A double dissociation of the acuity and crowding limits to letter identification, and the promise of improved visual screening

Abstract

Here, we systematically explore the size and spacing requirements for identifying a letter among other letters. We measure acuity for flanked and unflanked letters, centrally and peripherally, in normals and amblyopes. We find that acuity, overlap masking, and crowding each demand a minimum size or spacing for readable text. Just measuring flanked and unflanked acuity is enough for our proposed model to predict the observer's threshold size and spacing for letters at any eccentricity. We also find that amblyopia in adults retains the character of the childhood condition that caused it. Amblyopia is a developmental neural deficit that can occur as a result of either strabismus or anisometropia in childhood. Peripheral viewing during childhood due to strabismus results in amblyopia that is crowding limited, like peripheral vision. Optical blur of one eye during childhood due to anisometropia without strabismus results in amblyopia that is acuity limited, like blurred vision. Furthermore, we find that the spacing:acuity ratio of flanked and unflanked acuity can distinguish strabismic amblyopia from purely anisometropic amblyopia in nearly perfect agreement with lack of stereopsis. A scatter diagram of threshold spacing versus acuity, one point per patient, for several diagnostic groups, reveals the diagnostic power of flanked acuity testing. These results and two demonstrations indicate that the sensitivity of visual screening tests can be improved by using flankers that are more tightly spaced and letter like. Finally, in concert with Strappini, Pelli, Di Pace, and Martelli (submitted), we jointly report a double dissociation between acuity and crowding. Two clinical conditions-anisometropic amblyopia and apperceptive agnosia-each selectively impair either acuity A or the spacing:acuity ratio S/A, not both. Furthermore, when we specifically estimate crowding, we find a double dissociation between acuity and crowding. Models of human object recognition will need to accommodate this newly discovered independence of acuity and crowding.

Keywords: acuity; amblyopia; anisometropic; critical spacing; crowding; legibility; letter identification; object recognition; overlap masking; screening; spacing:acuity ratio; strabismic; threshold spacing.

Figure 1

Unflanked and flanked letter identification tasks. The observer is asked to fixate the center of the fixation mark and to identify the target letter, once it appears. The target letter is subsequently presented, briefly, either alone (unflanked) or surrounded (flanked) by four random letters of the same size. The center-to-center letter spacing in degrees between the target and the flankers scales with letter size and is s times the letter size, where s is usually 1.1. In each case, we use an adaptive procedure (QUEST) to determine the threshold size (covaried with spacing) for 50% correct identification. The letter spacing factor s is 1.1× for all data reported in all tables and figures, except where we indicate otherwise (in Tables 1 and 3 and Figures 2 and 3).

Figure 2

Flanked letter acuity (a & b) or threshold spacing (c & d) versus letter spacing factor (multiple of the letter size) for normal (a & c) and amblyopic (b & d) observers. Thresholds for the normal observer are measured at 0° (circle), 1.25° (triangle), and 5° (square) eccentricities; thresholds for the purely anisometropic amblyope (green) and the strabismic amblyopes (red, blue) are measured at fixation. The error bars on each data point indicate plus-or-minus one standard error. The same data and models are plotted twice: as flanked acuity A' in the upper graphs (a & b) and as threshold spacing S in the lower graphs (c & d). Threshold spacing S = sA' is the product of the letter spacing factor s and the flanked acuity A'. For each observer, the horizontal line (dashed) in the upper graphs (a & b) is the (unflanked) acuity A (rightmost point), and the horizontal line (dotted) in the lower graphs (c & d) is the critical spacing S_critical, estimated as the geometric mean of the points that lie above the (extended) dashed line. The data and lines are converted back and forth between upper and lower graphs by the relation log S = log s + log A'.

Figure 3

Scaling. Replotted from Figure 2. The parameters are acuity A, flanked acuity A', spacing S, and eccentricity φ. Left: normal vision; Right: amblyopia. Top: A' versus spacing S; Middle: normalized by acuity, plotting the acuity ratio A'/A versus spacing:acuity ratio S/A; Bottom: the acuity ratio A'/A versus spacing normalized by padded eccentricity S / (φ + φ_crowding). For normals (left) φ_crowding = 0.45° and several eccentricities φ are tested. For the amblyopes (right) the eccentricity is zero φ = 0 and each strabismic observer is assigned the best-fitting add-on φ_crowding, as specified in the legend. Each estimated add-on was adjusted to shift the curves horizontally to match the normal foveal curve. Note that the purely anisometropic amblyope (green) scales with acuity (Panel d) as expected for overlap masking, and the strabismic amblyopes (red and blue) scale with padded eccentricity (Panel f) as expected for crowding.

