Cardinal rules: visual orientation perception reflects knowledge of environmental statistics

Abstract

Humans are good at performing visual tasks, but experimental measurements have revealed substantial biases in the perception of basic visual attributes. An appealing hypothesis is that these biases arise through a process of statistical inference, in which information from noisy measurements is fused with a probabilistic model of the environment. However, such inference is optimal only if the observer's internal model matches the environment. We found this to be the case. We measured performance in an orientation-estimation task and found that orientation judgments were more accurate at cardinal (horizontal and vertical) orientations. Judgments made under conditions of uncertainty were strongly biased toward cardinal orientations. We estimated observers' internal models for orientation and found that they matched the local orientation distribution measured in photographs. In addition, we determined how a neural population could embed probabilistic information responsible for such biases.

Figure 1

Observer model for local 2D orientation estimation. Left panel: The environment. Each local edge has a true image orientation, θ. Central panel: The encoding stage, in which the observer obtains a visual measurement, m(θ), corrupted by sensory noise. Right panel: The decoding stage, in which a function (the estimator, black curve) is applied to the measurement to produce the estimated orientation, θ̂(m(θ)). Because of the sensory noise, the estimated orientation will exhibit variability across repeated presentation of the same stimulus, and may also exhibit a systematic bias relative to the true orientation.

Figure 2

Stimuli and experimental results. (a) Stimuli are arrays of oriented Gabor functions (contrast increased for illustrative purposes). Left: A low-noise stimulus (L). Right: A high-noise stimulus (H) with mean orientation slightly more clockwise. Observers indicated whether the right stimulus was oriented counter-clockwise or clockwise relative to the left stimulus. (b) Variability for the same-noise conditions for representative subject S1 (left column) and the mean subject (right column), expressed as the orientation discrimination threshold (i.e., the “just noticeable difference”, or JND). Mean subject values are computed by pooling raw choice data from all five subjects. Error bars indicate 95% confidence intervals. Dark gray and light gray curves are fitted rectified sinusoids, used to estimate the widths of the underlying measurement distributions. Pale gray regions indicate ± 1 s.d. of 1,000 bootstrapped fits. (c) Cross-noise (H vs. L) variability data (black circles). The horizontal axis is the orientation of the high-noise stimulus. (d) Relative bias, expressed as the angle by which the high-noise stimulus must be rotated counter-clockwise so as to be perceived as having the same mean orientation as the low-noise stimulus.

Figure 3

Recovered priors for subject S1 and mean subject. The control points of the piecewise cubic spline (see Methods) are indicated by black dots. The gray error region shows ± 1 s.d. of 1,000 bootstrapped estimated priors.

Figure 4

Natural image statistics. (a) Example natural scene from Fig. 1, with strongly oriented locations marked in red. (b) Orientation distribution for natural images (gray curve).

Figure 5

Comparison of human observers’ priors and environmental distribution for subject S1 (left column) and the mean subject (right column). (a) Human observers’ priors (black curves, from Fig. 3) and environmental distribution from natural images (medium gray curve, from Fig. 4b). (b) Cross-noise variability data (circles, from Fig. 2c) with predictions of the two Bayesian-observer models using each of the three priors shown in (a) and the uniform prior. The uniform prior predicts little or no effect of stimulus orientation on discrimination (light gray curves). In contrast, both the environmental prior (medium gray curves) and the recovered human observers’ priors (black curves) predict better discrimination at the cardinals, as seen in the human observers. (c) Relative bias data (circles, from Fig. 2d) with the predictions of Bayesian-observer models using three priors shown in (a). The uniform prior predicts no bias or a small bias in the opposite direction (e.g., Supplementary Fig. 1c, subject S2). In contrast, both non-uniform priors predict the bimodal bias exhibited by human observers. (d) Normalized log likelihood of the data for Bayesian-observer models using two different priors: environmental distribution (medium gray bars) and the recovered observer’s prior (dark gray bars). Error bars denote the 5th and 95th percentiles from 1,000 bootstrap estimates. Values greater than one indicate performance better than that of the raw psychometric fits, whereas values less than zero indicate performance worse than that obtained with a uniform prior.

Figure 6

Simulations of neural models with non-uniform encoder, and “population vector” decoder. (a) Tuning curves of an encoder population with non-uniform orientation preferences and non-uniform tuning widths based on neurophysiology (only a subset of neurons shown). Neurons preferring 45 deg and 90 deg stimuli are highlighted in black. (b) Tuning curves of a population with non-uniform preferences and uniform widths. (c) Tuning curves of a population with uniform preferences and non-uniform widths. (d–f) Variability for the same-noise conditions for the populations in (a–c): L vs. L (dark gray) and H vs. H (light gray). (g–i) Relative bias for the cross-noise condition (H vs. L) for the populations in (a–c). The fully non-uniform population (a) produces variability and bias curves similar to those exhibited by humans (Fig. 2b,d and Supplementary Fig. 1a,c). Girshick, Landy, Simoncelli

Figure 7

Derivation of the estimator θ̂(m(θ)). In all three grayscale panels, the horizontal axis is stimulus orientation θ, the vertical axis is the measured orientation m(θ), and the intensity corresponds to probability. Upper-left panel: The mean observer’s prior, raised to the power of 2.25 and re-normalized for visibility, is independent of the measurements (i.e., all horizontal slices are identical). Upper-right panel: The conditional distribution, p(m | θ). Vertical slices indicate measurement distributions, p(m | θ₁) and p(m | θ₂), for two particular stimuli θ₁ and θ₂. The widths of the measurement distributions are the average of those for the low- and high-noise conditions for the mean observer (multiplied by a factor of 10 for visibility). Horizontal slices, p(m₁ | θ) and p(m₂ | θ), describe the likelihood of the stimulus orientation, θ, for the particular measurements, m₁ and m₂. Note that the likelihoods are not symmetric, because the measurement distribution width depends on the stimulus orientation. Bottom panel: The posterior distribution is computed using Bayes’ rule, as the normalized product of the prior and likelihood (top two panels). Horizontal slices correspond to posterior distributions p(θ | m₁) and p(θ | m₂), which describe the probability of a stimulus orientation given two particular measurements. Red dots indicate MAP estimates (the modes of the posterior) for these two likelihoods, θ̂(m₁) and θ̂(m₂). Circular mean estimates yield similar results (see Supplementary Fig. 2). The red curve shows the estimator θ̂(m) computed for all measurements. An unbiased estimator would correspond to a straight line along the diagonal.

Figure 8

Example cross-noise comparison. The vertical axis is the measured orientation m(θ), and the horizontal axis is estimated stimulus orientation θ̂(m(θ)). Measurements corresponding to low-noise stimuli, m_L (θ_L) (dark gray), or high-noise stimuli, m_H (θ_H) (light gray), enter on the left. Each measurement is transformed by the appropriate nonlinear estimator (solid curves) into an estimate (bottom). The estimators correspond to those of the mean observer exaggerated for illustration as in Fig. 7. The high-noise estimator exhibits larger biases than the low-noise estimator. The sensory noise of the measurements propagates through the estimator, resulting in estimator distributions (note these should not be confused with the posteriors). Comparison of these distributions produces a single point on the psychometric function.