Amblyopia is a much-studied but poorly understood developmental visual disorder that reduces acuity, profoundly reducing contrast sensitivity for small targets. Here we use visual noise to probe the letter identification process and characterize its impairment by amblyopia. We apply five levels of analysis — threshold, threshold in noise, equivalent noise, optical MTF, and noise modeling — to obtain a two-factor model of the amblyopic deficit: substantially reduced efficiency for small letters and negligibly increased cortical noise. Cortical noise, expressed as an equivalent input noise, varies among amblyopes but is roughly 1.4× normal, as though only 1/1.4 the normal number of cortical spikes are devoted to the amblyopic eye. This raises threshold contrast for large letters by a factor of √1.4 = 1.2×, a negligible effect. All 16 amblyopic observers showed near-normal efficiency for large letters (> 4× acuity) and greatly reduced efficiency for small letters: 1/4 normal at 2× acuity and approaching 1/16 normal at acuity. Finding that the acuity loss represents a loss of efficiency rules out all models of amblyopia except those that predict the same sensitivity loss on blank and noisy backgrounds. One such model is the last-channel hypothesis, which supposes that the highest-spatial-frequency channels are missing, leaving the remaining highest-frequency channel struggling to identify the smallest letters. However, this hypothesis is rejected by critical band masking of letter identification, which shows that the channels used by the amblyopic eye have normal tuning for even the smallest letters. Finally, based on these results, we introduce a new “Dual Acuity” chart that promises to be a quick diagnostic test for amblyopia.

*N*

_{eq}and efficiency

*η*

^{+}(Pelli & Farell, 1999). Efficiency rates the observer’s computational performance (ability to identify faint letters) on an absolute scale. The equivalent noise represents the amount of visual noise added to the display that would account for the observer’s threshold on a blank screen.

*f*= 3/

*s*

^{2/3}to identify Bookman letters of size

*s*. This frequency is the bottom scale in most of our graphs.

^{2}). The noise began 496 ms before the signal and ended 496 ms after the end of the signal, and covered an area approximately twice as wide and twice as high as the average letter width and height. The noise consisted of independently generated square checks. The noise check duration was 45 ms. The power spectral density

*N*of the noise is the product of contrast power

*c*

^{2}

_{rms}, area, and duration of a check. The noise checks and letters were displayed at a fixed physical size. To create letters smaller than 1 deg, we increased viewing distance, with a fixed letter size (typographic x-height) of 0.94 cm at the display. For larger letters we increased the letter size on the screen, with a fixed viewing distance of 0.6 m. The noise check size scaled with the letter size so that there were always 14.5 checks per letter size, as in Figure 1. The letters and noise were generated by a Macintosh IIci or a PowerMac 6100, and displayed on a 12-in monochrome display screen using a video attenuator (Pelli & Zhang, 1991) and TextInNoise, a stand-alone C application that we wrote, which was a predecessor to the Psychophysics Toolbox extensions to MATLAB (Brainard, 1997; Pelli, 1997; http://psychtoolbox.org/).

*N*= 0) or noisy (

*N*> 0) background. The stimulus was then replaced by a response screen consisting of the 26 letters of the alphabet displayed at a high contrast. The observer responded by using the mouse to place the cursor on the matching letter, and clicking, which initiated feedback and a new trial. We used an extra large cursor, so that all the observers could see it. Each correct response was rewarded by a beep. In each run of 40 trials, the signal size and noise contrast were fixed. The signal contrast was controlled by an efficient staircase (improved QUEST — Watson & Pelli, 1983; King-Smith, Grigsby, Vingrys, Benes, & Supowit, 1994), which estimates the contrast required for 64% correct identification. We varied the letter size and noise contrast (usually 0 or 25%) between runs. Each threshold reported here is the geometric mean of 3 to 6 threshold estimates.

*C*

_{ideal}is measured independently of the human thresholds, by running the same tests on a computer program implementation of the ideal observer. The ideal observer knows the noisy stimulus and chooses the most likely signal (i.e., the one most similar to the stimulus) (Pelli, 1985; Geisler, 1989).

