nips nips2005 nips2005-3 knowledge-graph by maker-knowledge-mining

3 nips-2005-A Bayesian Framework for Tilt Perception and Confidence

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Author: Odelia Schwartz, Peter Dayan, Terrence J. Sejnowski

Abstract: The misjudgement of tilt in images lies at the heart of entertaining visual illusions and rigorous perceptual psychophysics. A wealth of ﬁndings has attracted many mechanistic models, but few clear computational principles. We adopt a Bayesian approach to perceptual tilt estimation, showing how a smoothness prior offers a powerful way of addressing much confusing data. In particular, we faithfully model recent results showing that conﬁdence in estimation can be systematically affected by the same aspects of images that affect bias. Conﬁdence is central to Bayesian modeling approaches, and is applicable in many other perceptual domains. Perceptual anomalies and illusions, such as the misjudgements of motion and tilt evident in so many psychophysical experiments, have intrigued researchers for decades.1–3 A Bayesian view4–8 has been particularly inﬂuential in models of motion processing, treating such anomalies as the normative product of prior information (often statistically codifying Gestalt laws) with likelihood information from the actual scenes presented. Here, we expand the range of statistically normative accounts to tilt estimation, for which there are classes of results (on estimation conﬁdence) that are so far not available for motion. The tilt illusion arises when the perceived tilt of a center target is misjudged (ie bias) in the presence of ﬂankers. Another phenomenon, called Crowding, refers to a loss in the conﬁdence (ie sensitivity) of perceived target tilt in the presence of ﬂankers. Attempts have been made to formalize these phenomena quantitatively. Crowding has been modeled as compulsory feature pooling (ie averaging of orientations), ignoring spatial positions.9, 10 The tilt illusion has been explained by lateral interactions11, 12 in populations of orientationtuned units; and by calibration.13 However, most models of this form cannot explain a number of crucial aspects of the data. First, the geometry of the positional arrangement of the stimuli affects attraction versus repulsion in bias, as emphasized by Kapadia et al14 (ﬁgure 1A), and others.15, 16 Second, Solomon et al. recently measured bias and sensitivity simultaneously.11 The rich and surprising range of sensitivities, far from ﬂat as a function of ﬂanker angles (ﬁgure 1B), are outside the reach of standard models. Moreover, current explanations do not offer a computational account of tilt perception as the outcome of a normative inference process. Here, we demonstrate that a Bayesian framework for orientation estimation, with a prior favoring smoothness, can naturally explain a range of seemingly puzzling tilt data. We explicitly consider both the geometry of the stimuli, and the issue of conﬁdence in the esti- 6 5 4 3 2 1 0 -1 -2 (B) Attraction Repulsion Sensititvity (1/deg) Bias (deg) (A) 0.6 0.5 0.4 0.3 0.2 0.1 -80 -60 -40 -20 0 20 40 60 80 Flanker tilt (deg) Figure 1: Tilt biases and sensitivities in visual perception. (A) Kapadia et al demonstrated the importance of geometry on tilt bias, with bar stimuli in the fovea (and similar results in the periphery). When 5 degrees clockwise ﬂankers are arranged colinearly, the center target appears attracted in the direction of the ﬂankers; when ﬂankers are lateral, the target appears repulsed. Data are an average of 5 subjects.14 (B) Solomon et al measured both biases and sensitivities for gratings in the visual periphery.11 On the top are example stimuli, with ﬂankers tilted 22.5 degrees clockwise. This constitutes the classic tilt illusion, with a repulsive bias percept. In addition, sensitivities vary as a function of ﬂanker angles, in a systematic way (even in cases when there are no biases at all). Sensitivities are given in units of the inverse of standard deviation of the tilt estimate. More detailed data for both experiments are shown in the results section. mation. Bayesian analyses have most frequently been applied to bias. Much less attention has been paid to the equally important phenomenon of sensitivity. This aspect of our model should be applicable to other perceptual domains. In section 1 we formulate the Bayesian model. The prior is determined by the principle of creating a smooth contour between the target and ﬂankers. We describe how to extract the bias and sensitivity. In section 2 we show experimental data of Kapadia et al and Solomon et al, alongside the model simulations, and demonstrate that the model can account for both geometry, and bias and sensitivity measurements in the data. Our results suggest a more uniﬁed, rational, approach to understanding tilt perception. 1 Bayesian model Under our Bayesian model, inference is controlled by the posterior distribution over the tilt of the target element. This comes from the combination of a prior favoring smooth conﬁgurations of the ﬂankers and target, and the likelihood associated with the actual scene. A complete distribution would consider all possible angles and relative spatial positions of the bars, and marginalize the posterior over all but the tilt of the central element. For simplicity, we make two benign approximations: conditionalizing over (ie clamping) the angles of the ﬂankers, and exploring only a small neighborhood of their positions. We now describe the steps of inference. Smoothness prior: Under these approximations, we consider a given actual conﬁguration (see ﬁg 2A) of ﬂankers f1 = (φ1 , x1 ), f2 = (φ2 , x2 ) and center target c = (φc , xc ), arranged from top to bottom. We have to generate a prior over φc and δ1 = x1 − xc and δ2 = x2 − xc based on the principle of smoothness. As a less benign approximation, we do this in two stages: articulating a principle that determines a single optimal conﬁguration; and generating a prior as a mixture of a Gaussian about this optimum and a uniform distribution, with the mixing proportion of the latter being determined by the smoothness of the optimum. Smoothness has been extensively studied in the computer vision literature.17–20 One widely (B) (C) f1 f1 β1 R Probability max smooth Max smooth target (deg) (A) 40 20 0 -20 c δ1 c -40 Φc f2 f2 1 0.8 0.6 0.4 0.2 0 -80 -60 -40 -20 0 20 40 Flanker tilt (deg) 60 80 -80 -60 -40 20 0 20 40 Flanker tilt (deg) 60 80 Figure 2: Geometry and smoothness for ﬂankers, f1 and f2 , and center target, c. (A) Example actual conﬁguration of ﬂankers and target, aligned along the y axis from top to bottom. (B) The elastica procedure can rotate the target angle (to Φc ) and shift the relative ﬂanker and target positions on the x axis (to δ1 and δ2 ) in its search for the maximally smooth solution. Small spatial shifts (up to 1/15 the size of R) of positions are allowed, but positional shift is overemphasized in the ﬁgure for visibility. (C) Top: center tilt that results in maximal smoothness, as a function of ﬂanker tilt. Boxed cartoons show examples for given ﬂanker tilts, of the optimally smooth conﬁguration. Note attraction of target towards ﬂankers for small ﬂanker angles; here ﬂankers and target are positioned in a nearly colinear arrangement. Note also repulsion of target away from ﬂankers for intermediate ﬂanker angles. Bottom: P [c, f1 , f2 ] for center tilt that yields maximal smoothness. The y axis is normalized between 0 and 1. used principle, elastica, known even to Euler, has been applied to contour completion21 and other computer vision applications.17 The basic idea is to ﬁnd the curve with minimum energy (ie, square of curvature). Sharon et al19 showed that the elastica function can be well approximated by a number of simpler forms. We adopt a version that Leung and Malik18 adopted from Sharon et al.19 We assume that the probability for completing a smooth curve, can be factorized into two terms: P [c, f1 , f2 ] = G(c, f1 )G(c, f2 ) (1) with the term G(c, f1 ) (and similarly, G(c, f2 )) written as: R Dβ 2 2 Dβ = β1 + βc − β1 βc (2) − ) where σR σβ and β1 (and similarly, βc ) is the angle between the orientation at f1 , and the line joining f1 and c. The distance between the centers of f1 and c is given by R. The two constants, σβ and σR , control the relative contribution to smoothness of the angle versus the spatial distance. Here, we set σβ = 1, and σR = 1.5. Figure 2B illustrates an example geometry, in which φc , δ1 , and δ2 , have been shifted from the actual scene (of ﬁgure 2A). G(c, f1 ) = exp(− We now estimate the smoothest solution for given conﬁgurations. Figure 2C shows for given ﬂanker tilts, the center tilt that yields maximal smoothness, and the corresponding probability of smoothness. For near vertical ﬂankers, the spatial lability leads to very weak attraction and high probability of smoothness. As the ﬂanker angle deviates farther from vertical, there is a large repulsion, but also lower probability of smoothness. These observations are key to our model: the maximally smooth center tilt will inﬂuence attractive and repulsive interactions of tilt estimation; the probability of smoothness will inﬂuence the relative weighting of the prior versus the likelihood. From the smoothness principle, we construct a two dimensional prior (ﬁgure 3A). One dimension represents tilt, the other dimension, the overall positional shift between target (B) Likelihood (D) Marginalized Posterior (C) Posterior 20 0.03 10 -10 -20 0 Probability 0 10 Angle Angle Angle 10 0 -10 -20 0.01 -10 -20 0.02 0 -0. 2 0 Position 0.2 (E) Psychometric function 20 -0. 2 0 0.2 -0. 