Figure 4

Effects of eccentricity and blur on threshold spacing in normal vision. (The reader is free to think of the vertical scale as 1.1A', instead of S, since S = sA' and the spacing factor s = 1.1.) Filled gray symbols represent the thresholds at fixation, and filled black symbols represent the thresholds at various eccentricities in the periphery of three normal observers. Refractive errors (if any) were fully corrected. Open circles represent the thresholds at fixation of one normal observer wearing blurring lenses of various refractive powers (see Methods). The dashed regression line for the no-blur data at various eccentricities is log S = 0.97 + 1.75 log A, where A is acuity in degrees and S is threshold spacing in degrees. The regression line (not shown) for the zero-eccentricity data at various blurs is log S = 0.13 + (0.99 ± 0.02) log A. Since that slope is insignificantly different from one, we fit and display a line (dotted) with unit slope log S = 0.13 + log A, i.e., S = 1.4A. Note that the three filled gray symbols are all at fixation, so the differences among them are not an effect of eccentricity. They all lie on the blur line (dotted), suggesting that this variation among normal individuals reflects differences in their blur (optical and neural). These two lines, dashed and dotted, are reproduced in several subsequent figures. The letter spacing factor s is 1.1× for all data reported in all tables and figures, except where we indicate otherwise (in Tables 1 and 3 and Figures 2 and 3). The observers are all young normals, so it is surprising that EJ's acuity is so much worse than that of the other two observers. However, AF and SS are emmetropic, while EJ is myopic, −2.50D and −3.00D. Her point lies near the line of optic blur, suggesting that her eyes may have uncorrected optical aberrations.

Figure 5

Threshold spacing versus acuity for (a) purely anisometropic (green) amblyopes and (b) strabismic (with or without anisometropia, blue and red) amblyopes. Amblyopic observers' threshold spacings are plotted against acuities. The dashed and the dotted lines are regression lines for normal eccentric and normal blurred results respectively, from Figure 4. The solid lines are the regression lines of (a) purely anisometropic amblyopes' thresholds, log S = 0.26 + 1.15 log A and (b) strabismic amblyopes' thresholds, log S = 0.90 + 1.67 log A. This graph also shows that the combination of threshold spacing and acuity is much better at distinguishing strabismic from purely anisometropic amblyopes than is threshold spacing or acuity alone. In other words, the two diagnostic categories cannot be reliably separated by a horizontal or vertical line, but are well separated by a diagonal line.

Figure 6

Effect of blur. Threshold spacing is less dependent on acuity (and blur) at greater eccentricity. We measured threshold spacing S versus acuity A for a normal observer (open symbols) and several amblyopes (filled symbols) with various amounts of blur. The normal was tested at several eccentricities; the amblyopes only at fixation. We fit a regression line to each observer at each eccentricity and then plotted the regression line slope (log-log slope of spacing vs. acuity) as a function of padded eccentricity φ + φ_crowding. (a) We measured the effect of optical blur. The normal observer (open symbols) was tested at eccentricities of 0° (dotted line), 1.25° (open triangle), and 5° (open square). Amblyopes (filled symbols) were tested at fixation (0°). Each measurement was obtained while the observer was wearing a blurring lens (see Methods) except for the leftmost data point for each observer, which was measured merely with the observer's refractive correction. The dotted regression line is a unit-slope line for a normal observer viewing directly with blur, from Figure 4. A regression is shown for the blur results (with a range of at least one diopter) for each observer and eccentricity tested (see Methods). (b) The lower graph plots the slope of each regression, as a function of the padded eccentricity (φ + φ_crowding). For the normal observer, the add-on is fixed φ_crowding = 0.45°, and several eccentricities φ are tested. For the amblyopes, eccentricity is zero φ = 0°, and we used the observer's add-on φ_crowding from Table 1. Regression lines. (a) SS at 1.25°: log S = 0.07 + 0.49 log A; SS at 5°: log S = 0.25 + 0.13 log A; GJ: log S = 0.06 + 0.57 log A; and AP: log S = 0.06 + 0.45 log A. (b) Regression line is: Slope = 1.0 – 0.19 (φ + φ_crowding).

Figure 7

Acuity and threshold spacing versus defocus in normal central vision. The gray disks, with or without a black edge, represent the threshold spacing or acuity, respectively. The fitted curve (solid) for threshold spacing is S = 0.45 ; the fitted curve (dotted) for acuity is A = 0.33 , where B is defocus in diopters. The inverse of the dotted curve for acuity is our equivalent-blur model of purely anisometropic amblyopia. The spacing:acuity ratio S/A = 1.4 ± 0.03 is not significantly different from m = 1.4, indicating that the threshold spacing is limited by overlap masking.

