Amblyopia results in a loss of visual acuity, contrast sensitivity, and position acuity. However, the nature of the neural losses is not yet fully understood. Here we report the results of experiments using noise to try to better understand the losses in amblyopia. Specifically, in one experiment we compared the performance of normal, amblyopic, and ideal observers for detecting a localized signal (a discrete frequency pattern or DFP) in *fixed contrast* white noise. In a second experiment, we used *visibility-scaled noise* and varied both the visibility of the noise (from 2 to 20 times the noise detection threshold) and the spatial frequency of the signal. Our results show a loss of efficiency for detection of known signals in noise that increases with the spatial frequency of the signal in observers with amblyopia. To determine whether the loss of efficiency was a consequence of a mismatched template, we derived classification images. We found that although the amblyopic observers' template was shifted to lower spatial frequencies, the shift was insufficient to account for their threshold elevation. Reduced efficiency in the amblyopic visual system may reflect a high level of internal noise, a poorly matched position template, or both. To analyze the type of internal noise we used an “N-pass” technique, in which observers performed the identical experiment *N* times (where *N* = 3 or 4). The amount of disagreement between the repeated trials enables us to parse the internal noise into random noise and consistent noise beyond that due to the poorly matched template. Our results show that the amblyopes' reduced efficiency for detecting signals in noise is explained in part by reduced template efficiency but to a greater extent by increased random internal noise. This loss is more or less independent of external noise contrast over a log unit range of external noise.

*N*

_{eq}), is shifted to the right in the amblyopic eye, suggesting increased additive noise. A rightward shift is associated with a larger threshold elevation at low than at high noise levels. In the lower panel, we have replotted the data with the noise specified in multiples of the noise detection threshold (i.e., noise threshold units or NTU). Plotted in this way, the curves are parallel, and

*N*

_{eq}is almost identical in the two eyes at approximately 1; that is, noise has little impact on performance until it is visible, a result that we find to be quite common. We will return to this point and its implications for the equivalent noise model in the Discussion section.

*additive*noise (Levi et al., 2007).

Observer | Age (years) | Gender | Strabismus (at 6 months) | Eye | Refractive error | Line letter acuity (single letter acuity) |
---|---|---|---|---|---|---|