2 0 0.2 Position Position -10 -5 0 Angle 5 10 Probability clockwise (A) Prior 20 1 0.8 0.6 0.4 0.2 0 -20 -10 0 10 20 Target angle (deg) Counter-clockwise Clockwise Figure 3: Bayes model for example ﬂankers and target. (A) Prior 2D distribution for ﬂankers set at 22.5 degrees (note repulsive preference for -5.5 degrees). (B) Likelihood 2D distribution for a target tilt of 3 degrees; (C) Posterior 2D distribution. All 2D distributions are drawn on the same grayscale range, and the presence of a larger baseline in the prior causes it to appear more dimmed. (D) Marginalized posterior, resulting in 1D distribution over tilt. Dashed line represents the mean, with slight preference for negative angle. (E) For this target tilt, we calculate probability clockwise, and obtain one point on psychometric curve. and ﬂankers (called ’position’). The prior is a 2D Gaussian distribution, sat upon a constant baseline.22 The Gaussian is centered at the estimated smoothest target angle and relative position, and the baseline is determined by the probability of smoothness. The baseline, and its dependence on the ﬂanker orientation, is a key difference from Weiss et al’s Gaussian prior for smooth, slow motion. It can be seen as a mechanism to allow segmentation (see Posterior description below). The standard deviation of the Gaussian is a free parameter. Likelihood: The likelihood over tilt and position (ﬁgure 3B) is determined by a 2D Gaussian distribution with an added baseline.22 The Gaussian is centered at the actual target tilt; and at a position taken as zero, since this is the actual position, to which the prior is compared. The standard deviation and baseline constant are free parameters. Posterior and marginalization: The posterior comes from multiplying likelihood and prior (ﬁgure 3C) and then marginalizing over position to obtain a 1D distribution over tilt. Figure 3D shows an example in which this distribution is bimodal. Other likelihoods, with closer agreement between target and smooth prior, give unimodal distributions. Note that the bimodality is a direct consequence of having an added baseline to the prior and likelihood (if these were Gaussian without a baseline, the posterior would always be Gaussian). The viewer is effectively assessing whether the target is associated with the same object as the ﬂankers, and this is reﬂected in the baseline, and consequently, in the bimodality, and conﬁdence estimate. We deﬁne α as the mean angle of the 1D posterior distribution (eg, value of dashed line on the x axis), and β as the height of the probability distribution at that mean angle (eg, height of dashed line). The term β is an indication of conﬁdence in the angle estimate, where for larger values we are more certain of the estimate. Decision of probability clockwise: The probability of a clockwise tilt is estimated from the marginalized posterior: 1 P = 1 + exp (3) −α.∗k − log(β+η) where α and β are deﬁned as above, k is a free parameter and η a small constant. Free parameters are set to a single constant value for all ﬂanker and center conﬁgurations. Weiss et al use a similar compressive nonlinearity, but without the term β. We also tried a decision function that integrates the posterior, but the resulting curves were far from the sigmoidal nature of the data. Bias and sensitivity: For one target tilt, we generate a single probability and therefore a single point on the psychometric function relating tilt to the probability of choosing clockwise. We generate the full psychometric curve from all target tilts and ﬁt to it a cumulative 60 40 20 -5 0 5 Target tilt (deg) 10 80 60 40 20 0 -10 (C) Data -5 0 5 Target tilt (deg) 10 80 60 40 20 0 -10 (D) Model 100 100 100 80 0 -10 Model Frequency responding clockwise (B) Data Frequency responding clockwise Frequency responding clockwise Frequency responding clockwise (A) 100 -5 0 5 Target tilt (deg) 10 80 60 40 20 0 -10 -5 0 5 10 Target tilt (deg) Figure 4: Kapadia et al data,14 versus Bayesian model. Solid lines are ﬁts to a cumulative Gaussian distribution. (A) Flankers are tilted 5 degrees clockwise (black curve) or anti-clockwise (gray) of vertical, and positioned spatially in a colinear arrangement. The center bar appears tilted in the direction of the ﬂankers (attraction), as can be seen by the attractive shift of the psychometric curve. The boxed stimuli cartoon illustrates a vertical target amidst the ﬂankers. (B) Model for colinear bars also produces attraction. (C) Data and (D) model for lateral ﬂankers results in repulsion. All data are collected in the fovea for bars. Gaussian distribution N (µ, σ) (ﬁgure 3E). The mean µ of the ﬁt corresponds to the bias, 1 and σ to the sensitivity, or conﬁdence in the bias. The ﬁt to a cumulative Gaussian and extraction of these parameters exactly mimic psychophysical procedures.11 2 Results: data versus model We ﬁrst consider the geometry of the center and ﬂanker conﬁgurations, modeling the full psychometric curve for colinear and parallel ﬂanks (recall that ﬁgure 1A showed summary biases). Figure 4A;B demonstrates attraction in the data and model; that is, the psychometric curve is shifted towards the ﬂanker, because of the nature of smooth completions for colinear ﬂankers. Figure 4C;D shows repulsion in the data and model. In this case, the ﬂankers are arranged laterally instead of colinearly. The smoothest solution in the model arises by shifting the target estimate away from the ﬂankers. This shift is rather minor, because the conﬁguration has a low probability of smoothness (similar to ﬁgure 2C), and thus the prior exerts only a weak effect. The above results show examples of changes in the psychometric curve, but do not address both bias and, particularly, sensitivity, across a whole range of ﬂanker conﬁgurations. Figure 5 depicts biases and sensitivity from Solomon et al, versus the Bayes model. The data are shown for a representative subject, but the qualitative behavior is consistent across all subjects tested. In ﬁgure 5A, bias is shown, for the condition that both ﬂankers are tilted at the same angle. The data exhibit small attraction at near vertical ﬂanker angles (this arrangement is close to colinear); large repulsion at intermediate ﬂanker angles of 22.5 and 45 degrees from vertical; and minimal repulsion at large angles from vertical. This behavior is also exhibited in the Bayes model (Figure 5B). For intermediate ﬂanker angles, the smoothest solution in the model is repulsive, and the effect of the prior is strong enough to induce a signiﬁcant repulsion. For large angles, the prior exerts almost no effect. Interestingly, sensitivity is far from ﬂat in both data and model. In the data (Figure 5C), there is most loss in sensitivity at intermediate ﬂanker angles of 22.5 and 45 degrees (ie, the subject is less certain); and sensitivity is higher for near vertical or near horizontal ﬂankers. The model shows the same qualitative behavior (Figure 5D). In the model, there are two factors driving sensitivity: one is the probability of completing a smooth curvature for a given ﬂanker conﬁguration, as in Figure 2B; this determines the strength of the prior. The other factor is certainty in a particular center estimation; this is determined by β, derived from the posterior distribution, and incorporated into the decision stage of the model Data 5 0 -60 -40 -80 -60 -40 -20 0 20 40 Flanker tilt (deg) -20 0 20 40 Flanker tilt (deg) 60 60 80 -60 -40 0.6 0.5 0.4 0.3 0.2 0.1 -20 0 20 40 Flanker tilt (deg) 60 80 60 80 -80 -60 -40 -20 0 20 40 Flanker tilt (deg) -80 -60 -40 -20 0 20 40 Flanker tilt (deg) 60 80 -20 0 20 40 Flanker tilt (deg) 60 80 (F) Bias (deg) 10 5 0 -5 0.6 0.5 0.4 0.3 0.2 0.1 -80 (D) 10 -10 -10 80 Sensitivity (1/deg) -80 5 0 -5 -80 -80 -60 -60 -40 -40 -20 0 20 40 Flanker tilt (deg) -20 0 20 40 Flanker tilt (deg) 60 -10 80 (H) 60 80 Sensitivity (1/deg) Sensititvity (1/deg) Bias (deg) 0.6 0.5 0.4 0.3 0.2 0.1 (G) Sensititvity (1/deg) 0 -5 (C) (E) 5 -5 -10 Model (B) 10 Bias (deg) Bias (deg) (A) 10 0.6 0.5 0.4 0.3 0.2 0.1 -80 -60 -40 Figure 5: Solomon et al data11 (subject FF), versus Bayesian model. (A) Data and (B) model biases with same-tilted ﬂankers; (C) Data and (D) model sensitivities with same-tilted ﬂankers; (E;G) data and (F;H) model as above, but for opposite-tilted ﬂankers (note that opposite-tilted data was collected for less ﬂanker angles). Each point in the ﬁgure is derived by ﬁtting a cummulative Gaussian distribution N (µ, σ) to corresponding psychometric curve, and setting bias 1 equal to µ and sensitivity to σ . In all experiments, ﬂanker and target gratings are presented in the visual periphery. Both data and model stimuli are averages of two conﬁgurations, on the left hand side (9 O’clock position) and right hand side (3 O’clock position). The conﬁgurations are similar to Figure 1 (B), but slightly shifted according to an iso-eccentric circle, so that all stimuli are similarly visible in the periphery. (equation 3). For ﬂankers that are far from vertical, the prior has minimal effect because one cannot ﬁnd a smooth solution (eg, the likelihood dominates), and thus sensitivity is higher. The low sensitivity at intermediate angles arises because the prior has considerable effect; and there is conﬂict between the prior (tilt, position), and likelihood (tilt, position). This leads to uncertainty in the target angle estimation . For ﬂankers near vertical, the prior exerts a strong effect; but there is less conﬂict between the likelihood and prior estimates (tilt, position) for a vertical target. This leads to more conﬁdence in the posterior estimate, and therefore, higher sensitivity. The only aspect that our model does not reproduce is the (more subtle) sensitivity difference between 0 and +/- 5 degree ﬂankers. Figure 5E-H depict data and model for opposite tilted ﬂankers. The bias is now close to zero in the data (Figure 5E) and model (Figure 5F), as would be