Figure 8

Double dissociation of A and S/A (a) A scatter diagram of threshold spacing S versus acuity A. Normal observers are solid black at ecc. 0° and open circles at eccs. 1.25°, 2.5°, 5°, 10°. Normals with optical or digital blur are + or ×. (0.5 D blur brings normal observer SS to the threshold acuity 0.15°.) Patients are in color. Strabismic amblyopes are red disks. Purely anisometropic amblyopes are green discs. Apperceptive agnosics are blue diamonds. (To prevent occlusion, several agnosics have been shifted up or down ±0.3° along the S axis.) The four clinical groups are quite well separated by the two lines, A = 0.15° and S/A = 1.84. Nearly all (18/20) of the apperceptive agnosics have near-normal acuity, A < 0.15°. Nearly all (11/12) strabismic amblyopes have poor acuity A > 0.15°. (The diagnosis of amblyopia requires impaired acuity, relative to the fellow eye. Strabismic amblyope JS has an acuity of 0.102 in his amblyopic eye, which is normal, but still qualifies as an amblyopic impairment because it's much worse than the unusually good acuity of his fellow eye.) (b) The same lines and data are replotted as spacing:acuity ratio S/A versus acuity A, which makes the dividing lines vertical and horizontal. (c) Summary of the double dissociation. All three normals have good spacing:acuity ratio S/A < 1.84 and acuity A < 1.15. All 20 apperceptive agnosics have high spacing:acuity ratio S/A > 1.84 and nearly all (18/20) have near-normal acuity A < 1.15°. All six purely strabismic amblyopes have impaired acuity A > 1.15° and near-normal spacing:acuity ratio S/A < 1.84. Thus, these two clinical conditions, purely anisometropic amblyopia and apperceptive agnosia, independently affect A and S/A. See Figure 10 for further methodological details of testing the normals and amblyopes.

Figure 9

Characterizing each observer and condition by the two add-ons, φ_A and φ_crowding. This plots estimated parameters (φ_crowding and φ_A from Equations 15 and 16) instead of the raw data (S and A). This makes the diagnostic categories more obvious than in the raw plot of Figure 8a. Here, S and A are measured at fixation, φ = 0. φ_A is proportional to A, but φ_crowding is nonlinearly related to S and A. At high S/A, φ_crowding is proportional to S, but at low S/A, φ_crowding is constant. The nonlinearity pushes all the normal, blur, and purely anisometropic points down to the floor. This analysis and presentation show that purely anisometropic amblyopia (and blur) affect only φ_A, not φ_crowding; strabismic amblyopia impairs both φ_A and φ_crowding; and apperceptive agnosia greatly impairs φ_crowding and typically spares φ_A.

Figure 10

The boundary between crowded and uncrowded. (These data also appear in Figure 8.) Spacing:acuity ratio versus acuity for normal and amblyopic observers. White letters are displayed on a CRT at medium contrast (0.6) and 1.1× spacing for 200 ms. (See Figure 15 in Appendix A for more conditions.) The observers are normals (black symbols) or strabismic (red) or purely anisometropic (green) amblyopes. Normals are tested at eccentricity zero (filled black circles) and 1.25°, 2.5°, 5°, and 10° (circles, small to large). The symbols + and × represent results with optical and digital blur, respectively. All but one of the strabismic-amblyope and normal peripheral ratios are above the 1.84 criterion (dot-dashed line). All of the purely anisometropic-amblyope and normal central ratios, including those with optical and digital blur, are below the 1.84 criterion. Thus, all the conditions limited by crowding are above the line, and the rest are below the line.

Figure 11

Spacing:acuity ratio versus status of stereopsis for amblyopes. Amblyopes who failed the stereopsis test are plotted as “Fail”; everyone who has some amount of stereopsis is plotted as “Pass.” The vertical dashed line divides the amblyopes into “Fail” and “Pass” groups, and the horizontal dashed line, S/A = 1.84, divides the amblyopes into two groups with large and small spacing:acuity ratio.

Figure 12

The pass/fail regions for screening by (a) single-letter acuity or flanked acuity with (b) loose or (c) tight spacing. (Data from Figure 8a.) The single-letter acuity test passes anyone who reads 0.15° letters. Thus, it passes the normals, and detects most of the patients, but fails to detect the patients who have normal acuity. The flanked acuity tests are more demanding. We relaxed the size criterion slightly, to 0.18°, but observers must read the 0.18° letters at 0.36° spacing on the loose chart (b) or 0.2° spacing on the tight chart (c). Flanked acuity passes all three normals. With loose spacing (b) it detects all but one of the patients (a strabismic amblyope). With tight spacing (c) it detects all the patients.