*c*

_{0}, and one with a high noise

*c*.

*Equivalent noise*is , where

*N*is the spectral density of the noise. , where

*C*

_{rms}is the root mean square noise contrast,

*A*

_{check}is the area of a noise check, and

*T*

_{check}is its duration.

*Efficiency*is where

*c*

_{ideal}is the ideal observer’s threshold contrast in noise. Pelli and Farell (1999) call this high-noise efficiency

*η** =

*η*

^{+}.

*c*

_{0}(without noise) and

*c*(with noise) for identification. The other quantities (

*E*

_{0},

*E*,

*η*

^{+},

*N*

_{eq}, etc.) are all derived from these two measurements and the stimulus parameters. Each measure gives a slightly different view of the results, as we will now see.

*c*

_{0}= 1) under the precise test conditions of this experiment: isolated letter, brief presentation, etc. We estimated the acuity for each observer directly from the data in Figure 3A by finding the horizontal shift of the normal curve that best fits each observer’s data, taking the extrapolation of the normal curve to

*c*

_{0}= 1 as the acuity estimate. We shall call this “acuity.” We call the channel spatial frequency 3/

*s*

^{2/3}corresponding to this letter size

*s*the “spatial frequency acuity”

*f*

_{acuity}. The observers in Table 2 are ordered and color-coded by acuity, which is a handy index of the severity of amblyopic deficit: green for the mildest (spatial frequency acuity 14 to 15 c/deg), blue for moderate (9.5 to 13 c/deg), and red for severe (6 to 9 c/deg). Gray is the average of three normal observers, tested monocularly (acuity 17 c/deg).

*c*when the noise level is high (i.e.,

*c*>>

*c*

_{0}), as in our experiments. Equivalent noise can be expressed as . Thus efficiency is inversely proportional to the threshold in noise, , and the equivalent noise is proportional to the ratio of thresholds without and with noise, .

*s*

^{2/3}) (bottom scale). Threshold goes up as the letter gets smaller. The colored curves are amblyopes. The gray curve, for comparison, is the average of thresholds from three normal observers.

*η*

^{+}and equivalent noise

*N*

_{eq}. Figure 4A shows efficiency as a function of letter size. Recall that efficiency is inversely proportional to the threshold in noise, , so it is essentially an upside-down version of the threshold in noise graph (Figure 3B), showing a family of curves that are flat and near normal at large letter size and which fall off in a similar way for small letters, with more severe amblyopes (red) falling off sooner (at a larger letter size). Dividing by the normal efficiency (gray curve) and replotting the data as a function of normalized letter size (Figure 4B) reveals that, as we anticipated, normalizing by acuity shifts the curves into horizontal alignment. All but one of the amblyopes have similar efficiency deficit (re normal) as a function of acuity-scaled letter size. DS is the exception, lying about a factor of 3 to the left of the rest. Each observer’s efficiency re normal is low-pass: unimpaired for large letter sizes and dropping steeply for small letters approaching the acuity limit. All amblyopic observers showed near-normal efficiencies for large letters (> 4× acuity), and a marked loss of efficiency at small letter sizes, 1/4 at 2×, and approaching a factor of 1/16 as we near the acuity limit.

*less*than normal equivalent noise —

*quite surprising*! — and each higher degree of amblyopia has a higher equivalent noise level. We are surprised because we presume that the normal visual system is optimized and that a disease state would be worse, not better.

*N*

_{eq}is well described as the sum of cortical and photon noises (Pelli, 1990; Raghavan, 1995). At the retina, the cortical noise spectrum is

*f*

^{−2}, and the photon noise spectrum is frequency independent

*f*

^{0}, where

*f*is spatial frequency. Figure 5A is the observer’s equivalent input noise at the display. To apply their analysis we must estimate the equivalent input noise at the retina. To convert, we multiply by the contrast power of the eye’s optics

*H*

^{2}(

*f*). The optical MTF

*H*(

*f*) is the ratio of image contrast at the retina to the object contrast in the visual field.