Strabismic | ||||||

MR | 22 | F | L EsoT 14 ^{Δ} | R | −3.25/−2.50 × 175 | 20/20 ^{+2} |

L | −3.25/−3.25 × 175 | 20/32 ^{−1} | ||||

JT | 52 | F | L EsoT 5 ^{Δ} | R | −1.00/−0.50 × 10 | 20/16 ^{+2} |

L | −0.75/−0.50 × 90 | 20/63 ^{−1} (20/25 ^{−2}) | ||||

SF | 20 | F | L EsoT 6-8 ^{Δ} | R | −2.75/−0.25 × 90 | 20/20 ^{+2} |

L | −2.00 | 20/50 (20/40 ^{−1}) | ||||

RH | 32 | M | L EsoT 2 ^{Δ} | R | −1.00/−0.50 × 170 | 20/15 |

L | −1.50/−1.50 × 10 | 20/59 | ||||

DH | 20 | M | L EsoT 5 ^{Δ} | R | +0.25/−0.50 × 100 | 20/15 |

L | +0.25/−1.25 × 80 | 20/200 | ||||

AH | 19 | F | R EsoT 12 ^{Δ} | R | +0.50/−1.00 × 93 | 20/50 |

L | pl/−0.75 × 100 | 20/25 | ||||

Anisometropic | ||||||

JW | 22 | F | None | R | +1.75 | 20/80 ^{−2} (20/80 ^{+1}) |

L | −2.00 | 20/20 | ||||

SC | 27 | M | None | R | +0.50 | 20/16 ^{+2} |

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

VG | 31 | M | None | R | +4.25/−4.00 × 03 | 20/50 ^{+2} |

L | −0.25/−1.50 × 177 | 20/20 ^{+2} | ||||

MLR | 44 | F | None | R | +4.00/−1.00 × 31 | 20/125 ^{−2} (20/100) |

L | +0.75 | 20/20 | ||||

RJ | 53 | M | None | R | +0.50/−0.25 × 95 | 20/14 |

L | +2.50/−0.75 × 125 | 20/57 | ||||

AM | 22 | F | None | R | +2.50/−1.00 × 10 | 20/50 |

L | −0.25 | 20/20 | ||||

Strabismus and anisometropia | ||||||

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

L | −2.00 | 20/16 ^{−2} | ||||

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

L | +5.00 | 20/125 (20/125+2) | ||||

AW | 22 | F | R EsoT 4–6 ^{Δ} & hypoT 4 ^{Δ} | R | +2.75/−1.00 × 160 | 20/80 ^{−1} (20/50 ^{−1}) |

L | −1.00/−0.50 × 180 | 20/16 ^{−1} | ||||

DM | 40 | F | L ExoT 3 ^{Δ} | R | +0.50/−0.25 × 92 | 20/20 |

L | +2.50/−1.0 × 160 | 20/80 | ||||

DS | 26 | M | R EsoT 8 ^{Δ} | R | +2.25 DS | 20/40 |

L | +0.50 DS | 20/20 |

*c*cos

^{10}(

*π*y) cos(2

*π*6y) (see top curve of Figure 3b) and composed of 11 harmonics all added in cosine phase (see top curves of Figures 3a and 3b for the spatial frequency and spatial profiles). Our choice of cos

^{10}(

*π*y) as the envelope was to limit the signal to eleven harmonics. The noise is a one-dimensional grating consisting of the same 11 harmonics with phases and amplitudes randomized with each harmonic having equal variance. In Experiment 1, the standard deviation of each noise component had a fixed contrast of 0.04 (see Levi & Klein, 2002), and the DFP and the noise contained spatial frequencies from 1 to 11 c/deg. In the second experiment, we used visibility-scaled noise (described below), and the stimuli were viewed from one of three viewing distances so that they provided three ranges of signal and noise spatial frequencies (0.5 to 5.5 c/deg; 1 to 11 c/deg; 2 to 22 c/deg). The target and the noise (inset in Figure 1) were presented for 0.75 s, in a square field with a mean luminance of 42 cd/m

^{2}with a dark surround.

*d*′—which is a measure of the observers' signal-to-noise ratio) and linear regression to compute the classification coefficients (Levi & Klein, 2002, 2003). The signal to be detected was either a blank (0 contrast) or one of three near-threshold stimuli, and the observer responded with numbers from 1 (confident the signal was a blank) to 4 (highest perceived contrast). Observers were given auditory feedback following each trial. The contrast levels were chosen to yield

*d*′s between approximately 0.5 and 2, based on preliminary trials. Thresholds, corresponding to the contrast at which

*d*′ = 1, were determined by fitting a transducer (power) function to the

*d*′ vs. contrast data using Matlabs lsqnonlin. One advantage of our rating-scale method is that it enables us to collect data over a reasonable range of

*d*′ values.

*N*= 2 (twice) or 4. However, we found that

*N*= 3 provided approximately equal reliability with fewer trials, and most of the response consistency data reported here was with

*N*= 3 passes.

*q*

^{2}that is the ratio of consistent response variance to total response variance (discussed in detail in Levi et al., 2007). We use

*q*

^{2}rather than

*q*as a reminder that we are discussing ratios of variances. The first method involves finding a best fit to the data in terms of the estimated

*d*′ values for the different stimulus levels and the placement of criteria. To estimate the correlation,

*q*

^{2}, we replaced the assumptions of independent Gaussian noise on each pass with bivariate Gaussian noise that was correlated across pairs of passes. The correlation was a free parameter that was varied to get the best fits to the double pass data. All pairings of multiple passes were examined, and the

*q*

^{2}from each pairing was averaged. The search for the optimal

*q*

^{2}was done with the

*d*′ and criteria constrained to their best values.

*q*

^{2}was much simpler. We simply calculated the standard correlations of the responses for all pairs of passes,

*p*. This analysis was done separately for each stimulus level. We found that the two methods gave similar estimates of

*q*

^{2}, so our estimates of the ratio of consistent to total noise are based on the average of the two methods.