Figure 13

Flankers should be target like. The clinical literature on screening for amblyopia has often implicitly assumed that “contour interaction” (overlap masking) and crowding are the same thing, but they are not, as this figure demonstrates. In the normal fovea, where flanked acuity is limited by overlap masking, surrounding the target with nearly contiguous bars (“contours”) or letters has the same effect, which you may witness by comparing these three eye charts. (In the first column, use only the middle letter for testing; the outer letters are not fully flanked.) In any row, across the three charts, all three targets have the same size, which drops by a factor of from row to row. The left column has letter flankers (R, Z), the middle column has bar flankers, and the right column is unflanked. The conclusions of this demo depend only on comparing charts, side by side, at any viewing distance. If you like, viewing from at least 2 m will eliminate any concern that you might be limited by the resolution of this page. As a normal observer, looking directly at each target, you will find that both kinds of flanker are equally effective. Letters and bars (left and middle columns) raise threshold one row above that for unflanked acuity (right column). For a given gap between target and flanker, you have the same flanked acuity (threshold row) with letter and bar flankers. This is overlap masking. Simulating a strabismic amblyope, please fix your gaze on a + sign and peripherally view the target (to right or left). As a (simulated) strabismic amblyope, you are limited by crowding. Unlike overlap masking, crowding is very sensitive to the degree of similarity of target and flanker. The letter flankers are much more effective than the bars, because they are more similar to the target, even though, having the same gap, they are farther away, center to center. Thus, in existing tests, replacing bar flankers by more target-like flankers will worsen the flanked acuity of the strabismic amblyopes without affecting the flanked acuity of normals. This will increase the separation of the two populations, increasing the power of the test to detect strabismic amblyopes among normals.

Figure 14

Spacing should be tight. Two charts with different spacing factors (i.e., spacing/size): a tight 1.1× (which we recommend) and a loose 2× (which is typical of the commercially available tests). Each line of each chart displays a target letter between two flankers (R,Z). Acuity imposes a floor on the spacing threshold: An observer with acuity A reading a chart with spacing factor s cannot read any letters with spacing below sA. The 2A floor of a 2× chart hardly increases the spacing at flanked acuity threshold of a strabismic amblyope (for whom S_crowding > 1.84A), but greatly increases that of a normal observer (for whom S_masking ≈ 1.4A if s = 1.1, and may be even lower with slightly larger s). A few commercially available letter-flanked acuity tests have a spacing factor of 1.5×, and the rest have 2×. Assuming you have normal vision, you can test this spacing-factor effect on yourself, first foveally—as a normal—and then peripherally—modeling a strabismic amblyope. Each column has a different spacing factor: 1.1× on the left and 2× on the right. The columns are aligned so that both targets in each row have the same center-to-center spacing S of target to flanker. The whole three-letter triplet shrinks by a factor of from row to row. As a normal observer, look directly at each target, the middle letter of each triplet. Notice that, viewing directly, you read one more row (smaller spacing) with the tighter spacing (left column). On the left, you are limited by overlap masking by the flankers (threshold spacing 1.4A), well above the spacing floor of 1.1A imposed by the chart, given your acuity A. On the right, the flankers have no effect and the spacing at your flanked acuity threshold is at the 2A floor imposed by acuity with s = 2. Thus threshold with the loose chart is 2/1.4 higher, which is roughly , one line on these charts. Simulating a strabismic amblyope, fixate the central + sign in the top row. While still fixating, try to identify the target to the left and the target to the right. If you succeed, then proceed to the next row down, until you fail. Notice that, as a strabismic amblyope, limited by crowding, you have the same spacing threshold (row) with both charts (left and right). Thus, tightening the spacing (from 2× to 1.1×) reduced normal threshold spacing (above) but does not affect threshold of the strabismic amblyope. This increases the separation of the two populations, increasing the power of the test to detect strabismic amblyopes among normals. If you like, viewing from at least 2 m will eliminate any concern that you might be limited by the resolution of this page. In fact, the point demonstrated here is independent of the source of the acuity-limiting blur. No matter whether the limiting blur arises in the chart, the retinal image, or the neural representation, the more tightly spaced chart is better at detecting strabismic amblyopia.