expected (since the maximally smooth angle is now always roughly vertical). Perhaps more surprisingly, the sensitivities continue to to be non-ﬂat in the data (Figure 5G) and model (Figure 5H). This behavior arises in the model due to the strength of prior, and positional uncertainty. As before, there is most loss in sensitivity at intermediate angles. Note that to ﬁt Kapadia et al, simulations used a constant parameter of k = 9 in equation 3, whereas for the Solomon et al. simulations, k = 2.5. This indicates that, in our model, there was higher conﬁdence in the foveal experiments than in the peripheral ones. 3 Discussion We applied a Bayesian framework to the widely studied tilt illusion, and demonstrated the model on examples from two different data sets involving foveal and peripheral estimation. Our results support the appealing hypothesis that perceptual misjudgements are not a consequence of poor system design, but rather can be described as optimal inference.4–8 Our model accounts correctly for both attraction and repulsion, determined by the smoothness prior and the geometry of the scene. We emphasized the issue of estimation conﬁdence. The dataset showing how conﬁdence is affected by the same issues that affect bias,11 was exactly appropriate for a Bayesian formulation; other models in the literature typically do not incorporate conﬁdence in a thoroughly probabilistic manner. In fact, our model ﬁts the conﬁdence (and bias) data more proﬁciently than an account based on lateral interactions among a population of orientationtuned cells.11 Other Bayesian work, by Stocker et al,6 utilized the full slope of the psychometric curve in ﬁtting a prior and likelihood to motion data, but did not examine the issue of conﬁdence. Estimation conﬁdence plays a central role in Bayesian formulations as a whole. Understanding how priors affect conﬁdence should have direct bearing on many other Bayesian calculations such as multimodal integration.23 Our model is obviously over-simpliﬁed in a number of ways. First, we described it in terms of tilts and spatial positions; a more complete version should work in the pixel/ﬁltering domain.18, 19 We have also only considered two ﬂanking elements; the model is extendible to a full-ﬁeld surround, whereby smoothness operates along a range of geometric directions, and some directions are more (smoothly) dominant than others. Second, the prior is constructed by summarizing the maximal smoothness information; a more probabilistically correct version should capture the full probability of smoothness in its prior. Third, our model does not incorporate a formal noise representation; however, sensitivities could be inﬂuenced both by stimulus-driven noise and conﬁdence. Fourth, our model does not address attraction in the so-called indirect tilt illusion, thought to be mediated by a different mechanism. Finally, we have yet to account for neurophysiological data within this framework, and incorporate constraints at the neural implementation level. However, versions of our computations are oft suggested for intra-areal and feedback cortical circuits; and smoothness principles form a key part of the association ﬁeld connection scheme in Li’s24 dynamical model of contour integration in V1. Our model is connected to a wealth of literature in computer vision and perception. Notably, occlusion and contour completion might be seen as the extreme example in which there is no likelihood information at all for the center target; a host of papers have shown that under these circumstances, smoothness principles such as elastica and variants explain many aspects of perception. The model is also associated with many studies on contour integration motivated by Gestalt principles;25, 26 and exploration of natural scene statistics and Gestalt,27, 28 including the relation to contour grouping within a Bayesian framework.29, 30 Indeed, our model could be modiﬁed to include a prior from natural scenes. There are various directions for the experimental test and reﬁnement of our model. Most pressing is to determine bias and sensitivity for different center and ﬂanker contrasts. As in the case of motion, our model predicts that when there is more uncertainty in the center element, prior information is more dominant. Another interesting test would be to design a task such that the center element is actually part of a different ﬁgure and unrelated to the ﬂankers; our framework predicts that there would be minimal bias, because of segmentation. Our model should also be applied to other tilt-based illusions such as the Fraser spiral and Z¨ llner. Finally, our model can be applied to other perceptual domains;31 and given o the apparent similarities between the tilt illusion and the tilt after-effect, we plan to extend the model to adaptation, by considering smoothness in time as well as space. Acknowledgements This work was funded by the HHMI (OS, TJS) and the Gatsby Charitable Foundation (PD). We are very grateful to Serge Belongie, Leanne Chukoskie, Philip Meier and Joshua Solomon for helpful discussions. References [1] J J Gibson. Adaptation, after-effect, and contrast in the perception of tilted lines. Journal of Experimental Psychology, 20:553–569, 1937. [2] C Blakemore, R H S Carpentar, and M A Georgeson. Lateral inhibition between orientation detectors in the human visual system. Nature, 228:37–39, 1970. [3] J A Stuart and H M Burian. A study of separation difﬁculty: Its relationship to visual acuity in normal and amblyopic eyes. American Journal of Ophthalmology, 53:471–477, 1962. [4] A Yuille and H H Bulthoff. Perception as bayesian inference. In Knill and Whitman, editors, Bayesian decision theory and psychophysics, pages 123–161. Cambridge University Press, 1996. [5] Y Weiss, E P Simoncelli, and E H Adelson. Motion illusions as optimal percepts. Nature Neuroscience, 5:598–604, 2002. [6] A Stocker and E P Simoncelli. Constraining a bayesian model of human visual speed perception. Adv in Neural Info Processing Systems, 17, 2004. [7] D Kersten, P Mamassian, and A Yuille. Object perception as bayesian inference. Annual Review of Psychology, 55:271–304, 2004. [8] K Kording and D Wolpert. Bayesian integration in sensorimotor learning. Nature, 427:244–247, 2004. [9] L Parkes, J Lund, A Angelucci, J Solomon, and M Morgan. Compulsory averaging of crowded orientation signals in human vision. Nature Neuroscience, 4:739–744, 2001. [10] D G Pelli, M Palomares, and N J Majaj. Crowding is unlike ordinary masking: Distinguishing feature integration from detection. Journal of Vision, 4:1136–1169, 2002. [11] J Solomon, F M Felisberti, and M Morgan. Crowding and the tilt illusion: Toward a uniﬁed account. Journal of Vision, 4:500–508, 2004. [12] J A Bednar and R Miikkulainen. Tilt aftereffects in a self-organizing model of the primary visual cortex. Neural Computation, 12:1721–1740, 2000. [13] C W Clifford, P Wenderoth, and B Spehar. A functional angle on some after-effects in cortical vision. Proc Biol Sci, 1454:1705–1710, 2000. [14] M K Kapadia, G Westheimer, and C D Gilbert. Spatial distribution of contextual interactions in primary visual cortex and in visual perception. J Neurophysiology, 4:2048–262, 2000. [15] C C Chen and C W Tyler. Lateral modulation of contrast discrimination: Flanker orientation effects. Journal of Vision, 2:520–530, 2002. [16] I Mareschal, M P Sceniak, and R M Shapley. Contextual inﬂuences on orientation discrimination: binding local and global cues. Vision Research, 41:1915–1930, 2001. [17] D Mumford. Elastica and computer vision. In Chandrajit Bajaj, editor, Algebraic geometry and its applications. Springer Verlag, 1994. [18] T K Leung and J Malik. Contour continuity in region based image segmentation. In Proc. ECCV, pages 544–559, 1998. [19] E Sharon, A Brandt, and R Basri. Completion energies and scale. IEEE Pat. Anal. Mach. Intell., 22(10), 1997. [20] S W Zucker, C David, A Dobbins, and L Iverson. The organization of curve detection: coarse tangent ﬁelds. Computer Graphics and Image Processing, 9(3):213–234, 1988. [21] S Ullman. Filling in the gaps: the shape of subjective contours and a model for their generation. Biological Cybernetics, 25:1–6, 1976. [22] G E Hinton and A D Brown. Spiking boltzmann machines. Adv in Neural Info Processing Systems, 12, 1998. [23] R A Jacobs. What determines visual cue reliability? Trends in Cognitive Sciences, 6:345–350, 2002. [24] Z Li. A saliency map in primary visual cortex. Trends in Cognitive Science, 6:9–16, 2002. [25] D J Field, A Hayes, and R F Hess. Contour integration by the human visual system: evidence for a local “association ﬁeld”. Vision Research, 33:173–193, 1993. [26] J Beck, A Rosenfeld, and R Ivry. Line segregation. Spatial Vision, 4:75–101, 1989. [27] M Sigman, G A Cecchi, C D Gilbert, and M O Magnasco. On a common circle: Natural scenes and gestalt rules. PNAS, 98(4):1935–1940, 2001. [28] S Mahumad, L R Williams, K K Thornber, and K Xu. Segmentation of multiple salient closed contours from real images. IEEE Pat. Anal. Mach. Intell., 25(4):433–444, 1997. [29] W S Geisler, J S Perry, B J Super, and D P Gallogly. Edge co-occurence in natural images predicts contour grouping performance. Vision Research, 6:711–724, 2001. [30] J H Elder and R M Goldberg. Ecological statistics of gestalt laws for the perceptual organization of contours. Journal of Vision, 4:324–353, 2002. [31] S R Lehky and T J Sejnowski. Neural model of stereoacuity and depth interpolation based on a distributed representation of stereo disparity. 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Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract The misjudgement of tilt in images lies at the heart of entertaining visual illusions and rigorous perceptual psychophysics. [sent-7, score-0.717]