*r*

_{acuity}for the whole pool of observers: the ratio of the cut-off frequency of the MTF (for incoherent light) to the spatial frequency acuity of the observer’s eye. After trying several values, we set

*r*

_{acuity}to 1.3, but it is not critical: Results are very similar for values of 1.1 and 1.5. This estimated MTF is shown in Figure 5B. Note that the MTF affects our calculations only at frequencies at which we have data (i.e., up to about half the acuity in c/deg).

*N*

_{photon}corresponding to 2% transduction efficiency at 555 nm, and an

*f*

^{−2}cortical noise

*N*

_{cortical}=

*k*

_{cortical}

*AT*with a power spectral density independent of luminance but proportional to letter area

*A*= (3/

*f*)

^{2}and duration

*T*. Under most conditions, either one noise or the other dominates, and that is true for most of our observers. However, for a few of our observers, our letters (45 cd/m

^{2}, duration

*T*= 0.2 s, size

*s*= 0.3 to 4 deg) straddle both domains. For larger letters the cortical noise dominates; for a few observers, for smaller letters, the photon noise dominates. We fit the Raghavan and Pelli model to our data, taking

*N*

_{photon}and

*k*

_{cortical}as degrees of freedom.

*N*

_{eq}data in Figure 5C. This reveals order that was hard to discern in the raw

*N*

_{eq}data of Figure 5A. Returning to Figure 5D, on the left of the plot,

*N*

_{eq}has a log-log slope of −2, because it is dominated by cortical noise

*f*

^{−2}, and on the right, for the normals and two amblyopes,

*N*

_{eq}has a log-log slope of zero, because it is dominated by photon noise

*f*

^{0}. At low spatial frequencies, where the slope is −2, the amblyopes are all near or above normal, indicating that they have higher-than-normal cortical noise. The low-frequency portion of the curve is very little affected by our admittedly crude MTF correction. Using a different approach, Levi and Klein (2003) reached a similar conclusion. At high spatial frequency, all but one of the amblyopes are below normal, indicating that their photon noise is below normal. All but 2 of the levels of cortical noise (i.e., on the left) are above normal. They span a range of 0.8 log units, and do not seem to be correlated with acuity (i.e., the different color-coded acuities are all jumbled up). The next figure offers a closer look.

*N*

_{cortical}and

*N*

_{photon}re normal. The scatter diagram in Figure 6 plots

*N*

_{cortical}re normal, one point per observer, as a function of the observer’s acuity. There is quite a bit of scatter across observers, especially among the most severe amblyopes, but the regression line through the data has nearly zero slope, indicating practically no correlation between log

*N*

_{cortical}and log acuity.

*N*

_{cortical}and

*N*

_{photon}are made after applying our estimate of the MTF (Figure 5B). However, the MTF is nearly 1 at low and mid spatial frequencies, varying significantly only at the high spatial frequencies. The estimate of

*N*

_{cortical}depends primarily on the low and mid frequencies, and thus is insensitive to the MTF, and the estimate of

*N*

_{photon}depends primarily on the high spatial frequencies and is thus extremely sensitive to the assumed MTF.

*Q*is the photon flux (i.e., retinal illuminance expressed as 555 nm quanta s

^{−1}deg

^{−2}), which is the product of luminance and pupil area. Raghavan and Pelli used an artificial pupil and found that the transduction efficiency is about 2% (at 555 nm in the fovea). The transduction efficiency estimate for our normal observers is 0.7%. We use the pupil size of 4.5 mm, measured on SC’s eye, but note that every observer will have a different pupil size and that this variation in retinal illuminance would explain the variance in

*N*

_{photon}and transduction efficiency.

