articles Motion illusions as optimal percepts 2002 Nature Publishing Group http neurosci nature com Yair Weiss1 Eero P Simoncelli2 and Edward H Adelson3 1 School of Computer Science and Engineering Hebrew University of Jerusalem Givat Ram Campus Jerusalem 91904 Israel 2 Howard Hughes Medical Institute Center for Neural Science and Courant Institute of Mathematical Sciences New York University 4 Washington Place New York New York 10003 USA 3 Brain and Cognitive Sciences Department Massachusetts Institute of Technology 77 Massachusetts Ave Cambridge Massachusetts 02139 USA Correspondence should be addressed to Y W yweiss cs huji ac il Published online 20 May 2002 DOI 10 1038 nn858 The pattern of local image velocities on the retina encodes important environmental information Although humans are generally able to extract this information they can easily be deceived into seeing incorrect velocities We show that these illusions arise naturally in a system that attempts to estimate local image velocity We formulated a model of visual motion perception using standard estimation theory under the assumptions that i there is noise in the initial measurements and ii slower motions are more likely to occur than faster ones We found that specific instantiation of such a velocity estimator can account for a wide variety of psychophysical phenomena The human ability to analyze visual motion in general scenes far exceeds the capabilities of the most sophisticated computer vision algorithms Yet psychophysical experiments show that humans also make some puzzling mistakes misjudging speed or direction of very simple stimuli In this paper we propose that such mistakes of human motion perception represent the best solution of a rational system designed to operate in the presence of uncertainty In both biological and artificial vision systems motion analysis begins with local measurements such as the output of direction selective cells in primary visual cortex1 or of spatial and temporal derivative operators in artificial systems2 3 These are then integrated to generate larger more global motion descriptions The integration process is essential because the initial local motion measurements are ambiguous For example in the vicinity of a contour only the motion component perpendicular to the contour can be determined a phenomenon referred to as the aperture problem 2 4 7 Such an integration stage seems to be consistent with much of the psychophysical8 11 and physiological8 12 14 data Despite the vast amount of psychophysical data published over the past two decades the nature of the integration scheme underlying human motion perception remains unclear This is true even for the simple and widely studied plaid stimulus in which two superimposed oriented gratings translate move without changing shape size or orientation in the image plane Fig 1a Due to the aperture problem each grating s motion is consistent with an infinite number of possible translational velocities lying on a constraint line in the space of all velocities Fig 1b When viewing a single drifting grating in isolation subjects typically perceive it as translating in a direction normal to its contours Fig 1b When two gratings are presented simultaneously subjects often perceive them as a coherent pattern translating with a single motion5 7 How is this coherent pattern motion estimated Most explanations are based on one of three rules7 intersection of constraints IOC vector average VA or feature tracking FT The IOC solution is the unique translation vector consistent with the 598 information of both gratings Graphically this corresponds to the point in velocity space that lies at the intersection of both constraint lines Fig 1b circle The VA solution is the average of the two normal velocities Graphically this corresponds to the point in velocity space that lies halfway between the two normal velocities Fig 1b square An FT solution corresponds to the velocity of some feature of the plaid intensity pattern for example the locations of maximum luminance at the grating intersections 15 16 For plaids the FT and IOC solutions both correspond to the veridical true pattern motion Which of the three rules best describes human perception The answer is not clear depending on the stimulus the perceived pattern motion can be nearly veridical consistent with IOC or FT or closer to the VA solution The relevant stimulus features include relative grating orientation and speed17 19 contrast20 presentation time17 and retinal location17 Similar effects have been reported with stimuli that appear quite different from plaids16 21 For a moving rhombus Fig 2 as for a plaid pattern the motion of each opposing pair of sides is consistent with a constraint line in the space of velocities As shown in the velocity space diagrams Fig 2c and f IOC or FT predicts horizontal motion whereas VA predicts diagonal motion Perceptually however the rhombus appears to move horizontally at high contrast and diagonally at low contrast To further complicate the situation the percept depends on the shape If the rhombus is fattened Fig 2d it appears to move horizontally at both contrasts To view these moving stimuli see http www cs huji ac il yweiss Rhombus One might reason that the visual system uses VA for a thin low contrast rhombus and IOC FT for a thin high contrast rhombus and for a fat rhombus Although a model based on this ad hoc combination of rules certainly fits the data it is clearly not a parsimonious explanation Furthermore each of the idealized rules is limited to stimuli containing straight structures at only two orientations and does not offer a method for computing the normal velocities of those structures One would prefer a single coherent model that could predict the perceived nature neuroscience volume 5 no 6 june 2002 articles a Fig 1 Intersection of constraints a Drifting gratings superimposed in the image plane produce a translating plaid pattern b Dotted lines indicate constraint lines arrows indicate perceived direction of grating viewed in isolation The IOC solution circle is the unique velocity consistent with the constraint lines of both gratings The VA solution square is the average of the two normal velocities There is experimental evidence for both types of combination rule bV y IOC 2002 Nature Publishing Group http neurosci nature com Vx VA velocity of any arbitrary spatiotemporal stimulus that appears to be translating We have developed such a model
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