2 We adopt a Bayesian approach to perceptual tilt estimation, showing how a smoothness prior offers a powerful way of addressing much confusing data. [sent-9, score-0.818]

3 Perceptual anomalies and illusions, such as the misjudgements of motion and tilt evident in so many psychophysical experiments, have intrigued researchers for decades. [sent-12, score-0.653]

4 1–3 A Bayesian view4–8 has been particularly inﬂuential in models of motion processing, treating such anomalies as the normative product of prior information (often statistically codifying Gestalt laws) with likelihood information from the actual scenes presented. [sent-13, score-0.221]

5 Here, we expand the range of statistically normative accounts to tilt estimation, for which there are classes of results (on estimation conﬁdence) that are so far not available for motion. [sent-14, score-0.637]

6 The tilt illusion arises when the perceived tilt of a center target is misjudged (ie bias) in the presence of ﬂankers. [sent-15, score-1.428]

7 Another phenomenon, called Crowding, refers to a loss in the conﬁdence (ie sensitivity) of perceived target tilt in the presence of ﬂankers. [sent-16, score-0.686]

8 9, 10 The tilt illusion has been explained by lateral interactions11, 12 in populations of orientationtuned units; and by calibration. [sent-19, score-0.751]

9 First, the geometry of the positional arrangement of the stimuli affects attraction versus repulsion in bias, as emphasized by Kapadia et al14 (ﬁgure 1A), and others. [sent-21, score-0.406]

10 Moreover, current explanations do not offer a computational account of tilt perception as the outcome of a normative inference process. [sent-25, score-0.645]

11 Here, we demonstrate that a Bayesian framework for orientation estimation, with a prior favoring smoothness, can naturally explain a range of seemingly puzzling tilt data. [sent-26, score-0.716]

12 1 -80 -60 -40 -20 0 20 40 60 80 Flanker tilt (deg) Figure 1: Tilt biases and sensitivities in visual perception. [sent-33, score-0.778]

13 (A) Kapadia et al demonstrated the importance of geometry on tilt bias, with bar stimuli in the fovea (and similar results in the periphery). [sent-34, score-0.781]

14 When 5 degrees clockwise ﬂankers are arranged colinearly, the center target appears attracted in the direction of the ﬂankers; when ﬂankers are lateral, the target appears repulsed. [sent-35, score-0.452]

15 14 (B) Solomon et al measured both biases and sensitivities for gratings in the visual periphery. [sent-37, score-0.323]

16 This constitutes the classic tilt illusion, with a repulsive bias percept. [sent-40, score-0.712]

17 In addition, sensitivities vary as a function of ﬂanker angles, in a systematic way (even in cases when there are no biases at all). [sent-41, score-0.158]

18 Sensitivities are given in units of the inverse of standard deviation of the tilt estimate. [sent-42, score-0.576]

19 The prior is determined by the principle of creating a smooth contour between the target and ﬂankers. [sent-49, score-0.307]

20 In section 2 we show experimental data of Kapadia et al and Solomon et al, alongside the model simulations, and demonstrate that the model can account for both geometry, and bias and sensitivity measurements in the data. [sent-51, score-0.377]

