{0}and σ

_{int}were adjusted to minimize the mean squared error between the fitted function and the data.

_{crit}and σ

_{int}in two steps. First, the Gaussian function representing equivalent intrinsic blur was converted from the spatial domain to the frequency domain. In the frequency domain, the SD of the Gaussian function is a measure of the high-frequency cutoff of the band of retinal frequencies mediating VA (cpd

_{crit}). The value of σ

_{int}was converted to cpd

_{crit}as follows:

_{t}as a function of log σ

_{s}_{tim}for one representative normally sighted subject (subject 10). For reference, the right

*y*-axis shows the corresponding Snellen equivalents of the logMAR

_{t}values. The curve represents the least-squares best fit of equation 1 to the data (

*R*

^{2}values for the individual subjects ranged from 0.8 to 1.0). According to equation 1, logMAR

_{t}is approximately constant for small values of log σ

_{s}_{tim}, whereas logMAR

_{t}increases linearly with a slope of 1 for substantially larger values of log σ

_{s}_{tim}. In this figure, logMAR

_{0}and log σ

_{int}are indicated by the arrows. For reference, the value of logMAR

_{crit}is also indicated. For this subject, MAR

_{0}and σ

_{int}were approximately 0.7 and 1.1 arcmin, respectively.

_{0}and log σ

_{int}for the 10 normally sighted subjects (circles) and five DM patients (squares). The right

*y*-axis shows the corresponding Snellen equivalents of the logMAR

_{0}values. The individual values of logMAR

_{0}and log σ

_{int}are listed in Table 2. The values of logMAR

_{0}in this table correlated significantly with the chart acuity values given in Table 1 (

*r*= 0.92,

*P*< 0.01), demonstrating the validity of using logMAR

_{0}as an index of VA.

Subject No. | LogMAR_{0} | σ_{int} (arcmin) | Retinal Frequency (cpd_{crit}) | Object Frequency (cpl_{crit}) |
---|---|---|---|---|

1 | −0.13 | 0.87 | 10.96 | 0.95 |

2 | −0.10 | 1.02 | 9.33 | 0.89 |

3 | −0.01 | 0.89 | 10.72 | 1.23 |

4 | 0.04 | 1.35 | 7.08 | 0.91 |

5 | 0.14 | 1.38 | 6.92 | 1.10 |

6 | 0.10 | 1.45 | 6.61 | 0.95 |

7 | −0.10 | 1.02 | 9.33 | 0.87 |

8 | 0.04 | 1.23 | 7.76 | 1.02 |

9 | −0.11 | 0.98 | 9.77 | 0.89 |

10 | −0.14 | 1.10 | 8.71 | 0.72 |

11 | 0.61 | 3.16 | 3.02 | 1.45 |

12 | 0.48 | 2.09 | 4.57 | 1.62 |

13 | 0.37 | 1.95 | 4.90 | 1.35 |

14 | 0.57 | 2.75 | 3.47 | 1.55 |

15 | 0.64 | 3.02 | 3.16 | 1.62 |

^{ 8,10,17 }logMAR

_{0}was related linearly to log σ

_{int}, such that subjects with lower values of logMAR

_{0}(better VA) had lower equivalent intrinsic blur. The data of the normally sighted subjects alone were best fit with a linear regression line with a slope of 1.0, which is in agreement with previous studies.

^{ 8,10,17 }However, when the patients with VA loss are included, the best-fit regression line (Fig. 2, solid line) has a slope of 1.47 (

*R*

^{2}= 0.94), which is significantly steeper than 1.0 (

*t*= 5.1,

*P*< 0.01).

_{crit}and logMAR

_{0}is shown in Figure 3, with the individual values of cpd

_{crit}given in Table 2. The dashed line in Figure 3 has a slope of −1.0 and describes the relationship between retinal spatial frequency and VA, assuming that scale invariance holds and that a logMAR of 0 (20/20 Snellen VA) is equivalent to a retinal spatial frequency of 30 cpd. It is apparent from Figure 3 that there was a linear relationship between log cpd

_{crit}and logMAR

_{0}, but the slope of the best-fit regression line (solid line,

*R*

^{2}= 0.94) was −0.64. This slope is significantly shallower than the slope of −1.0 (

*t*= −8.7,

*P*< 0.01) predicted by scale invariance (dashed line).

