Intrinsic uncertainty explains second responses

Spatial summation for motion detection

We used the psychophysical summation paradigm to reveal some spatial characteristics of the mechanism responsible for detecting a motion-defined visual target in central vision. There has been much previous work on spatial summation for motion detection and direction discrimination, but none has assessed it in terms of the velocity threshold or used velocity noise to provide a measure of the efficiency of the velocity processing mechanism. Motion-defined targets were centered within square fields of randomly selected gray levels. The motion was produced within the disk-shaped target region by shifting the pixels rightwards for 0.2 s. The uniform target motion was perturbed by Gaussian motion noise in horizontal strips of 16 pixels. Independent variables were field size, the diameter of the disk target, and the variance of an independent perturbation added to the (signed) velocity of each 16-pixel strip. The dependent variable was the threshold velocity for target detection. Velocity thresholds formed swoosh-shaped (descending, then ascending) functions of target diameter. Minimum values were obtained when targets subtended approximately 2 degrees of visual angle. The data were fit with a continuum of models, extending from the theoretically ideal observer through various inefficient and noisy refinements thereof. In particular, we introduce the concept of sparse sampling to account for the relative inefficiency of the velocity thresholds. The best fits were obtained from a model observer whose responses were determined by comparing the velocity profile of each stimulus with a limited set of sparsely sampled "DoG" templates, each of which is the product of a random binary array and the difference between two 2-D Gaussian density functions.

A unified framework for perceived magnitude and discriminability of sensory stimuli

The perception of sensory attributes is often quantified through measurements of sensitivity (the ability to detect small stimulus changes), as well as through direct judgments of appearance or intensity. Despite their ubiquity, the relationship between these two measurements remains controversial and unresolved. Here, we propose a framework in which they arise from different aspects of a common representation. Specifically, we assume that judgments of stimulus intensity (e.g., as measured through rating scales) reflect the mean value of an internal representation, and sensitivity reflects a combination of mean value and noise properties, as quantified by the statistical measure of Fisher information. Unique identification of these internal representation properties can be achieved by combining measurements of sensitivity and judgments of intensity. As a central example, we show that Weber's law of perceptual sensitivity can coexist with Stevens' power-law scaling of intensity ratings (for all exponents), when the noise amplitude increases in proportion to the representational mean. We then extend this result beyond the Weber's law range by incorporating a more general and physiology-inspired form of noise and show that the combination of noise properties and sensitivity measurements accurately predicts intensity ratings across a variety of sensory modalities and attributes. Our framework unifies two primary perceptual measurements-thresholds for sensitivity and rating scales for intensity-and provides a neural interpretation for the underlying representation.

An image-driven model for pattern detection, resistant to Birdsall linearisation

If detection were governed by an isolated (and possibly nonlinear) transducer, then a linearisation of the psychometric function (d-prime vs target amplitude) must accompany any threshold elevation due to the addition of external noise. This is the Birdsall theorem. From the fact that noise can elevate threshold without linearising the psychometric function, we can safely infer that detection is not governed by an isolated transducer. Heretofore, image-driven models, which accept images or numerical descriptions thereof as input, have proven incompatible with this failure of Birdsall linearisation, unless they incorporate the principle of intrinsic uncertainty, which asserts that detection is governed by the maximum activity in several independent (noisy) sensors. One image-driven model incompatible with the failure of Birdsall linearisation is Watson and Solomon's (J. Opt. Soc. Am. A, 14 (1997), 2379) model of visual contrast gain control and pattern masking. Here I report a simple modification - pooling sensor outputs before, instead of after the comparison of input images - allowing that model to predict curved psychometric functions, even when external noise elevates threshold by more than 20 dB, without any detrimental effect to the quality of its fit to pattern-masking thresholds in the absence of noise. The failure of Birdsall linearisation, therefore, does not necessarily imply independent samples of performance-limiting noise in multiple visual sensors. Instead, performance-limiting noise may arise after the visual system combines output from mutually inhibitory sensors.

Improvement of contrast sensitivity with practice is not compatible with a sensory threshold account

In forced-choice detection, incorrect responses are routinely ascribed to internal noise, because experienced psychophysical observers do not act as if they have a sensory threshold, below which all perceived intensities would be identical. To determine whether inexperienced observers have sensory thresholds, we examined psychometric functions (percent correct versus log contrast) for detection and detection in full-screen, dynamic visual noise. Over five days, neither type of psychometric function changed shape, but both shifted leftwards, indicating increased sensitivity. These results are not consistent with a lowered sensory threshold, which would decrease psychometric slope. Our results can be understood within the context of Dosher and Lu's "stochastic" perceptual template model [Vis. Res.40, 1269 (2000)], augmented to allow intrinsic uncertainty. Specifically, our results are consistent with a combination of reduced internal additive noise and improved filtering of external noise.