*internal*noise may depend upon the visibility of the external noise.

- At each scale, the amblyopic eye thresholds are uniformly elevated, and this elevation is generally most marked at the higher spatial frequencies. For example, JD (blue circles) shows about a 2.5-fold loss (relative to normal controls) at 0.5–5.5 c/deg, a ≈6-fold loss at 1–11 c/deg, and an 8-fold loss at 2–22 c/deg. Note that many of the amblyopes could not be tested at the highest spatial frequency range because they needed more contrast to reach threshold than we could generate. At each spatial scale, the uniform threshold elevation (i.e., thresholds are elevated about the same amount at each noise level) implies that the loss is at least in part due to multiplicative noise. In the sections that follow, we explore to what extent this noise contrast independent loss is due to a mismatched template or to other sources of noise.
- The kink in the curves representing the additive equivalent internal noise (
*N*_{eq}) occurs at about the noise detection threshold (≈1 NTU) for all observers. Figure 1 (lower panel) shows an example of this. The mean value of*N*_{eq}was 1.18 ± 0.04 NTU (*SD*= 0.23NTU), based on the 29 fits (12 amblyopic eye fits, 5 normal eye fits, and 12 non-amblyopic eye fits—note that Figure 5 shows only fits to the mean normal and non-amblyopic eye data). This finding, that the noise becomes effective as a masker when it becomes visible, is not terribly surprising, but it was not inevitable and surprisingly, it has not been pointed out before. We will return to this point in the Discussion. - At the two highest spatial frequencies, thresholds for the non-amblyopic eyes were slightly elevated relative to the normal controls.

*identical*noise sequences shown in a randomized order, we estimated

*q*

^{2}, the ratio of consistent to total noise power. The amount of response disagreement between the

*N*tests allows the system's total noise to be parsed into random noise that is independent across multiple presentations of the identical stimulus and noise that is consistent (100% correlated) across multiple presentations (see 1 for details).

*q*

^{2}increases from about 40% to 60%, as external noise contrast increases from 2 to 20 times threshold. While some of the amblyopes show similar increases in the lowest spatial frequency range, at the higher frequencies both the non-amblyopic eyes (open squares) and the amblyopic eyes show much lower values of

*q*

^{2}, particularly as noise contrast increases. This indicates that the amblyopic visual system has increased random noise (relative to normal). Note that at 0 NTU,

*q*

^{2}must fall to zero. The small symbols in Figure 13 show the contribution (if any) of consistent noise to human performance (beyond the effects of consistent noise resulting from the template). Consistent noise results in little or no loss beyond the mismatched template, so the small symbols are rarely seen. The negligible role for consistent noise other than a mismatched template rules out the possibility that there are unusual suppression effects whereby some inappropriate spatial frequencies or phases disrupt the visibility of a DFP in a consistent but nonlinear manner across multiple passes.

*N*

_{eq}≈

*N*

_{th}can be used to eliminate the class of models in which the independent additive noise follows multiplicative noise. Suppose the amblyopic loss is purely multiplicative. In that case, thresholds would be elevated at large

*N*

_{ext}. Assuming the multiplicative noise comes before the additive noise thresholds would be untouched at low

*N*

_{ext}. The result is that the kink point,

*N*

_{eq}of the TvN curve would shift leftward (see Equation 3). However, given that

*N*

_{eq}is tied to

*N*

_{th}that shift is to the right for degraded amblyopic vision. This simple logic is a problem for models with independent late additive noise such as the PTM model. One can always rescue the model by saying that the late additive noise is not independent and must increase as the multiplicative noise increases, in which case it is indistinguishable from a model with early additive noise. In a model with independent early additive noise, an increase in multiplicative noise, such as a mismatched template, lifts the log–log TvN plot vertically. Additional additive noise would increase the TvN floor and thereby shift

*N*

_{eq}rightward as would be expected from a concomitant increase in

*N*

_{th}.