21 Our results suggest a more uniﬁed, rational, approach to understanding tilt perception. [sent-52, score-0.576]

22 1 Bayesian model Under our Bayesian model, inference is controlled by the posterior distribution over the tilt of the target element. [sent-53, score-0.756]

23 This comes from the combination of a prior favoring smooth conﬁgurations of the ﬂankers and target, and the likelihood associated with the actual scene. [sent-54, score-0.22]

24 A complete distribution would consider all possible angles and relative spatial positions of the bars, and marginalize the posterior over all but the tilt of the central element. [sent-55, score-0.739]

25 For simplicity, we make two benign approximations: conditionalizing over (ie clamping) the angles of the ﬂankers, and exploring only a small neighborhood of their positions. [sent-56, score-0.111]

26 Smoothness prior: Under these approximations, we consider a given actual conﬁguration (see ﬁg 2A) of ﬂankers f1 = (φ1 , x1 ), f2 = (φ2 , x2 ) and center target c = (φc , xc ), arranged from top to bottom. [sent-58, score-0.234]

27 We have to generate a prior over φc and δ1 = x1 − xc and δ2 = x2 − xc based on the principle of smoothness. [sent-59, score-0.145]

28 17–20 One widely (B) (C) f1 f1 β1 R Probability max smooth Max smooth target (deg) (A) 40 20 0 -20 c δ1 c -40 Φc f2 f2 1 0. [sent-62, score-0.224]

29 2 0 -80 -60 -40 -20 0 20 40 Flanker tilt (deg) 60 80 -80 -60 -40 20 0 20 40 Flanker tilt (deg) 60 80 Figure 2: Geometry and smoothness for ﬂankers, f1 and f2 , and center target, c. [sent-66, score-1.31]

30 (B) The elastica procedure can rotate the target angle (to Φc ) and shift the relative ﬂanker and target positions on the x axis (to δ1 and δ2 ) in its search for the maximally smooth solution. [sent-68, score-0.5]

31 Small spatial shifts (up to 1/15 the size of R) of positions are allowed, but positional shift is overemphasized in the ﬁgure for visibility. [sent-69, score-0.114]

32 (C) Top: center tilt that results in maximal smoothness, as a function of ﬂanker tilt. [sent-70, score-0.621]

33 Note attraction of target towards ﬂankers for small ﬂanker angles; here ﬂankers and target are positioned in a nearly colinear arrangement. [sent-72, score-0.399]

34 Note also repulsion of target away from ﬂankers for intermediate ﬂanker angles. [sent-73, score-0.241]

35 Bottom: P [c, f1 , f2 ] for center tilt that yields maximal smoothness. [sent-74, score-0.621]

36 The two constants, σβ and σR , control the relative contribution to smoothness of the angle versus the spatial distance. [sent-82, score-0.274]

37 Figure 2C shows for given ﬂanker tilts, the center tilt that yields maximal smoothness, and the corresponding probability of smoothness. [sent-87, score-0.621]

38 For near vertical ﬂankers, the spatial lability leads to very weak attraction and high probability of smoothness. [sent-88, score-0.184]

39 These observations are key to our model: the maximally smooth center tilt will inﬂuence attractive and repulsive interactions of tilt estimation; the probability of smoothness will inﬂuence the relative weighting of the prior versus the likelihood. [sent-90, score-1.523]

40 From the smoothness principle, we construct a two dimensional prior (ﬁgure 3A). [sent-91, score-0.19]

41 One dimension represents tilt, the other dimension, the overall positional shift between target (B) Likelihood (D) Marginalized Posterior (C) Posterior 20 0. [sent-92, score-0.194]

42 2 Position Position -10 -5 0 Angle 5 10 Probability clockwise (A) Prior 20 1 0. [sent-101, score-0.124]

43 2 0 -20 -10 0 10 20 Target angle (deg) Counter-clockwise Clockwise Figure 3: Bayes model for example ﬂankers and target. [sent-105, score-0.124]

44 (B) Likelihood 2D distribution for a target tilt of 3 degrees; (C) Posterior 2D distribution. [sent-109, score-0.686]

45 All 2D distributions are drawn on the same grayscale range, and the presence of a larger baseline in the prior causes it to appear more dimmed. [sent-110, score-0.115]

46 (E) For this target tilt, we calculate probability clockwise, and obtain one point on psychometric curve. [sent-113, score-0.222]

47 22 The Gaussian is centered at the estimated smoothest target angle and relative position, and the baseline is determined by the probability of smoothness. [sent-116, score-0.298]

48 The baseline, and its dependence on the ﬂanker orientation, is a key difference from Weiss et al’s Gaussian prior for smooth, slow motion. [sent-117, score-0.116]

49 Likelihood: The likelihood over tilt and position (ﬁgure 3B) is determined by a 2D Gaussian distribution with an added baseline. [sent-120, score-0.668]

50 22 The Gaussian is centered at the actual target tilt; and at a position taken as zero, since this is the actual position, to which the prior is compared. [sent-121, score-0.293]

51 Posterior and marginalization: The posterior comes from multiplying likelihood and prior (ﬁgure 3C) and then marginalizing over position to obtain a 1D distribution over tilt. [sent-123, score-0.216]

52 Other likelihoods, with closer agreement between target and smooth prior, give unimodal distributions. [sent-125, score-0.167]

53 Note that the bimodality is a direct consequence of having an added baseline to the prior and likelihood (if these were Gaussian without a baseline, the posterior would always be Gaussian). [sent-126, score-0.224]

54 We deﬁne α as the mean angle of the 1D posterior distribution (eg, value of dashed line on the x axis), and β as the height of the probability distribution at that mean angle (eg, height of dashed line). [sent-128, score-0.249]

55 Decision of probability clockwise: The probability of a clockwise tilt is estimated from the marginalized posterior: 1 P = 1 + exp (3) −α. [sent-130, score-0.728]

56 Bias and sensitivity: For one target tilt, we generate a single probability and therefore a single point on the psychometric function relating tilt to the probability of choosing clockwise. [sent-135, score-0.798]

57 (A) Flankers are tilted 5 degrees clockwise (black curve) or anti-clockwise (gray) of vertical, and positioned spatially in a colinear arrangement. [sent-138, score-0.318]

58 The center bar appears tilted in the direction of the ﬂankers (attraction), as can be seen by the attractive shift of the psychometric curve. [sent-139, score-0.251]

59 The boxed stimuli cartoon illustrates a vertical target amidst the ﬂankers. [sent-140, score-0.231]

60 11 2 Results: data versus model We ﬁrst consider the geometry of the center and ﬂanker conﬁgurations, modeling the full psychometric curve for colinear and parallel ﬂanks (recall that ﬁgure 1A showed summary biases). [sent-147, score-0.386]

61 Figure 4A;B demonstrates attraction in the data and model; that is, the psychometric curve is shifted towards the ﬂanker, because of the nature of smooth completions for colinear ﬂankers. [sent-148, score-0.409]

62 The smoothest solution in the model arises by shifting the target estimate away from the ﬂankers. [sent-151, score-0.204]

63 This shift is rather minor, because the conﬁguration has a low probability of smoothness (similar to ﬁgure 2C), and thus the prior exerts only a weak effect. [sent-152, score-0.26]

64 The above results show examples of changes in the psychometric curve, but do not address both bias and, particularly, sensitivity, across a whole range of ﬂanker conﬁgurations. [sent-153, score-0.199]

65 Figure 5 depicts biases and sensitivity from Solomon et al, versus the Bayes model. [sent-154, score-0.223]

66 In ﬁgure 5A, bias is shown, for the condition that both ﬂankers are tilted at the same angle. [sent-156, score-0.154]

67 The data exhibit small attraction at near vertical ﬂanker angles (this arrangement is close to colinear); large repulsion at intermediate ﬂanker angles of 22. [sent-157, score-0.457]

