We measure acuity, crowding, and reading in amblyopic observers to answer four questions. (1) Is reading with the amblyopic eye impaired because of larger required letter size (i.e., worse acuity) or larger required spacing (i.e., worse crowding)? The size or spacing required to read at top speed is called “critical”. For each eye of seven amblyopic observers and the preferred eyes of two normal observers, we measure reading rate as a function of the center-to-center spacing of the letters in central and peripheral vision. From these results, we estimate the critical spacing for reading. We also measured traditional acuity for an isolated letter and the critical spacing for identifying a letter among other letters, which is the classic measure of crowding. For both normals and amblyopes, in both central and peripheral vision, we find that the critical spacing for reading equals the critical spacing for crowding. The identical critical spacings, and very different critical sizes, show that crowding, not acuity, limits reading. (2) Does amblyopia affect peripheral reading? No. We find that amblyopes read normally with their amblyopic eye except that abnormal crowding in the fovea prevents them from reading fine print. (3) Is the normal periphery a good model for the amblyopic fovea? No. Reading centrally, the amblyopic eye has an abnormally large critical spacing but reads all larger spacings at normal rates. This is unlike the normal periphery, in which both critical spacing and maximum reading rate are severely impaired relative to the normal fovea. (4) Can the uncrowded-span theory of reading rate explain amblyopic reading? Yes. The case of amblyopia shows that crowding limits reading solely by determining the uncrowded span: the number of characters that are not crowded. Characters are uncrowded if and only if their spacing is more than critical. The text spacing may be uniform, but the observer's critical spacing increases with distance from fixation, so the uncrowded span extends out to where the spacing is critical. Amblyopes have normal critical spacing in the periphery, so, when the uncrowded span extends into the periphery, it has normal extent, which predicts our finding that reading rate is normal too. This confirms the theory that reading rate is determined by the width of the uncrowded span, independent of the critical spacing within the span. The uncrowded-span model of normal reading fits the amblyopic results well, with a roughly fivefold increase in the critical spacing at fixation. Thus, the entire amblyopic reading deficit is accounted for by crowding.

*Journal of Vision*.

*visual span*: the number of characters that can be identified without moving one's eyes. Pelli et al. (2007, this issue) show that the visual span is the

*uncrowded span*: the number of character positions in a line of text that are uncrowded, that is, spaced apart more than the critical spacing. This number depends on the letter spacing and the vertical position in the visual field. Crowding is more extensive at greater eccentricity, so, for a given vertical eccentricity, the uncrowded region extends left and right to the horizontal eccentricities at which the observer's critical spacing equals the letter spacing of the text. The uncrowded-span theory is that reading rate is proportional to the uncrowded span. This gives a good account for how reading rate depends on letter spacing in normal observers. Here, we apply it to reading by amblyopes. Pelli et al. show that crowding limits reading in normal vision. Does crowding also limit reading in amblyopic vision?

*u*is the number of uncrowded letter positions for the given text's letter spacing and vertical eccentricity.

*r*is proportional to the uncrowded span,

*ρ*is a proportionality constant with a value on the order of 100 words/min. This is Legge's “visual span” conjecture, updated to reflect the proof that the visual span is actually the uncrowded span. Equation 1 is the key assumption of the uncrowded-span model. Two ancillary assumptions, Bouma's law and dither, enhance the model to predict realistic reading rate curves without extensive measurements of critical spacing.

*s*of horizontal text grows with eccentricity

*ϕ*and is

*s*=

*s*

_{0}+

*bϕ*on the horizontal midline and

*s*=

*s*

_{0}+

*bϕ*/

*ɛ*at the vertical midline, where

*s*

_{0}is the critical spacing at zero eccentricity,

*ɛ*= 2 is the ellipticity, and

*b*was once thought to be a constant but is now known to depend on eccentricity

*b*=

*b*

_{1}+

*b*

_{2}

*ϕ*(Pelli et al., 2007). Given Bouma's law, a geometric analysis shows that the uncrowded span is

*s*is spacing and

*ϕ*

_{v}is the vertical eccentricity.

*r*versus spacing

*s*produces a curve consisting of a nearly vertical cliff at the critical spacing

*r*

_{max}may be slightly higher than the asymptote.

*dither*: random trial-to-trial variation of the critical spacing, with a uniform probability distribution over a 0.55 log-unit range of spacing. The model's predicted reading rate is the average of the instantaneous rate

*r*over the distribution of possible values of the critical spacing (Equation C1 in Pelli et al.). This smears out the step-like reading rate function of Equations 1 and 2 along the log spacing axis to yield a ramp-like function that fits the human data well. The model's estimated critical spacing is the

*P*th quantile of the dithering range, where

*P*is the (arbitrary) critical probability of letter identification, corrected for guessing. Table 2 summarizes each fit (Equations C1 and B10 in Pelli et al., 2007) by its critical spacing (Equation 3) and maximum reading rate.