_{crit}as a function of logMAR

_{0}for the normally sighted subjects and DM patients. The individual values of cpl

_{crit}are given in Table 2. The solid line in Figure 4 represents the best-fit regression line (

*R*

^{2}= 0.92), which has a slope of 0.36. This slope is significantly steeper than a slope of 0 (

*t*= 8.7,

*P*< 0.01). This non-0 slope indicates a dependence of object frequency on VA, such that observers with worse VA (higher logMAR

_{0}) used higher object frequencies than subjects with better VA. This lack of scale invariance accounts for the absence of a one-to-one relationship between log cpd

_{crit}and logMAR

_{0}(Fig. 3).

^{ 18 }

_{crit}) of the band of object frequencies mediating VA for subjects with different VA levels was derived using an equivalent intrinsic blur paradigm. An estimate of equivalent intrinsic blur was obtained for each subject, and this estimate was used to derive the high-frequency cutoff of the band of retinal frequencies (cpd

_{crit}) mediating VA. The value of cpd

_{crit}was then converted to the corresponding object frequency (cpl

_{crit}). The results showed that the object frequency information mediating VA is not scale invariant. Subjects with worse VA (higher values of logMAR

_{0}) had higher values of cpl

_{crit}than subjects with better VA. Because of this lack of scale invariance, VA defined in terms of MAR and VA defined in terms of equivalent retinal frequency were not proportionally related by a slope of −1.0 (Fig. 3). For example, subjects with MAR values that differed by 1.0 log unit differed in equivalent retinal frequency by only 0.66 log units.

_{crit}for our sample of subjects ranged from 0.7 to 1.6 (Fig. 4). These values are lower than those reported in a previous study that investigated the effect of low-pass and high-pass filtering on orientation judgments of the tumbling E in the normal visual field periphery.

^{ 4 }That study reported that object frequencies between approximately 1.25 and 2.25 cpl mediated performance for the tumbling E. However, a direct comparison of our results with those of this previous study is complicated by the fact that, whereas the previous investigators based their estimate of object frequency on the filter cutoffs required to affect VA, the values reported here are dependent on the point on the function relating logMAR

_{t}to log σ

_{stim}that is selected for analysis (Fig. 1). We chose to use the point at which MAR

_{t}was elevated by √2 above MAR

_{0}, which corresponds to the standard measure of equivalent intrinsic blur. Selecting a lower point on the curve would result in a higher estimate of cpl

_{crit}, which would be more similar to the values reported previously.

^{ 4 }It is important to note, however, that altering the chosen point on the curve would not affect the non-0 slope of the line relating log cpl

_{crit}and logMAR

_{0}(Fig. 4). Consequently, there is a lack of scale invariance for VA, regardless of the point that is selected as the basis for the derivation of critical object frequency.

_{crit}and logMAR

_{0}was approximately ⅓ (Fig. 4). A similar slope has been reported in previous studies of contrast sensitivity for broadband optotypes, including orientation judgments of the Sloan N (a two-alternative, forced-choice task similar to that of the present study), letter detection, letter discrimination, and letter identification tasks.

^{ 5 –7,11,12 }The contrast sensitivity data of these previous studies together with the VA data of the present study suggest that an increase in object frequency with increasing target size is a general characteristic of the measurement of visual function with broadband optotypes. A linear relationship between object frequency and letter size with a slope of approximately one third was also observed in a study of contrast sensitivity in amblyopic subjects.

^{ 19 }This latter finding suggests that the lack of scale invariance for VA found in the present study would probably generalize to other patient populations beyond the DM patients studied here. However, additional work is needed to confirm this hypothesis.

^{ 20,21 }but these stimuli are typically unfamiliar to patients, and VA measurements made in the periphery with these stimuli can be affected by spurious resolution and aliasing.

^{ 22 }VA has also been measured with “vanishing optotypes” that have pseudo–high-pass spatial characteristics.

^{ 23,24 }However, untrained subjects, patients with central field loss, or patients with unsteady fixation may have trouble localizing these targets in space at sizes near the acuity limit, which would increase spatial uncertainty. Thus, further study is needed to identify optotypes that maintain the desirable characteristics of letters but conform to the expectations of scale invariance, which would provide a better assessment of VA.