Following six consecutive days of repeated testing (at least 900 trials per day), five of our seven observers showed a sizeable improvement (a reduction in contrast threshold) in their letter identification performance at the trained eccentric retinal location (10° eccentricity in the inferior visual field). This finding is consistent with that of Chung et al. (2004)
in which percent-correct performance of letter identification at 10° eccentricity was shown to improve with training.
showed that peripheral Landolt C acuity does not benefit from training. While these results may seem to be at odds with ours, there are several reasons why they might be expected to differ. First, we measured contrast thresholds for identifying letters from amongst a large array (26); a more demanding task than identifying the orientation of a C. Second, our letters were a fixed size, about twice the acuity limit. Because the slope of the high spatial frequency limb of the contrast sensitivity function is very steep, a 20% change in sensitivity (i.e., a 20% change along the contrast axis) translates into a very small (≈6.4%) change along the size (spatial frequency) axis. Thus, even if learning occurred, it would be expected to produce only very tiny changes in acuity. Third, it is also possible that Westheimer did not see learning because the size of the stimulus (and therefore the visual information (i.e. template) needed to perform the task) was the parameter used to increase task difficulty. This would make it difficult to learn a specific template since it would be changing over time. Finally, peripheral acuity is likely to be limited by anatomical constraints of the retina, e.g. the density of photoreceptors and the convergence of photoreceptors onto ganglion cells (e.g. Levi, Klein, & Aitsebaomo, 1985
), therefore, improvement in acuity might not be possible because the retinal anatomy is unlikely to change with training.
Recently, Dosher and Lu (2004)
examined whether perceptual learning at the fovea occurred at the first or second stage of visual processing. They measured contrast thresholds for discriminating the letter K from its mirror images for first (luminance-defined) and second- order (texture-defined) stimuli embedded in external Gaussian noise following five days of training. For first-order stimuli, they reported no improvement in contrast threshold. However, for second-order stimuli, there was a reduction (improvement) in contrast threshold, with the magnitude of improvement similar to the average magnitude reported in the present paper. Dosher and Lu argued that their results hint at a site for perceptual learning to be located at the second (nonlinear) stage of visual processing, at least for foveal tasks. Within the general context of a linear–nonlinear –linear model, our results suggest that for peripheral (first-order) letter identification task, learning is possible at the first (linear) stage of visual processing. Because nonlinearities can be combined to form a linear system, our results, although parsimoniously modeled by linear factors alone, do not guarantee the site of learning must be at a linear stage of visual processing, nor do they guarantee that the physiological site of learning must be early. Although differences in perceptual learning between the fovea and periphery (e.g. Chung, 2002
) may be due to difference in the learning sites, they may simply reflect the fact that foveal letter identification is over-trained through years of reading, while peripheral vision is inexperienced in this task. In other words, with regard to letter-identification, peripheral vision may be amblyopic.
We found that the improvement in contrast threshold for letter identification following training occurs almost uniformly across all levels of external noise. To derive the nature of the processes underlying the improvement, we applied the LAM to analyze our data. Analysis using the LAM shows that training leads to an increase in the relative sampling efficiency, but no changes to the equivalent internal noise. As a result, the optimal contrast threshold was reduced. These findings are very similar to those reported by Gold et al. (1999)
, who showed that learning to identify faces and random texture in the fovea is due to an increase in the sampling efficiency 2
but not a reduction in the equivalent noise. Our results are also similar to the changes seen in foveal position discrimination following learning (Li et al., 2004
). Like Gold et al. (1999)
, Li et al. found that practice improved performance more or less uniformly across (positional) noise levels––consistent with an improvement in sampling efficiency. In a second experiment, Li et al. (2004)
measured the observer’s perceptual template using the classification image technique, and found that learning re-tuned the observer’s template (i.e., it became more ideal) resulting in improved sampling efficiency. Here, using a different task (letter identification) and in a peripheral retinal location (10° eccentricity inferior visual field), we also attribute learning to an increase in the sampling efficiency. In plain words, an increase in sampling efficiency means that observers’ template for the task becomes closer to that of the ideal observer, so that the template is more capable of extracting the crucial information from the signal.