*both*eyes of amblyopes show a reduced amplitude of the pupil light reflex. For patterned stimuli, the amblyopic eye’s pupil response amplitude is, on average, about 1.8× smaller than normal. That is in the right direction, but pupil response amplitude is a nonlinear function of luminance, so we could not calculate, from the Barbur et al. results, how much bigger than normal our amblyopes’ pupils were. Photon noise that is 0.4× normal would result from a pupil area 1/0.4 = 2.5× normal, or a pupil size √2.5 = 1.6× normal. We checked the plausibility of this increased-pupil-size explanation by measuring the pupil size of four normal and five amblyopic observers under the conditions of our experiment. The means are 4.9 mm (normal), 4.3 mm (amblyopic), and 3.9 mm (non-amblyopic) with a

*SE*of 0.4 mm. To account for the low photon noise estimate, the amblyopic pupil should have been much larger than normal, but in fact is insignificantly smaller. More precisely, the difference in log pupil size should have been 0.23, but the measured difference (mean ±

*SE*) in log pupil size between amblyopic and normal eyes is −0.05 ± 0.06. Having ruled out a sufficiently large increase in pupil size, we tentatively ascribe our low photon-noise estimate to underestimating the optical MTF at high spatial frequencies (Figure 5B). Using a small artificial pupil (with a correspondingly brighter display) would eliminate this uncertainty. Our assumed value for the MTF (Figure 5B) has a large effect on the estimated photon noise, but has little or no effect on the estimated cortical noise, which depends only on the low- and mid-frequency results, where the MTF is nearly 1.

*x*-height) and slightly increased cortical noise. These results are not included with those of the other observers because the artificial pupil resulted in a much lower retinal illuminance.

- Efficiency. All but one of our 16 amblyopic observers show the same low-pass loss of efficiency re normal as a function of spatial frequency re acuity, dropping to 0.5 at half acuity. Thus even our mildest amblyopes, with near-normal acuity, are qualitatively different from normal. Threshold for a small letter in noise is diagnostic.
- Cortical noise. On average, the amblyopes have 1.4 × higher than normal cortical noise, suggesting that, on average, only 1/1.4 the normal number of neural spikes are devoted to that eye, but there are large individual differences and some severe amblyopes have normal cortical noise. This is a very small effect, raising threshold contrast by a negligible factor of √1.4 = 1.2×.
- MTF. Noting that the contrast sensitivity functions of amblyopes and normals are alike once scaled by acuity, we suppose that the optical point spread is proportional to acuity. This provides a one-parameter account for the splaying of
*N*_{eq}for different degrees of amblyopia, and allows us to estimate the equivalent noise spectrum at the retina, which is much simpler than at the display. - Photon noise. The estimated photon noise for amblyopes is 2.5 × below normal, but is relevant only at the highest spatial frequencies. It seems unlikely that amblyopes transduce a larger-than-normal fraction of the photons incident on the retina, and we ruled out larger-than-normal pupils, so we tentatively ascribe our low photon-noise estimate to underestimating the optical MTF (Figure 5B) at the highest spatial frequency by a factor of √2.5 = 1.6×. This has no effect on the estimation of efficiency or cortical noise.

*s*

^{2/3}of the letter. The measured channel frequency follows the predicted channel frequency increases as nominal letter spatial frequency in both the non-amblyopic and amblyopic eyes (Figure 8A). This agrees with the relationship (gray line) found in normal observers (Majaj et al., 2002). While there are fairly large individual differences, the channel used by an amblyopic eye appears to have the same center frequency as that used in normal eyes.

*f*= 3/

*s*

^{2/3}, and this predicted (i.e., normal) channel frequency is displayed as the bottom scale of each graph. Figure 8A plots channel frequency. For non-amblyopic eyes (gray circles) the channel frequency is normal, within measurement error of the dotted line. For two of the five amblyopic eyes, RH and CB, the channel frequency is less than normal, but increases over the measured range, with no hint of saturating. For the other three, DS, AJ, and JB, the channel frequency does seem to hit a ceiling, as predicted by the last-channel hypothesis. However, the hypothesis predicts that the less-than-normal efficiency is a direct result of less-than-normal channel frequency. For RH and DS, efficiency (Figure 8B) drops precipitously while channel frequency (Figure 8A) continues to increase in parallel with the normal dotted line. For JB and AJ, efficiency is about half normal despite the fact that channel frequency is approximately normal. Thus, we cannot attribute the low efficiency for small letters of any of our observers to abnormally low channel frequency. Two recent studies have reached similar conclusions, for letters close to the acuity limit (Hess, Dakin, Tewfik, & Brown, 2001) and for letters up to about 20 times the acuity limit (Chung, Levi, Legge, & Tjan, 2002b).