*lowering*noise detection thresholds, the early noise model may be preferable. For now we remain agnostic on this issue.

*N*

_{ext}the external noise, and

*d*′ is the signal detection measure of detection. The contrast response function ICRF(

*d*′) (inverse contrast response function) specifies how the threshold depends on the

*d*′ that is chosen for defining threshold. The special feature of the class of models specified by Equation 1 is that the dependence on

*N*

_{ext}and

*d*′ is separable. Although later in this discussion, we will show that our data provide strong evidence against the separability shown in Equation 1, this class of models provides a very useful framework for discussing TvN data.

*N*

_{eq}, Th

_{0}, and

*p*, are evident:

*N*

_{eq}, the equivalent input noise, shifts the TvN curve horizontally on log–log axes. This parameter is has been popularized by Barlow (1957) and Pelli (1990). The threshold parameter Th

_{0}and the associated ICRF shifts the curve vertically on log–log axes. We typically normalize the CRF(

*d*′) function as CRF(1) = 1 so that Equation 1 gives

_{0}is the threshold for zero external noise at

*d*′ = 1. For

*d*′ values other than unity, the ICRF simply shifts the TvN curve vertically on log–log axes without changing the shape of the function. The Minkowski exponent,

*p*, has a slight effect on the shape of the TvN function when

*N*

_{ext}is close to

*N*

_{eq}, but the effect is so small that an unreasonable amount of data is needed to pin it down.

*p*, is coupled to the transducer exponent of the CRF. Another version of Equation 1 is the linear uncertainty model (LUM) of Eckstein et al. (1997) in which the pooling exponent is

*p*= 2 and is decoupled from the CRF. As stated above, it is very difficult to pin down the pooling exponent,

*p*, based on the shape of the TvN function. Lu and Dosher provide evidence against a version of LUM with a linear CRF (the linear amplifier model, LAM). However, it is well known that the inverse contrast response function is nonlinear and is approximately ICRF(

*d*′) =

*d*′

^{1/2}for low values of

*d*′ and it accelerates above

*d*′ = 1 (Nachmias & Sansbury, 1974; Stromeyer & Klein, 1974). This behavior of the inverse CRF (contrast as a function of

*d*′) is the inverse behavior of the usual contrast response function where

*d*′ is a function of contrast.

*N*

_{eq}a parameter from the LUM. From the numerator of Equation 16 of Lu and Dosher (1999), we see that

*N*

_{eq}can be written in terms of PTM parameters as

*N*

_{A}is the magnitude of the PTM additive noise,

*N*

_{M}is the magnitude of the PTM multiplicative noise, and the exponent

*p*= 2

*γ*where

*γ*is the power of the nonlinear stage of the PTM. Note that Lu and Dosher use the same convention we do with

*N*standing for noise contrast rather than noise power (for Pelli, on the other hand,

*N*means power).

*p*of Equation 1 with

*p*= 2 since that Minkowski exponent is very hard to pin down from the shape of the TvN curve. We have also added a consistent noise term, arbitrarily chosen to add linearly, that may contribute, especially at low levels of external noise (Levi et al., 2005). Further clarification of Equation 4 with plots is presented in 1.

*d*′ to stimulus contrast, also called the ICRF in Equation 1) for each observer and condition in the present study. We found no significant or systematic differences across the three spatial frequency ranges, so we averaged them. These exponents are plotted as a function of noise contrast (in NTU) in Figure 15. While the exponents are substantially higher (≈2.5) when there is no noise, in nonzero external noise the exponent is ≈1.3–1.6 and shows only a slight change as external noise increases from 2 to 20 NTU. The sharp decrease in the CRF exponent when noise is present violates the separability of the class of models (PTM, LUM) specified by Equation 1.

*q*

^{2}(the ratio of consistent to total noise power) for detection (abscissa) versus

*q*

^{2}for position.