4.1. The perceptual template model
The LAM analysis, although popular (e.g. Gold et al., 1999
; Legge, Kersten, & Burgess, 1987
; Tjan, Braje, Legge, & Kersten, 1995
), may be an over simplification. First, when threshold contrast energy is plotted as a function of external noise variance, the function is assumed to be a straight line. This restriction, however, has very little impact on our data since the deviation from linearity in our results was slight, and when presented, concentrated within the low external noise regime. Second, the LAM analysis is criterion-dependent, thus the changes in sampling efficiency and equivalent internal noise as analyzed by the model are also criterion- dependent. To ascertain that our qualitative findings are not specific to the criterion used, we reanalyzed our data using two approaches. First, we reanalyzed our data using LAM for thresholds specified at two other criteria––d
′ of 1.0 and 2.9, corresponding to 20% and 80% correct performance (after correction for guessing) on our psychometric functions. Analyses using these two criteria yielded qualitatively similar results (Tjan, Chung, & Levi, 2002
). Second, we reanalyzed our data using the perceptual template model
(PTM: Dosher & Lu, 1998
; Dosher & Lu, 1999
; Lu & Dosher, 2004
). The PTM extends LAM to include a nonlinear transducer function and a stimulus-dependent noise component. The added machinery allows us to fit the data at more than one criterion simultaneously. Details of this model can be found elsewhere (Dosher & Lu, 1998
; Dosher & Lu, 1999
; Lu & Dosher, 2004
). In brief, PTM attributes the improvement as a result of learning to three mechanisms in isolation or in combination: stimulus enhancement, external noise exclusion and internal multiplicative noise suppression (). According to the model, if learning is a consequence of stimulus enhancement
, which is equivalent to turning- up the gain of the perceptual template (filter) or equivalently reducing the internal additive (stimulus-independent
) noise, then it will be reflected as an improvement primarily at low noise levels. With respect to the model, there will be a reduction in the parameter Aa
(proportional change in stimulus-independent internal noise) following learning. Another possibility is that learning results from external noise exclusion, which simply means that the perceptual template becomes more appropriately tuned to include only the signal, thus eliminating the irrelevant noise in the stimulus. In this case, the improvement due to learning will occur primarily at high noise levels. Accordingly, parameter Af
(proportional change in the effective external noise perceived by the observer) will become smaller in value following learning. The third possibility for an improvement in performance is a result of a reduction in the internal multiplicative (stimulus-dependent) noise, which will lead to improvement at both low and high noise levels. If so, then the parameter Am
(proportional change in stimulus-dependent internal noise) will be reduced following learning. The following is the mathematical description of the model:
is a proportional constant, γ
is the transducer nonlinearity, Nm
are the variance (or spectral density) of the multiplicative (stimulus-dependent) and the additive (stimulus-independent) internal noise, respectively.
Fig. 8 A schematic figure showing the perceptual template model (PTM), and how practice can lead to an improvement in performance by modifying the various components in the model. The bottom panels illustrate the three scenarios (contrast threshold vs. external (more ...)
When applying the PTM to analyze our data, the curve-fit to our plots of contrast threshold vs. external noise contrast was in general, very good. compares the goodness-of-fit of the curves fitted by LAM and PTM, taking into account the different degrees of freedom in each model.
To summarize the major findings of the PTM analysis (), we found that learning leads to a reduction in both the ratios Aa
(internal additive noise: t(df=6)
= 3.78, p
= 0.009) and Af
(external additive noise: t(df=6)
= 8.24, p
= 0.0002), but not Am
(internal multiplicative noise: t(df = 6)
= 0.05, p
= 0.96). These findings are identical to those of Dosher and Lu (1998
even though the task was different (they used an orientation discrimination task). According to the model, a reduction in Aa
following training is consistent with an improvement in observers’ ability to enhance the stimulus by reducing the additive
internal noise. A reduction in Af
following training means that observers are more capable of excluding the external noise in the stimulus, by fine-tuning the shape of the perceptual template. Therefore, even with the PTM which is criterion-free, we obtained the same conclusion for our data on learning to identify letters in peripheral vision––that the improvements in learning is primarily attributable to an increased in the observers’ ability to extract and use the relevant information in the stimulus. However, we also found a reduction in Aa
following learning, which according to the model, is consistent with an improvement in observers’ ability to enhance the stimulus by reducing the additive
internal noise. This seems to contradict the results from the LAM and the double-pass analyses, which suggested no changes in internal noise. As we shall elaborate next, this apparent contradiction is superficial.