^{2}, which is at the middle of the monitor's range (Pelli & Zhang, 1991).

*not*the letter size (Hariharan et al., 2005; Levi et al., 2002a, 2002b; Pelli et al., 2004, 2007; Strasburger et al., 1991; Tripathy & Cavanagh, 2002). As noted above, measuring the size threshold with two different spacings enables us to distinguish the effects of letter size and spacing.

*Loves Music, Loves to Dance*and was shown in the original word order. No observer saw the same text twice. The average word length was five characters. The text was presented in the Courier font (Bitstream “Courier 10 Pitch Bold”) as black letters on the green screen.

Observer | Age (years) | Gender | Strabismus (at 6 m) | Eye | Refractive error (diopter) | Line letter acuity (isolated letter acuity) ^{a} |
---|---|---|---|---|---|---|

Strabismic | ||||||

A.P. | 19 | F | L EsoT 4 ^{Δ} and L hyper 2 ^{Δ} | R | −1.50/−0.50 × 180 | 20/12.5 ^{−2} |

L | −0.75/−0.25 × 5 | 20/50 (20/32 ^{+1}) | ||||

J.S. | 22 | F | L EsoT 6–8 ^{Δ} and hyperT 4–6 ^{Δ} | R | +1.25 | 20/16 |

L | +1.00 | 20/40 (20/32 ^{+1}) | ||||

| ||||||

Anisometropic | ||||||

S.C. | 27 | M | None | R | +0.50 | 20/16 ^{+2} |

L | +3.25/−0.75 × 60 | 20/50 ^{+2} (20/40 ^{−2}) | ||||

C.J. | 22 | M | None | R | −15.00/−1.25 × 150 | 20/125 ^{−4} (20/125 ^{+1}) |

L | −6.00 | 20/16 ^{−2} | ||||

| ||||||

Strabismic and anisometropic | ||||||

S.M. | 55 | F | Alt. ExoT 18 ^{Δ} | R | +2.75/−1.25 × 135 | 20/40 (20/25 ^{+1}) |

L | −2.00 | 20/16 ^{−2} | ||||

J.D. | 19 | M | L EsoT 3 ^{Δ} | R | +2.50 | 20/16 |

L | +5.00 | 20/125 (20/125 ^{+2}) | ||||

A.W. | 22 | F | R EsoT 4–6 ^{Δ} and hypoT 4 ^{Δ} | R | +2.75/−1.0 × 160 | 20/80 ^{−1} (20/50 ^{−1}) |

L | −1.00/−0.50 × 180 | 20/16 ^{−1} |

*x,*

*y*) coordinates of the point (critical size or spacing and maximum reading rate) and the slope of the cliff.

Eccentricity (deg) | Eye | Observer | Maximum reading rate | Critical spacing for reading | Critical print size (CPS) | Ratio | RMS error in log reading rate | Critical spacing for crowding | Acuity | Uncrowded-span model parameters | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Uncrowded-span model fit (words/min) | Two-line fit (words/min) | Ratio | Uncrowded-span model fit (deg) | Two-line fit (deg) | Uncrowded-span model fit | Two-line fit | Flanked letter acuity × 1.1 (deg) | Isolated letter acuity (deg) | b | ρ (words/min) | s _{0} (deg) | ||||

0 | Norm. | A.F. | 738 | 581 | 0.79 | 0.08 | 0.21 | 2.63 | 0.06 | 0.03 | 0.12 | 0.08 | 0.50 | 126 | 0.08 |

0 | Norm. | E.J. | 493 | 321 | 0.65 | 0.12 | 0.20 | 1.68 | 0.09 | 0.05 | 0.19 | 0.14 | 0.50 | 99 | 0.12 |

