Where is the light? Bayesian perceptual priors for lighting direction

J.V. Stone ) I.S. Kerrigan J. Porrill Subject collections 0 School of Psychology, University of Southampton , Southampton SO17 1BJ , UK 1 Department of Psychology, University of Sheffield , Sheffield S10 2TP , UK Articles on similar topics can be found in the following collections Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here - Email alerting service To subscribe to Proc. R. Soc. B go to: http://rspb.royalsocietypublishing.org/subscriptions Where is the light? Bayesian perceptual priors for lighting direction Perception of shaded three-dimensional figures is inherently ambiguous, but this ambiguity can be resolved if the brain assumes that figures are lit from a specific direction. Under the Bayesian framework, the visual system assigns a weighting to each possible direction, and these weightings define a prior probability distribution for light-source direction. Here, we describe a non-parametric maximum-likelihood estimation method for finding the prior distribution for lighting direction. Our results suggest that each observer has a distinct prior distribution, with non-zero values in all directions, but with a peak which indicates observers are biased to expect light to come from above left. The implications of these results for estimating general perceptual priors are discussed. 1. INTRODUCTION Perception consists of interpreting two-dimensional retinal images of a three-dimensional world. The process of projecting a three-dimensional scene onto a twodimensional retina necessarily discards information about the three-dimensional structure of that scene. This makes it impossible, in principle, to deduce all of the three-dimensional structure of a scene, and perception is therefore a classic example of an ill-posed problem (Poggio et al. 1985). However, even though such problems cannot be solved by deduction, acceptable solutions can be found using statistical inference. This involves using additional information, usually based on prior experience, to interpret two-dimensional retinal images, where this additional information takes the form of heuristics (rules of thumb) or constraints (rules which exclude certain illegal solutions). Within the Bayesian framework, this extra information is realized in the form of prior distributions. For example, the image marked with a cross in figure 1 can be interpreted as either convex or concave. The particular perception evoked by this image depends only on the direction in which the light source is assumed to originate ( Rittenhouse 1786; Brewster 1847; Oppel 1856; Kleffner & Ramachandran 1992). If the light source is assumed to originate from below, then the image is interpreted as convex, but if the light source is assumed to originate from above, then the image is interpreted as concave. As this is the usual interpretation made by human observers, it implies that we implicitly assume light originates from above. However, such demonstrations provide only a qualitative impression of where we assume the light source to be. In reality, it is unlikely that human observers make the simplistic assumption that light comes only from above or below. More realistically, each observer assigns a probability to each possible light-source direction, which may be based on prior experience of the directions in which light sources originate. These probability values collectively define a prior probability density function, which can be visualized using a polar plot, where the radial distance in a given direction indicates the relative probability that the light originates from that direction (as in figure 2c). In this paper, we show how it is possible to estimate the overall form of this prior, which, for reasons that will become obvious, we call the light-from-above prior. For the sake of clarity, note that we do not seek the prior for lighting direction, which could be obtained empirically, but the prior as used by a given observer. Our general strategy is closely related to that described in Paninski (2006). However, in the simulated experiment described by Paninski, the observer estimates a continuous parameter, and so each trial provides an equality constraint on the prior. Here, we concentrate on the more common case in which the observer makes a forced choice, so that each trial provides a weaker, inequality constraint on the prior. 2. RESULTS The shape information in our images is a function of two parameters, the direction q of the light source and the three-dimensional shape c of the imaged surface, which specifies whether the stimulus is concave cZc1 or convex cZc0. On each trial, the observer is presented with an image x, and makes a binary response rZ1 if the stimulus appears concave or rZ0 if the stimulus appears convex (see appendix A). We assume that the observers perceived shape c^ of a shape c depends on two quantities: the posterior probability density function and the loss function. First, the probability (density) that the shape has value c and that the light source is in direction q given an image x defines the joint posterior probability density function p(c,qjx). Second, the cost of perceiving a shape as c^, when it is actually c, is defined by the loss function Dc^; c. 1798 J. V. Stone et al. Where is the light? pc; q j xDc^; cdq dc; The observers perception is assumed to correspond to the shape c^, which minimizes the expected loss, where this expectation is taken over all possible values of q and c dqK qx p c0; q dq; Z pc0; qx=px: 2:9 If the observer assumes that the stimulus shape and the lighting direction are independent, then the joint prior distribution p(c0,qx) factorizes to yield p c0 j x Z p c0pqqx=p x; where pq(qx) is the prior over lighting direction and p(c0) is the prior for the shape c0. A similar calculation for cZc1 yields p c1 j x Z p c1pqqx=px: Regardless of the value of qx and qx, each observer perceives the stimulus as either convex c0 or concave c1, and responds accordingly. Thus, together, p(c1) and p(c0) is a pair of co-determined observer-specific scalar priors, such that p(c1)Cp(c0)Z1. We call the prior p(c1) the concavity preference for a given observer, which can be estimated using the same method (described below) for estimating the prior pq(q). We choose the zero/one loss function to model the forced choice task, i.e. Dc^; cZ 0 for a correct decision and Dc^; cZ 1 for an incorrect decision (Bishop 1996); the optimal decision rule under this loss function minimizes the number of misclassified stimuli. Substituting this loss function into equation (2.5), we find that the observer should respond rZ0 (convex) if the log posterior ratio Using Bayes rule, the posterior is given by pc; q j x Z px j c; qpc; q=px; where the observers prior expectations about shapes and lighting directions define the joint prior distribution p(c, q), and where the probability of the observed imag (...truncated)