Fig. 9 The three parameters of PTM are plotted as a function of training day: Aa (top), Af (middle) and Am (bottom). Data are shown for each individual observer. Thick curves drawn through the data points represent the values averaged across the seven observers. (more ...)
4.2. Differences between LAM and PTM
According to LAM, there is no change in the equivalent noise (assumed to be additive), whereas the PTM analysis leads to the conclusion that there is a reduction in the additive
internal noise (but not the multiplicative noise). As has been shown elsewhere (Gold, Sekuler, & Bennett, 2004
; Tjan et al., 2002
), this difference in results is superficial and arises solely because of the relative placement of the various components in the two models ––an argument we shall restate here.
The position of the equivalent noise component (Neq) in the LAM () and the additive internal noise component (Na) in the PTM () are not the same in their respective models; therefore, these two components cannot be treated as identical. The traditional formulation of LAM places the internal noise at the stimulus. This is why it is referred to as the “equivalent noise”––a noise if added to the stimulus is equivalent to the noise internal to an observer. In this formulation, sampling efficiency corresponds to the fraction of the net signal-to- noise ratio (SNR) utilized by an otherwise ideal observer to make perceptual decisions, where net SNR equals signal energy (E) divided by the sum of the spectral densities of the external noise (Next) and the equivalent internal noise (Neq). That is, the effective SNR utilized by an observer is equal to sampling efficiency (η) times net SNR, or, effective SNR = η(E/(Next + Neq)). Because the noise and the sampling processes in LAM are linear operators, their relative positions can be swapped relative to each other. Therefore, an equivalent formulation of LAM is that a fraction (equal to the sampling efficiency) of the stimulus SNR is passed on to the observer. The observer’s internal noise, which shall be denoted as Nint to distinguish it from Neq, is then added to the sampled input. Thus, in this formulation, the effective SNR is equal to E/(Next/η + Nint). The difference between these two formulations of LAM is illustrated in . Mathematically, the two formulations are identical up to a change of the variable (Nint = Neq/η), and no empirical test can distinguish between the two. However, since Nint = Neq/η, if sampling efficiency increases but no change is found in equivalent noise according to the first formulation, then the internal noise of the second formulation will show an increase by a ratio precisely equal to the reciprocal of the ratio of change in sampling efficiency. Refer to , the second formulation of LAM (panel b) resembles that of PTM () in that sampling (template operation) precedes internal additive noise. That is, the reason that PTM shows a decrease in internal additive noise (Na) is because this component corresponds to Nint in the second formulation of LAM, which decreases when sampling efficiency increases if Neq is held constant.
The two formulations of the LAM, with different placements of the internal noise relative to the sampling process.
Although there are similarities in conclusions drawn from the analyses presented above, there are also substantial differences between PTM and LAM from a model point of view. PTM differs from LAM (either formulation) because of its non-linear components (nonlinear transducer function and internal multiplicative noise). With these non-linear components, it is not possible to have an equivalent formulation of PTM by moving its additive internal noise component before the template computation to provide a component closely resembling Neq
in the first formulation of LAM. Also, PTM models the entire psychometric function and thus its conclusions (reduction in Aa
, no change in Am
) can be generalized qualitatively and quantitatively to all performance criteria (Tjan et al., 2002
; Lu & Dosher, 2004
). In contrast, the conclusion based on LAM (increase in sampling efficiency, no change in equivalent noise) is quantitatively true only at the criterion tested. There is a restricted set of conditions that if met, will allow conclusions from LAM to generalize qualitatively to other criterion levels as well. The set of conditions and their derivations are outside the scope of the present paper.
Despite the fundamental differences between LAM and PTM, both models imply that the mechanism underlying perceptual learning of letter identification in peripheral vision is a consequence of the template (or filter) becoming more capable of extracting the crucial information from the stimulus.