68 5 and 45 degrees from vertical; and minimal repulsion at large angles from vertical. [sent-158, score-0.227]

69 For intermediate ﬂanker angles, the smoothest solution in the model is repulsive, and the effect of the prior is strong enough to induce a signiﬁcant repulsion. [sent-160, score-0.181]

70 For large angles, the prior exerts almost no effect. [sent-161, score-0.12]

71 In the data (Figure 5C), there is most loss in sensitivity at intermediate ﬂanker angles of 22. [sent-163, score-0.227]

72 5 and 45 degrees (ie, the subject is less certain); and sensitivity is higher for near vertical or near horizontal ﬂankers. [sent-164, score-0.211]

73 1 -20 0 20 40 Flanker tilt (deg) 60 80 60 80 -80 -60 -40 -20 0 20 40 Flanker tilt (deg) -80 -60 -40 -20 0 20 40 Flanker tilt (deg) 60 80 -20 0 20 40 Flanker tilt (deg) 60 80 (F) Bias (deg) 10 5 0 -5 0. [sent-173, score-2.304]

74 1 -80 (D) 10 -10 -10 80 Sensitivity (1/deg) -80 5 0 -5 -80 -80 -60 -60 -40 -40 -20 0 20 40 Flanker tilt (deg) -20 0 20 40 Flanker tilt (deg) 60 -10 80 (H) 60 80 Sensitivity (1/deg) Sensititvity (1/deg) Bias (deg) 0. [sent-179, score-1.152]

75 1 -80 -60 -40 Figure 5: Solomon et al data11 (subject FF), versus Bayesian model. [sent-191, score-0.126]

76 (A) Data and (B) model biases with same-tilted ﬂankers; (C) Data and (D) model sensitivities with same-tilted ﬂankers; (E;G) data and (F;H) model as above, but for opposite-tilted ﬂankers (note that opposite-tilted data was collected for less ﬂanker angles). [sent-192, score-0.227]

77 Each point in the ﬁgure is derived by ﬁtting a cummulative Gaussian distribution N (µ, σ) to corresponding psychometric curve, and setting bias 1 equal to µ and sensitivity to σ . [sent-193, score-0.308]

78 In all experiments, ﬂanker and target gratings are presented in the visual periphery. [sent-194, score-0.179]

79 For ﬂankers that are far from vertical, the prior has minimal effect because one cannot ﬁnd a smooth solution (eg, the likelihood dominates), and thus sensitivity is higher. [sent-198, score-0.277]

80 The low sensitivity at intermediate angles arises because the prior has considerable effect; and there is conﬂict between the prior (tilt, position), and likelihood (tilt, position). [sent-199, score-0.437]

81 This leads to uncertainty in the target angle estimation . [sent-200, score-0.235]

82 For ﬂankers near vertical, the prior exerts a strong effect; but there is less conﬂict between the likelihood and prior estimates (tilt, position) for a vertical target. [sent-201, score-0.291]

83 The only aspect that our model does not reproduce is the (more subtle) sensitivity difference between 0 and +/- 5 degree ﬂankers. [sent-203, score-0.132]

84 The bias is now close to zero in the data (Figure 5E) and model (Figure 5F), as would be expected (since the maximally smooth angle is now always roughly vertical). [sent-205, score-0.268]

85 Perhaps more surprisingly, the sensitivities continue to to be non-ﬂat in the data (Figure 5G) and model (Figure 5H). [sent-206, score-0.136]

86 As before, there is most loss in sensitivity at intermediate angles. [sent-208, score-0.141]

87 3 Discussion We applied a Bayesian framework to the widely studied tilt illusion, and demonstrated the model on examples from two different data sets involving foveal and peripheral estimation. [sent-213, score-0.621]

88 4–8 Our model accounts correctly for both attraction and repulsion, determined by the smoothness prior and the geometry of the scene. [sent-215, score-0.358]

89 11 Other Bayesian work, by Stocker et al,6 utilized the full slope of the psychometric curve in ﬁtting a prior and likelihood to motion data, but did not examine the issue of conﬁdence. [sent-219, score-0.326]

90 18, 19 We have also only considered two ﬂanking elements; the model is extendible to a full-ﬁeld surround, whereby smoothness operates along a range of geometric directions, and some directions are more (smoothly) dominant than others. [sent-224, score-0.136]

91 Second, the prior is constructed by summarizing the maximal smoothness information; a more probabilistically correct version should capture the full probability of smoothness in its prior. [sent-225, score-0.303]

92 Third, our model does not incorporate a formal noise representation; however, sensitivities could be inﬂuenced both by stimulus-driven noise and conﬁdence. [sent-226, score-0.136]

93 Fourth, our model does not address attraction in the so-called indirect tilt illusion, thought to be mediated by a different mechanism. [sent-227, score-0.693]

94 However, versions of our computations are oft suggested for intra-areal and feedback cortical circuits; and smoothness principles form a key part of the association ﬁeld connection scheme in Li’s24 dynamical model of contour integration in V1. [sent-229, score-0.199]

95 The model is also associated with many studies on contour integration motivated by Gestalt principles;25, 26 and exploration of natural scene statistics and Gestalt,27, 28 including the relation to contour grouping within a Bayesian framework. [sent-232, score-0.149]

96 Most pressing is to determine bias and sensitivity for different center and ﬂanker contrasts. [sent-235, score-0.241]

97 As in the case of motion, our model predicts that when there is more uncertainty in the center element, prior information is more dominant. [sent-236, score-0.145]

98 Finally, our model can be applied to other perceptual domains;31 and given o the apparent similarities between the tilt illusion and the tilt after-effect, we plan to extend the model to adaptation, by considering smoothness in time as well as space. [sent-239, score-1.462]

99 Constraining a bayesian model of human visual speed perception. [sent-259, score-0.131]

100 Crowding and the tilt illusion: Toward a uniﬁed account. [sent-274, score-0.576]

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tfidf for this paper:

wordName wordTfidf (topN-words)

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same-paper 1 0.99999988 3 nips-2005-A Bayesian Framework for Tilt Perception and Confidence