Figure 15

Threshold spacing versus acuity for normal observers (see Figure 10 for amblyopes). (a) All conditions. (b) Expanded view of lower left. Contrast is denoted by gray level, lighter for lower contrast. Results for white/black letters are represented by open/filled symbols. Duration is denoted by size; “∞” in the legend indicates static presentation. Spacing is 1.1× (empty and filled circles, +) or 1.5× (squares, triangles, C, N). We assess the effect of image quality at fixation by comparing results with a 2 mm pinhole (triangle) or optical or digital blur (hourglass symbol). Dashed lines represent the crowding limits at 0° and 5° eccentricity. The dotted line represents the overlap masking limit with m = 1.4 (Equation 3). Except for the Cambridge Crowding Cards (C), and the “N-Z” test (N or Z), the rest of the data are for computerized testing on a CRT. The N represents the results for identifying one of four symbols (N, reflected N, Z, reflected Z) with or without four flankers on a printed card that that is otherwise like the Cambridge Crowding Cards (s = 1.5). The Z is similar, but for a tightly spaced chart (s = 1.1). Foveal data points from the literature are plotted as yellow symbols. Danilova and Bondarko (, yellow X) measured foveal acuities and critical spacings for several subjects under multiple conditions: Landolt C with flankers such as bars and gratings of different spatial frequencies, Tumbling E's, grating flanked by similar gratings of the same or higher spatial frequencies with fixed or random orientations. These conditions are not distinguished in the graph. Error bars show the standard errors across subjects. Note that the thresholds were not measured at a fixed criterion (e.g., 75% correct). Latham and Whitaker (, yellow square 1.5× and diamond 1.25×) measured foveal acuities and the critical spacings of the grating flanked by similar gratings at random orientations. The spacing factor s is fixed at either 1.5× (square) or 1.25× (diamond). The target and flankers scale proportionally. Their experimental condition is the most similar to ours. Since there are only two subjects, their data points were plotted individually instead of being averaged. Toet and Levi (, yellow circle) measured foveal acuities and the critical spacings of the tumbling letter T with similar flankers. Error bars show the standard errors across subjects.

Figure 16

Distinguishing crowding from overlap masking. When a target letter is among flanking letters, the flankers can impair target recognition in two ways: crowding and overlap masking. These effects have different explanations and phenomenology but have not always been easy to distinguish. Here we introduce a new diagnostic test. Making the flankers identical to the target abolishes crowding yet hardly affects overlap masking. Regan, Giaschi, Kraft, and Kothe (1992) introduced a repeated-letter chart to tolerate errors in fixation or selection. Here we exploit its immunity to crowding. The threshold spacing of peripheral vision is normally due to crowding. (The conclusions of this demo depend only on comparing the two charts, side by side, at any viewing distance. If you like, viewing from at least 2 m will eliminate any concern that you might be limited by the resolution of this page.) Please fixate the top plus sign and try to identify the middle letter in the vertical triplets to the left and right. Ignore the two flankers above and below each target letter. On the left, the flankers (R) are different from the target. On the right, they are identical to the target. You will find that you can read more lines in the right column than in the left column. Your left-column threshold is limited by the severe crowding of peripheral vision; your threshold spacing is several times larger than your acuity A. Your right-column threshold is limited by overlap masking, a spacing of 1.4A, which is less than twice your acuity, allowing you to read more lines. In central vision, we saw above that crowding and overlap masking are both viable explanations for the measured threshold spacing of 1.4A. Using central vision to directly fixate each target we find that we can read down to the same level on both sides of the chart. There is no effect of making the flankers identical to the target. Since abolishing crowding has no effect, any crowding present must have a threshold spacing less than or equal to that of the overlap masking. Note that most charts probe critical spacing radially from the point of fixation; the target and flankers all lie on the same radial line. This chart probes critical spacing tangentially; the target and flankers lie on a line that is orthogonal to the radial line connecting target to fixation. Critical spacing tangentially is about half the critical spacing radially (Toet & Levi, 1992).

Figure 17

Assessing the blur model of purely anisometropic amblyopia by measuring effects of blur. As a function of defocus, for normal (gray) and purely anisometropic-amblyopic (green) central vision, symbols with and without black edges represent threshold spacing and acuity, respectively. Symbols with a central dot indicate that a pinhole was used. The natural pupil diameter is 3–4 mm and the pinhole diameter is 1.5 mm. The dotted curve is our equivalent-blur model for purely anisometropic amblyopia. Normal data are plotted against defocus. Purely anisometropic-amblyopic data are plotted against the corresponding equivalent blurs (to match acuity of the normal when both have natural pupils) for that observer. The spacing:acuity ratios S/A for the blurred normal (1.4 ± 0.03) and the amblyope (1.4) are not significantly different from m = 1.4, indicating that the threshold spacing is limited by overlap masking.