Fixed versus variable internal noise in contrast transduction: The significance of Whittle's data

A longstanding issue in vision research concerns whether the internal noise involved in contrast transduction is fixed or variable in relation to contrast magnitude. Previous attempts to resolve the issue have focused on the analysis of contrast discrimination data, despite the fact that the effects of internal noise on thresholds are necessarily compounded by the shape of the underlying transducer function. An alternative approach is to compare data obtained from a particular class of scaling experiment - one based on a comparison of perceived contrast differences - with data from discrimination experiments gathered across the full range of contrast. Data from two studies by the late Paul Whittle provide the basis for such an analysis, pointing to the conclusion that contrast internal noise is fixed not variable.

Four theorems on the psychometric function

In a 2-alternative forced-choice (2AFC) discrimination task, observers choose which of two stimuli has the higher value. The psychometric function for this task gives the probability of a correct response for a given stimulus difference, . This paper proves four theorems about the psychometric function. Assuming the observer applies a transducer and adds noise, Theorem 1 derives a convenient general expression for the psychometric function. Discrimination data are often fitted with a Weibull function. Theorem 2 proves that the Weibull “slope” parameter, , can be approximated by , where is the of the Weibull function that fits best to the cumulative noise distribution, and depends on the transducer. We derive general expressions for and , from which we derive expressions for specific cases. One case that follows naturally from our general analysis is Pelli's finding that, when , . We also consider two limiting cases. Theorem 3 proves that, as sensitivity improves, 2AFC performance will usually approach that for a linear transducer, whatever the actual transducer; we show that this does not apply at signal levels where the transducer gradient is zero, which explains why it does not apply to contrast detection. Theorem 4 proves that, when the exponent of a power-function transducer approaches zero, 2AFC performance approaches that of a logarithmic transducer. We show that the power-function exponents of 0.4–0.5 fitted to suprathreshold contrast discrimination data are close enough to zero for the fitted psychometric function to be practically indistinguishable from that of a log transducer. Finally, Weibull reflects the shape of the noise distribution, and we used our results to assess the recent claim that internal noise has higher kurtosis than a Gaussian. Our analysis of for contrast discrimination suggests that, if internal noise is stimulus-independent, it has lower kurtosis than a Gaussian.

The absolute threshold of cone vision

We report measurements of the absolute threshold of cone vision, which has been previously underestimated due to suboptimal conditions or overly strict subjective response criteria. We avoided these limitations by using optimized stimuli and experimental conditions while having subjects respond within a rating scale framework. Small (1' fwhm), brief (34 ms), monochromatic (550 nm) stimuli were foveally presented at multiple intensities in dark-adapted retina for 5 subjects. For comparison, 4 subjects underwent similar testing with rod-optimized stimuli. Cone absolute threshold, that is, the minimum light energy for which subjects were just able to detect a visual stimulus with any response criterion, was 203 ± 38 photons at the cornea, ~0.47 log unit lower than previously reported. Two-alternative forced-choice measurements in a subset of subjects yielded consistent results. Cone thresholds were less responsive to criterion changes than rod thresholds, suggesting a limit to the stimulus information recoverable from the cone mosaic in addition to the limit imposed by Poisson noise. Results were consistent with expectations for detection in the face of stimulus uncertainty. We discuss implications of these findings for modeling the first stages of human cone vision and interpreting psychophysical data acquired with adaptive optics at the spatial scale of the receptor mosaic.

Non-linear integration of crowded orientation signals

Crowding of oriented signals has been explained as linear, compulsory averaging of the signals from target and flankers [Parkes, L., Lund, J., Angelucci, A., Solomon, J. A., & Morgan, M. (2001). Compulsory averaging of crowded orientation signals in human vision. Nature Neuroscience, 4(7), 739-744]. On the other hand, a comparable search task with sparse stimuli is well modeled by a 'Signed-Max' rule that integrates non-linearly local tilt estimates [Baldassi, S., & Verghese, P. (2002). Comparing integration rules in visual search. Journal of Vision, 2(8), 559-570], as reflected by the bimodality of the distributions of reported tilts in a magnitude matching task [Baldassi, S., Megna, N., & Burr, D. C. (2006). Visual clutter causes high-magnitude errors. PLoS Biology, 4(3), e56]. This study compares the two models in the context of crowding by using a magnitude matching task, to measure distributions of perceived target angles and a localization task, to probe the degree of access to local information. Response distributions were bimodal, implying uncertainty, only in the presence of abutting flankers. Localization of the target is relatively preserved but it quantitatively falls in between the predictions of the two models, possibly suggesting local averaging followed by a max operation. This challenges the notion of global averaging and suggests some conscious access to local orientation estimates.

A 'dipper' function for texture discrimination based on orientation variance

We measured the just-noticeable difference (JND) in orientation variance between two textures (Figure 1) as we varied the baseline (pedestal) variance present in both textures. JND's first fell as pedestal variance increased and then rose, producing a 'dipper' function similar to those previously reported for contrast, blur, and orientation-contrast discriminations. A dipper function (both facilitation and masking) is predicted on purely statistical grounds by a noisy variance-discrimination mechanism. However, for two out of three observers, the dipper function was significantly better fit when the mechanism was made incapable of discriminating between small sample variances. We speculate that a threshold nonlinearity like this prevents the visual system from including its intrinsic noise in texture representations and suggest that similar thresholds prevent the visibility of other artifacts that sensory coding would otherwise introduce, such as blur.