0 | NAE | A.P. | 499 | 314 | 0.50 | 0.19 | 0.24 | 1.09 | 0.22 | 0.08 | 0.27 | 0.17 | 0.50 | 100 | 0.19 |

0 | NAE | A.W. | 648 | 316 | 0.49 | 0.17 | 0.20 | 1.18 | 0.11 | 0.04 | 0.17 | 0.11 | 0.50 | 130 | 0.17 |

0 | NAE | C.J. | 610 | 430 | 0.70 | 0.09 | 0.16 | 1.78 | 0.07 | 0.03 | 0.17 | 0.11 | 0.50 | 122 | 0.09 |

0 | NAE | J.D. | 353 | 275 | 0.78 | 0.07 | 0.17 | 2.43 | 0.09 | 0.07 | 0.12 | 0.09 | 0.50 | 71 | 0.07 |

0 | NAE | J.S. | 519 | 305 | 0.59 | 0.16 | 0.22 | 1.38 | 0.11 | 0.03 | 0.13 | 0.08 | 0.50 | 104 | 0.16 |

0 | NAE | S.C. | 607 | 299 | 0.49 | 0.16 | 0.19 | 1.19 | 0.10 | 0.06 | 0.11 | 0.09 | 0.50 | 121 | 0.16 |

0 | NAE | S.M. | 431 | 213 | 0.49 | 0.20 | 0.26 | 1.30 | 0.10 | 0.09 | 0.27 | 0.13 | 0.50 | 86 | 0.20 |

0 | AE | A.P. | 300 | 230 | 0.58 | 0.53 | 0.92 | 1.48 | 0.16 | 0.09 | 0.64 | 0.22 | 0.50 | 60 | 0.53 |

0 | AE | A.W. | 645 | 265 | 0.41 | 0.89 | 1.08 | 1.21 | 0.17 | 0.06 | 0.72 | 0.28 | 0.50 | 129 | 0.89 |

0 | AE | C.J. | 961 | 316 | 0.33 | 1.19 | 1.18 | 0.99 | 0.13 | 0.08 | 0.93 | 0.51 | 0.50 | 192 | 1.19 |

0 | AE | J.D. | 361 | 279 | 0.77 | 0.59 | 1.43 | 2.42 | 0.09 | 0.09 | 0.76 | 0.41 | 0.50 | 72 | 0.59 |

0 | AE | J.S. | 512 | 263 | 0.51 | 0.43 | 0.41 | 0.95 | 0.19 | 0.04 | 0.26 | 0.10 | 0.50 | 102 | 0.43 |

0 | AE | S.C. | 547 | 280 | 0.51 | 0.47 | 0.55 | 1.17 | 0.14 | 0.07 | 0.33 | 0.22 | 0.50 | 109 | 0.47 |

0 | AE | S.M. | 313 | 165 | 0.53 | 0.43 | 0.54 | 1.26 | 0.09 | 0.08 | 0.45 | 0.20 | 0.50 | 63 | 0.43 |

5 | Norm. | A.F. | 161 | 137 | 0.85 | 1.91 | 2.44 | 1.28 | 0.11 | 0.08 | 2.06 | 0.30 | 0.73 | 39 | 0.08 |

5 | Norm. | E.J. | 168 | 143 | 0.85 | 1.53 | 2.21 | 1.44 | 0.06 | 0.03 | 1.32 | 0.34 | 0.56 | 33 | 0.12 |

5 | NAE | A.W. | 226 | 199 | 0.88 | 1.41 | 2.08 | 1.48 | 0.07 | 0.06 | 1.66 | 0.38 | 0.50 | 40 | 0.17 |

5 | NAE | C.J. | – | – | – | – | – | – | – | – | 2.02 | 0.94 | – | – | – |

5 | NAE | J.D. | – | – | – | – | – | – | – | – | 0.49 ^{a} | 0.34 | – | – | – |

5 | NAE | J.S. | 127 | 113 | 0.89 | 1.39 | 1.93 | 1.39 | 0.07 | 0.05 | 1.15 | 0.31 | 0.49 | 22 | 0.16 |

5 | NAE | S.C. | 201 | 188 | 0.94 | 1.81 | 3.53 | 1.95 | 0.07 | 0.08 | 1.01 ^{a} | 0.52 | 0.66 | 45 | 0.16 |

5 | NAE | S.M. | 126 | 113 | 0.90 | 1.31 | 1.91 | 1.46 | 0.12 | 0.09 | 2.34 | 0.39 | 0.45 | 20 | 0.20 |

5 | AE | A.W. | 270 | 176 | 0.65 | 1.73 | 2.44 | 1.41 | 0.07 | 0.07 | 1.65 | 0.43 | 0.34 | 34 | 0.89 |

5 | AE | C.J. | – | – | – | – | – | – | – | – | 2.30 | – | – | – | – |

5 | AE | J.D. | – | – | – | – | – | – | – | – | 0.92 ^{a} | 0.61 | – | – | – |

5 | AE | J.S. | 151 | 115 | 0.76 | 1.53 | 2.07 | 1.35 | 0.11 | 0.05 | 1.32 | 0.26 | 0.44 | 24 | 0.43 |

5 | AE | S.C. | 217 | 176 | 0.81 | 1.82 | 2.67 | 1.47 | 0.06 | 0.03 | 0.90 ^{a} | 0.41 | 0.54 | 41 | 0.47 |