Author: Odelia Schwartz, Peter Dayan, Terrence J. Sejnowski

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Abstract: We extend a previously developed Bayesian framework for perception to account for sensory adaptation. We ﬁrst note that the perceptual effects of adaptation seems inconsistent with an adjustment of the internally represented prior distribution. Instead, we postulate that adaptation increases the signal-to-noise ratio of the measurements by adapting the operational range of the measurement stage to the input range. We show that this changes the likelihood function in such a way that the Bayesian estimator model can account for reported perceptual behavior. In particular, we compare the model’s predictions to human motion discrimination data and demonstrate that the model accounts for the commonly observed perceptual adaptation effects of repulsion and enhanced discriminability. 1 Motivation A growing number of studies support the notion that humans are nearly optimal when performing perceptual estimation tasks that require the combination of sensory observations with a priori knowledge. The Bayesian formulation of these problems deﬁnes the optimal strategy, and provides a principled yet simple computational framework for perception that can account for a large number of known perceptual effects and illusions, as demonstrated in sensorimotor learning [1], cue combination [2], or visual motion perception [3], just to name a few of the many examples. Adaptation is a fundamental phenomenon in sensory perception that seems to occur at all processing levels and modalities. A variety of computational principles have been suggested as explanations for adaptation. Many of these are based on the concept of maximizing the sensory information an observer can obtain about a stimulus despite limited sensory resources [4, 5, 6]. More mechanistically, adaptation can be interpreted as the attempt of the sensory system to adjusts its (limited) dynamic range such that it is maximally informative with respect to the statistics of the stimulus. A typical example is observed in the retina, which manages to encode light intensities that vary over nine orders of magnitude using ganglion cells whose dynamic range covers only two orders of magnitude. This is achieved by adapting to the local mean as well as higher order statistics of the visual input over short time-scales [7]. ∗ corresponding author. If a Bayesian framework is to provide a valid computational explanation of perceptual processes, then it needs to account for the behavior of a perceptual system, regardless of its adaptation state. In general, adaptation in a sensory estimation task seems to have two fundamental effects on subsequent perception: • Repulsion: The estimate of parameters of subsequent stimuli are repelled by those of the adaptor stimulus, i.e. the perceived values for the stimulus variable that is subject to the estimation task are more distant from the adaptor value after adaptation. This repulsive effect has been reported for perception of visual speed (e.g. [8, 9]), direction-of-motion [10], and orientation [11]. • Increased sensitivity: Adaptation increases the observer’s discrimination ability around the adaptor (e.g. for visual speed [12, 13]), however it also seems to decrease it further away from the adaptor as shown in the case of direction-of-motion discrimination [14]. In this paper, we show that these two perceptual effects can be explained within a Bayesian estimation framework of perception. Note that our description is at an abstract functional level - we do not attempt to provide a computational model for the underlying mechanisms responsible for adaptation, and this clearly separates this paper from other work which might seem at ﬁrst glance similar [e.g., 15]. 2 Adaptive Bayesian estimator framework Suppose that an observer wants to estimate a property of a stimulus denoted by the variable θ, based on a measurement m. In general, the measurement can be vector-valued, and is corrupted by both internal and external noise. Hence, combining the noisy information gained by the measurement m with a priori knowledge about θ is advantageous. According to Bayes’ rule 1 p(θ|m) = p(m|θ)p(θ) . (1) α That is, the probability of stimulus value θ given m (posterior) is the product of the likelihood p(m|θ) of the particular measurement and the prior p(θ). The normalization constant α serves to ensure that the posterior is a proper probability distribution. Under the assumpˆ tion of a squared-error loss function, the optimal estimate θ(m) is the mean of the posterior, thus ∞ ˆ θ(m) = θ p(θ|m) dθ . (2) 0 ˆ Note that θ(m) describes an estimate for a single measurement m. As discussed in [16], the measurement will vary stochastically over the course of many exposures to the same stimulus, and thus the estimator will also vary. We return to this issue in Section 3.2. Figure 1a illustrates a Bayesian estimator, in which the shape of the (arbitrary) prior distribution leads on average to a shift of the estimate toward a lower value of θ than the true stimulus value θstim . The likelihood and the prior are the fundamental constituents of the Bayesian estimator model. Our goal is to describe how adaptation alters these constituents so as to account for the perceptual effects of repulsion and increased sensitivity. Adaptation does not change the prior ... An intuitively sensible hypothesis is that adaptation changes the prior distribution. Since the prior is meant to reﬂect the knowledge the observer has about the distribution of occurrences of the variable θ in the world, repeated viewing of stimuli with the same parameter a b probability probability attraction ! posterior likelihood prior modified prior Ã θ θ ˆ θ' θ θadapt Figure 1: Hypothetical model in which adaptation alters the prior distribution. a) Unadapted Bayesian estimation conﬁguration in which the prior leads to a shift of the estimate ˆ θ, relative to the stimulus parameter θstim . Both the likelihood function and the prior distriˆ bution contribute to the exact value of the estimate θ (mean of the posterior). b) Adaptation acts by increasing the prior distribution around the value, θadapt , of the adapting stimulus ˆ parameter. Consequently, an subsequent estimate θ of the same stimulus parameter value θstim is attracted toward the adaptor. This is the opposite of observed perceptual effects, and we thus conclude that adjustments of the prior in a Bayesian model do not account for adaptation. value θadapt should presumably increase the prior probability in the vicinity of θadapt . Figure 1b schematically illustrates the effect of such a change in the prior distribution. The estimated (perceived) value of the parameter under the adapted condition is attracted to the adapting parameter value. In order to account for observed perceptual repulsion effects, the prior would have to decrease at the location of the adapting parameter, a behavior that seems fundamentally inconsistent with the notion of a prior distribution. ... but increases the reliability of the measurements Since a change in the prior distribution is not consistent with repulsion, we are led to the conclusion that adaptation must change the likelihood function. But why, and how should this occur? In order to answer this question, we reconsider the functional purpose of adaptation. We assume that adaptation acts to allocate more resources to the representation of the parameter values in the vicinity of the adaptor [4], resulting in a local increase in the signal-to-noise ratio (SNR). This can be accomplished, for example, by dynamically adjusting the operational range to the statistics of the input. This kind of increased operational gain around the adaptor has been effectively demonstrated in the process of retinal adaptation [17]. In the context of our Bayesian estimator framework, and restricting to the simple case of a scalar-valued measurement, adaptation results in a narrower conditional probability density p(m|θ) in the immediate vicinity of the adaptor, thus an increase in the reliability of the measurement m. This is offset by a broadening of the conditional probability density p(m|θ) in the region beyond the adaptor vicinity (we assume that total resources are conserved, and thus an increase around the adaptor must necessarily lead to a decrease elsewhere). Figure 2 illustrates the effect of this local increase in signal-to-noise ratio on the likeli- unadapted adapted θadapt p(m2| θ )' 1/SNR θ θ θ1 θ2 θ1 p(m2|θ) θ2 θ m2 p(m1| θ )' m1 m m p(m1|θ) θ θ θ θadapt p(m| θ2)' p(m|θ2) likelihoods p(m|θ1) p(m| θ1)' p(m|θadapt )' conditionals Figure 2: Measurement noise, conditionals and likelihoods. The two-dimensional conditional density, p(m|θ), is shown as a grayscale image for both the unadapted and adapted cases. We assume here that adaptation increases the reliability (SNR) of the measurement around the parameter value of the adaptor. This is balanced by a decrease in SNR of the measurement further away from the adaptor. Because the likelihood is a function of θ (horizontal slices, shown plotted at right), this results in an asymmetric change in the likelihood that is in agreement with a repulsive effect on the estimate. a b ^ ∆θ ^ ∆θ [deg] + 0 60 30 0 -30 - θ θ adapt -60 -180 -90 90 θadapt 180 θ [deg] Figure 3: Repulsion: Model predictions vs. human psychophysics. a) Difference in perceived direction in the pre- and post-adaptation condition, as predicted by the model. Postadaptive percepts of motion direction are repelled away from the direction of the adaptor. b) Typical human subject data show a qualitatively similar repulsive effect. Data (and ﬁt) are replotted from [10]. hood function. The two gray-scale images represent the conditional probability densities, p(m|θ), in the unadapted and the adapted state. They are formed by assuming additive noise on the measurement m of constant variance (unadapted) or with a variance that decreases symmetrically in the vicinity of the adaptor parameter value θadapt , and grows slightly in the region beyond. In the unadapted state, the likelihood is convolutional and the shape and variance are equivalent to the distribution of measurement noise. However, in the adapted state, because the likelihood is a function of θ (horizontal slice through the conditional surface) it is no longer convolutional around the adaptor. As a result, the mean is pushed away from the adaptor, as illustrated in the two graphs on the right. Assuming that the prior distribution is fairly smooth, this repulsion effect is transferred to the posterior distribution, and thus to the estimate. 