A common contrast pooling rule for suppression within and between the eyes

Recent work has revealed multiple pathways for cross-orientation suppression in cat and human vision. In particular, ipsiocular and interocular pathways appear to assert their influence before binocular summation in human but have different (1) spatial tuning, (2) temporal dependencies, and (3) adaptation after-effects. Here we use mask components that fall outside the excitatory passband of the detecting mechanism to investigate the rules for pooling multiple mask components within these pathways. We measured psychophysical contrast masking functions for vertical 1 cycle/deg sine-wave gratings in the presence of left or right oblique (±45 deg) 3 cycles/deg mask gratings with contrast C%, or a plaid made from their sum, where each component (i) had contrast 0.5Ci%. Masks and targets were presented to two eyes (binocular), one eye (monoptic), or different eyes (dichoptic). Binocular-masking functions superimposed when plotted against C, but in the monoptic and dichoptic conditions, the grating produced slightly more suppression than the plaid when Ci ≥ 16%. We tested contrast gain control models involving two types of contrast combination on the denominator: (1) spatial pooling of the mask after a local nonlinearity (to calculate either root mean square contrast or energy) and (2) “linear suppression” (Holmes & Meese, 2004, Journal of Vision4, 1080–1089), involving the linear sum of the mask component contrasts. Monoptic and dichoptic masking were typically better fit by the spatial pooling models, but binocular masking was not: it demanded strict linear summation of the Michelson contrast across mask orientation. Another scheme, in which suppressive pooling followed compressive contrast responses to the mask components (e.g., oriented cortical cells), was ruled out by all of our data. We conclude that the different processes that underlie monoptic and dichoptic masking use the same type of contrast pooling within their respective suppressive fields, but the effects do not sum to predict the binocular case.

Contrast discrimination: second responses reveal the relationship between the mean and variance of visual signals

To explain the relationship between first- and second-response accuracies in a detection experiment, Swets, Tanner, and Birdsall [Swets, J., Tanner, W. P., Jr., & Birdsall, T. G. (1961). Decision processes in perception. Psychological Review, 68, 301–340] proposed that the variance of visual signals increased with their means. However, both a low threshold and intrinsic uncertainty produce similar relationships. I measured the relationship between first- and second-response accuracies for suprathreshold contrast discrimination, which is thought to be unaffected by sensory thresholds and intrinsic uncertainty. The results are consistent with a slowly increasing variance.

Binocular contrast interactions: dichoptic masking is not a single process

To decouple interocular suppression and binocular summation we varied the relative phase of mask and target in a 2IFC contrast-masking paradigm. In Experiment I, dichoptic mask gratings had the same orientation and spatial frequency as the target. For in-phase masking, suppression was strong (a log-log slope of approximately 1) and there was weak facilitation at low mask contrasts. Anti-phase masking was weaker (a log-log slope of approximately 0.7) and there was no facilitation. A two-stage model of contrast gain control [Meese, T.S., Georgeson, M.A. and Baker, D.H. (2006). Binocular contrast vision at and above threshold. Journal of Vision, 6: 1224-1243] provided a good fit to the in-phase results and fixed its free parameters. It made successful predictions (with no free parameters) for the anti-phase results when (A) interocular suppression was phase-indifferent but (B) binocular summation was phase sensitive. Experiments II and III showed that interocular suppression comprised two components: (i) a tuned effect with an orientation bandwidth of approximately +/-33 degrees and a spatial frequency bandwidth of >3 octaves, and (ii) an untuned effect that elevated threshold by a factor of between 2 and 4. Operationally, binocular summation was more tightly tuned, having an orientation bandwidth of approximately +/-8 degrees , and a spatial frequency bandwidth of approximately 0.5 octaves. Our results replicate the unusual shapes of the in-phase dichoptic tuning functions reported by Legge [Legge, G.E. (1979). Spatial frequency masking in human vision: Binocular interactions. Journal of the Optical Society of America, 69: 838-847]. These can now be seen as the envelope of the direct effects from interocular suppression and the indirect effect from binocular summation, which contaminates the signal channel with a mask that has been suppressed by the target.

Stochastic re-calibration: contextual effects on perceived tilt

The human visual system exaggerates the difference between the tilts of adjacent lines or grating patches. In addition to this tilt illusion, we found that oblique flanks reduced acuity for small changes of tilt in the centre of the visual field. However, no flanks--regardless of their tilts--decreased sensitivity to contrast. Thus, the foveal tilt illusion should not be attributed to orientation-selective lateral inhibition. Nor is it similar to conventional crowding, which typically does not impair letter recognition in the fovea. Our observers behaved as though the reference orientation (horizontal) had a small tilt in the direction of the flanks. We suggest that the extent of this re-calibration varies randomly over trials, and we demonstrate that this stochastic re-calibration can explain flank-induced acuity loss in the fovea.