*q*

^{2}for both tasks are high and approximately equal (≈0.6). The proportion of random noise under these conditions (top and right axes) is roughly 0.4. However, as DFP spatial frequency increases (smaller open circles) or noise contrast is reduced, the proportion of random noise increases (i.e.,

*q*decreases) for both tasks, but more for the position task and the data fall well below the 1:1 line (thick dotted line), indicating that random noise is larger for position discrimination than for detection at low external noise levels and at high target spatial frequencies, as would be predicted by the noisy template model outlined above. For many of the amblyopic cases, the increased random noise, especially for position discrimination (where it exceeds 80%) is obvious.

*q*

^{2}, indicating increased random noise, particularly for high spatial frequencies.

*both*the amblyopic and the preferred eyes suggests that this elevated random noise is central and is likely related to the absence of correlated binocular visual experience early in life (Kind et al., 2002; McKee, Levi, & Movshon, 2003).

*a*

_{I}is obtained by a mathematical ideal observer calculation. For detection of signals in white noise (the present situation given our stimuli), the matched filter template is ideal so the calculation is quite simple (Levi & Klein, 2002). The value

*a*

_{T}is given by

*a*

_{T}=

*a*

_{I}/

*r*, where

*r*is the correlation between the ideal template and the human template. The values of

*a*

_{R},

*a*

_{0}, and

*a*

_{C}are obtained from measuring the human threshold and from measuring the ratio of consistent to total noise in a multi-pass experiment. The human threshold at zero external noise is a measure of

*a*

_{0}. The parameter

*a*

_{C}is determined by ratio C(

*N*

_{ext}) / H(

*N*

_{ext}) =

*q*(

*N*

_{ext}), where

*q*

^{2}is the correlation between responses of the two passes of a double pass experiment, as discussed in the text. The forms of the “black box” curves in Equations A1c and A1d were chosen for their simplicity and their compatibility with the data rather than because of any deeper principle.

*a*

_{cons}=

*a*

_{C};

*N*

_{eq}=

*a*

_{T}

^{2}+

*a*

_{R}

^{2}.

*N*

_{ext}, the human threshold is

*a*

_{0}= 1. At asymptotically large

*N*

_{ext}, the human threshold is (

*a*

_{I}

^{2}+

*a*

_{R}

^{2})

^{0.5}= 0.141

*N*

_{ext}. Thus, the asymptotic behavior is captured by the function Thresh =

*a*

_{0}(1 + (

*a*

_{I}

^{2}+

*a*

_{R}

^{2})

*N*

_{ext}

^{2})

^{1/2}=

*a*

_{0}(1 +

*N*

_{ext}

^{2}/

*N*

_{eq}

^{2})

^{1/2}, where

*N*

_{eq}= 0.02

^{−1/2}= 7.1 is the equivalent input noise. Thus, the kink point of the TvN curve of Figure A1a is defined to be at an external noise square of 1/ (

*a*

_{I}

^{2}+

*a*

_{R}

^{2}) = 50. As discussed in the text,

*N*

_{eq}has been found to be very close to the threshold for seeing the noise. We have found it useful for many of our plots to have the abscissa expressed in noise threshold units. That effectively places

*N*

_{eq}= 1. Panel b of Figure A1 is identical to panel a, except that the

*x*axis has been shifted by a factor of 50 so that the abscissa is expressed in NTU

^{2}units (noise threshold units squared). Panels c and d are also expressed in NTU (but not squared).

*N*

_{ext}regime more visible. For example, the slightly shallower slope of the consistent noise contribution becomes visible. The log–log axes have the disadvantage of masking the near linearity of TvN functions when plotted as threshold squared vs. noise squared as in panels a and b. Panel d is similar to panel c except that it is root efficiency that is on the ordinate. Root efficiency is the ratio of the thresholds to the ideal observer's threshold. It can be seen that above 2NTU, the root efficiencies are approximately constant.

*N*

_{ext}and the ideal and template observers, the rms power is easily obtained because there is a direct connection between the template output (an ideal template for the ideal observer) and

*d*′. This is so even if the task is one of orientation or spatial frequency discrimination because these can be measured by the output of the appropriate set of contrast domain templates just as is possible with contrast discrimination. The human observer response is obtained by measuring the signal strength (threshold) that gives

*d*′ = 1.

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