3 Simulation Results We have qualitatively demonstrated that an increase in the measurement reliability around the adaptor is consistent with the repulsive effects commonly seen as a result of perceptual adaptation. In this section, we simulate an adapted Bayesian observer by assuming a simple model for the changes in signal-to-noise ratio due to adaptation. We address both repulsion and changes in discrimination threshold. In particular, we compare our model predictions with previously published data from psychophysical experiments examining human perception of motion direction. 3.1 Repulsion In the unadapted state, we assume the measurement noise to be additive and normally distributed, and constant over the whole measurement space. Thus, assuming that m and θ live in the same space, the likelihood is a Gaussian of constant width. In the adapted state, we assume a simple functional description for the variance of the measurement noise around the adapter. Speciﬁcally, we use a constant plus a difference of two Gaussians, a b relative discrimination threshold relative discrimination threshold 1.8 1 θ θadapt 1.6 1.4 1.2 1 0.8 -40 -20 θ adapt 20 40 θ [deg] Figure 4: Discrimination thresholds: Model predictions vs. human psychophysics. a) The model predicts that thresholds for direction discrimination are reduced at the adaptor. It also predicts two side-lobes of increased threshold at further distance from the adaptor. b) Data of human psychophysics are in qualitative agreement with the model. Data are replotted from [14] (see also [11]). each having equal area, with one twice as broad as the other (see Fig. 2). Finally, for simplicity, we assume a ﬂat prior, but any reasonable smooth prior would lead to results that are qualitatively similar. Then, according to (2) we compute the predicted estimate of motion direction in both the unadapted and the adapted case. Figure 3a shows the predicted difference between the pre- and post-adaptive average estimate of direction, as a function of the stimulus direction, θstim . The adaptor is indicated with an arrow. The repulsive effect is clearly visible. For comparison, Figure 3b shows human subject data replotted from [10]. The perceived motion direction of a grating was estimated, under both adapted and unadapted conditions, using a two-alternative-forced-choice experimental paradigm. The plot shows the change in perceived direction as a function of test stimulus direction relative to that of the adaptor. Comparison of the two panels of Figure 3 indicate that despite the highly simpliﬁed construction of the model, the prediction is quite good, and even includes the small but consistent repulsive effects observed 180 degrees from the adaptor. 3.2 Changes in discrimination threshold Adaptation also changes the ability of human observers to discriminate between the direction of two different moving stimuli. In order to model discrimination thresholds, we need to consider a Bayesian framework that can account not only for the mean of the estimate but also its variability. We have recently developed such a framework, and used it to quantitatively constrain the likelihood and the prior from psychophysical data [16]. This framework accounts for the effect of the measurement noise on the variability of the ˆ ˆ estimate θ. Speciﬁcally, it provides a characterization of the distribution p(θ|θstim ) of the estimate for a given stimulus direction in terms of its expected value and its variance as a function of the measurement noise. As in [16] we write ˆ ∂ θ(m) 2 ˆ var θ|θstim = var m ( ) |m=θstim . (3) ∂m Assuming that discrimination threshold is proportional to the standard deviation, ˆ var θ|θstim , we can now predict how discrimination thresholds should change after adaptation. Figure 4a shows the predicted change in discrimination thresholds relative to the unadapted condition for the same model parameters as in the repulsion example (Figure 3a). Thresholds are slightly reduced at the adaptor, but increase symmetrically for directions further away from the adaptor. For comparison, Figure 4b shows the relative change in discrimination thresholds for a typical human subject [14]. Again, the behavior of the human observer is qualitatively well predicted. 4 Discussion We have shown that adaptation can be incorporated into a Bayesian estimation framework for human sensory perception. Adaptation seems unlikely to manifest itself as a change in the internal representation of prior distributions, as this would lead to perceptual bias effects that are opposite to those observed in human subjects. Instead, we argue that adaptation leads to an increase in reliability of the measurement in the vicinity of the adapting stimulus parameter. We show that this change in the measurement reliability results in changes of the likelihood function, and that an estimator that utilizes this likelihood function will exhibit the commonly-observed adaptation effects of repulsion and changes in discrimination threshold. We further conﬁrm our model by making quantitative predictions and comparing them with known psychophysical data in the case of human perception of motion direction. Many open questions remain. The results demonstrated here indicate that a resource allocation explanation is consistent with the functional effects of adaptation, but it seems unlikely that theory alone can lead to a unique quantitative prediction of the detailed form of these effects. Speciﬁcally, the constraints imposed by biological implementation are likely to play a role in determining the changes in measurement noise as a function of adaptor parameter value, and it will be important to characterize and interpret neural response changes in the context of our framework. Also, although we have argued that changes in the prior seem inconsistent with adaptation effects, it may be that such changes do occur but are offset by the likelihood effect, or occur only on much longer timescales. Last, if one considers sensory perception as the result of a cascade of successive processing stages (with both feedforward and feedback connections), it becomes necessary to expand the Bayesian description to describe this cascade [e.g., 18, 19]. For example, it may be possible to interpret this cascade as a sequence of Bayesian estimators, in which the measurement of each stage consists of the estimate computed at the previous stage. Adaptation could potentially occur in each of these processing stages, and it is of fundamental interest to understand how such a cascade can perform useful stable computations despite the fact that each of its elements is constantly readjusting its response properties. References [1] K. K¨ rding and D. Wolpert. Bayesian integration in sensorimotor learning. o 427(15):244–247, January 2004. Nature, [2] D C Knill and W Richards, editors. Perception as Bayesian Inference. Cambridge University Press, 1996. [3] Y. Weiss, E. Simoncelli, and E. Adelson. Motion illusions as optimal percept. Nature Neuroscience, 5(6):598–604, June 2002. [4] H.B. Barlow. Vision: Coding and Efﬁciency, chapter A theory about the functional role and synaptic mechanism of visual after-effects, pages 363–375. Cambridge University Press., 1990. [5] M.J. Wainwright. Visual adaptation as optimal information transmission. Vision Research, 39:3960–3974, 1999. [6] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling maximizes information transmission. Neuron, 26:695–702, June 2000. [7] S.M. Smirnakis, M.J. Berry, D.K. Warland, W. Bialek, and M. Meister. Adaptation of retinal processing to image contrast and spatial scale. Nature, 386:69–73, March 1997. [8] P. Thompson. Velocity after-effects: the effects of adaptation to moving stimuli on the perception of subsequently seen moving stimuli. Vision Research, 21:337–345, 1980. [9] A.T. Smith. Velocity coding: evidence from perceived velocity shifts. Vision Research, 25(12):1969–1976, 1985. [10] P. Schrater and E. Simoncelli. Local velocity representation: evidence from motion adaptation. Vision Research, 38:3899–3912, 1998. [11] C.W. Clifford. Perceptual adaptation: motion parallels orientation. Trends in Cognitive Sciences, 6(3):136–143, March 2002. [12] C. Clifford and P. Wenderoth. Adaptation to temporal modulaton can enhance differential speed sensitivity. Vision Research, 39:4324–4332, 1999. [13] A. Kristjansson. Increased sensitivity to speed changes during adaptation to ﬁrst-order, but not to second-order motion. Vision Research, 41:1825–1832, 2001. [14] R.E. Phinney, C. Bowd, and R. Patterson. Direction-selective coding of stereoscopic (cyclopean) motion. Vision Research, 37(7):865–869, 1997. [15] N.M. Grzywacz and R.M. Balboa. A Bayesian framework for sensory adaptation. Neural Computation, 14:543–559, 2002. [16] A.A. Stocker and E.P. Simoncelli. Constraining a Bayesian model of human visual speed perception. In Lawrence K. Saul, Yair Weiss, and L´ on Bottou, editors, Advances in Neural Infore mation Processing Systems NIPS 17, pages 1361–1368, Cambridge, MA, 2005. MIT Press. [17] D. Tranchina, J. Gordon, and R.M. Shapley. Retinal light adaptation – evidence for a feedback mechanism. Nature, 310:314–316, July 1984. ´ [18] S. Deneve. Bayesian inference in spiking neurons. In Lawrence K. Saul, Yair Weiss, and L eon Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), vol 17, Cambridge, MA, 2005. MIT Press. [19] R. Rao. Hierarchical Bayesian inference in networks of spiking neurons. In Lawrence K. Saul, Yair Weiss, and L´ on Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), e vol 17, Cambridge, MA, 2005. MIT Press.

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