5 | AE | S.M. | 115 | 94 | 0.82 | 1.46 | 2.26 | 1.55 | 0.09 | 0.07 | 1.83 | 0.77 | 0.41 | 17 | 0.43 |

10 | NAE | J.D. | 161 | 168 | 1.04 | 2.39 | 3.99 | 1.67 | 0.15 | 0.07 | 2.07 | 0.57 | 0.46 | 27 | 0.07 |

10 | NAE | S.C. | 49 | 37 | 0.76 | 2.44 | 3.17 | 1.30 | 0.05 | 0.11 | 2.99 | 0.65 | 0.46 | 8 | 0.16 |

10 | NAE | A.W. | – | – | – | – | – | – | – | – | 3.40 | 0.64 | – | – | – |

10 | AE | J.D. | 206 | 152 | 0.74 | 2.50 | 3.21 | 1.28 | 0.09 | 0.03 | 1.98 | 0.82 | 0.38 | 29 | 0.59 |

10 | AE | S.C. | 33 | 27 | 4.61 | 2.40 | 3.42 | 1.43 | 0.09 | 0.08 | 2.81 | 0.43 | 0.39 | 5 | 0.47 |

10 | AE | A.W. | – | – | – | – | – | – | – | – | 3.51 | 0.58 | – | – | – |

*r*=

*ρu*, where

*r*is reading rate,

*ρ*is a proportionality constant with a value on the order of 100 words/min, and

*u*is the uncrowded span ( Equation 1). Specifically, we fit Equations C1 and B10 in Pelli et al. (2007) to the reading rate versus spacing data (our Figures 3 and 4) to estimate the three model parameters,

*s*

_{0},

*b*, and

*ρ*. We set the threshold criterion

*P*to 0.44 to match our letter-identification task criterion of 50%, after correction for guessing (0.1 for the 10-letter Sloan alphabet). We optimized

*s*

_{0}and fixed

*b*= 0.5 in our fits to reading rates at 0° eccentricity. For each eye, the fits to the data collected at 5° and 10° eccentricity optimized

*b*and fixed

*s*

_{0}to the value estimated from the data collected at 0°. The fitted value of

*s*

_{0}was about 0.1° in normals, roughly twice that in nonamblyopic eyes, and roughly five times normal (range = 0.52° to 1.5°) in the amblyopic eyes (Table 2). For each fit, Table 2 also provides the maximum reading rate and the critical spacing (Equation 3). The MATLAB program FitAmblyopes.m that fit the uncrowded-span model to our reading rate versus spacing data is available at http://psych.nyu.edu/pelli/software.html.

*s*

_{0}(see Pelli et al., 2007; Figure 8). Corrected for guessing, this probability (criterion for letter identification) is the parameter

*P*of the uncrowded span model.

*P*affects the dither and thus shifts the model's curve (and critical spacing) along the log spacing axis. However, while the model does deal with the letter-identification criterion, the model does not explicitly refer to the performance criterion of the reading task. It simply assumes that the measured reading rate is proportional to the uncrowded span (Equation 1). One would expect the proportionality constant to depend on the criterion, but this would not affect the estimated critical spacing. For reading by human observers, lowering the threshold criterion (from their 80% to our 50%) increases reading rate but, remarkably, does not affect the critical spacing. This may be seen by comparing the Chung et al. (1998; Figures 3 and 7) reading rate curves for 80% and 50% threshold criteria.

*s*=

*s*

_{0}+

*b*

_{1}

*ϕ*+

*b*

_{2}

*ϕ*

^{2}(Pelli et al.), though developed for normal vision, also fits our amblyopic results well.

*x*and

*y*position of a template curve.

*y*= (1.0 ± 0.3)

*x*+ 0.1 ± 0.2 with

*R*

^{2}= 0.85. Again, the slope is 1 and the intercept is 0. This is the same result—equality—that we saw in the full data set.

*R*

^{2}= 0.9 in Figures 2 and 8)? Common cause. It has long been known that acuity and critical spacing depend on eccentricity, so it is no surprise that they are correlated in a data set that includes a range of eccentricities. The eccentricity dependence of both suggests that critical spacing and acuity might be independent effects of a common eccentricity-dependent factor (e.g., in the development of the retina).

*y*= (1.01 ± 0.06)

*x*+ 0.16 ± 0.03 with

*R*

^{2}= 0.96 and the regression of log CPS against log isolated acuity is

*y*= (1.48 ± 0.12)

*x*+ 0.84 ± 0.08 with

*R*

^{2}= 0.92. These regression lines (not shown) have practically the same slopes as the regression lines for the critical spacing for reading ( Figures 7 and 8). Again, the relation with flanked acuity has a slope (1) consistent with equality or proportionality, and the relation with isolated letter acuity has a much greater slope (1.5 > 1). The weakness of this assumption-free test is that it only tests the slope of the regression line (1 for equality), not the intercept (0 for equality). The intercept (0.16 ± 0.03) is the log ratio of the CPS to flanker acuity. It depends on the criterion for letter identification, increasing threefold over a criterion range of 0.3 to 0.9 (Pelli et al., 2007; Figure 8). Thus, this test is not as stringent as the model-based one, because it tests only the slope, not the intercept, of the regression line. Even so, it confirms our equality result without assuming any model.