- Measure threshold in noise (Figure 3B). Unlike 1., above, simple acuity scaling fails. Compute efficiency (Figure 4). Efficiency re normal does scale (i.e., is the same low-pass function of letter size re acuity) among amblyopes, though there is a difference between amblyopes and normals (Figure 4B).
- Compute the equivalent input noise (Figure 5A). This exposes what is left unaccounted for in the no-noise thresholds after factoring out the efficiency loss. This graph is bewilderingly complicated: the curves tend to hook upward and splay out at high spatial frequencies.

- Efficiency re normal of amblyopes is a low-pass function of letter size re acuity, 1 for large letters (> 4× acuity), dropping to 1/4 at 2× acuity, and approaching 1/16 at acuity.
- Cortical noise is roughly 1.4× normal (Figure 6B). This raises threshold contrast without noise by roughly √1.4 = 1.2× at all frequencies at which cortical noise dominates. Cortical noise dominates at low spatial frequencies; photon noise dominates at high spatial frequencies.

^{2}luminance), but, even so, we do not understand why their amblyopes should differ so much more than ours. Note that 2 of their 3 amblyopes had extremely poor acuity (1/60 and 2/60), well outside the range of our observers and the range of acuities that are typical of amblyopia (Ciuffreda et al., 1991).

Theory of Pelli-Levi Dual Acuity Chart

The goal is to detect observers with lower-than-normal efficiency for small letters. The right side of the chart is at a fixed signal-to-noise ratio *E/N* = *R* that was chosen to be just above normal threshold, just barely readable by an observer with normal efficiency. It is below threshold (i.e., unreadable) for an observer whose efficiency is much lower than normal. In general, the observer’s efficiency and equivalent noise are both letter-size-dependent. The chart noise is high at the top and drops from line to line. For optical reasons discussed in the text, the observer’s equivalent noise increases for the smallest letters. The observer’s equivalent noise adds to the chart, and at some line is equal, in effect, to the noise on the chart. That *dividing line* is the key to understanding this chart. The triangular diagram above shows this horizontal dividing line and specifies the chart’s effective signal-to-noise ratio above and below it. Our analysis is all relative to the location of this dividing line, whose location on the chart depends on the observer and viewing distance, but the observer’s final score does not depend on where the dividing line is because, in the end, we consider only the difference between left and right acuities. In the *upper* part of the chart, above the dividing line, the chart noise is absent on the left and dominates on the right, producing an effective signal-to-noise ratio of *R* on the right, and greater than *R* on the left; in the *lower* part, below that line, the observer’s equivalent noise dominates both left and right sides, producing practically the same effective signal-to-noise ratio, less than *R*, on both sides. Thus the left and right sides are functionally different in the upper part and functionally identical in the lower part. One measures acuity by finding the smallest letter size at which the letters can be read (i.e., at which the chart’s effective signal-to-noise ratio is above the observer’s threshold). Any observer with normal efficiency for large letters will be able to read the top line, and thus will have a measurable acuity on both sides of the chart. Observers with normal efficiency for small letters will progress further down in noise (right side) than less efficient observers. If the in-noise acuity is below the line, then the no-noise acuity will be the same, because there’s effectively the same noise level on left and right. If the observer’s in-noise (right side) acuity is above the dividing line, then the no-noise (left side) acuity will be better, because there is less noise on the left. Thus the chart noise (right side) worsens the acuity of observers with low efficiency for small letters, but has no effect on the acuity of observers with normal efficiency for small letters. This acuity difference, occurring only for observers with low efficiency for small letters, is proposed as a diagnostic test for amblyopia. The acuity difference is relatively immune to optical blur, as increasing the blur simply increases the equivalent noise, shifting the dividing line up, with little effect on the left-right acuity difference.