A Bayesian framework for cue integration in multistable grouping: Proximity, collinearity, and orientation priors in zigzag lattices Laboratory of Experimental Psychology, Department of Psychology, University of Leuven, Belgium,& Institute of Biomedical Sciences, University of São Paulo, Brazil Peter M. E. Claessens Laboratory of Experimental Psychology, Department of Psychology, University of Leuven, Belgium Johan Wagemans Integration of proximity and good continuation cues is analyzed as a probabilistic inference problem in contour grouping. A Bayesian framework was tested in a multistable dot lattice experiment. In rectangular lattices, distance ratio and global orientation of rows and columns were manipulated. Discollinearity was introduced by imposing zigzag in one orientation, by either fixed or stochastic displacement of elements. Results indicate that proximity and good continuation are generally treated as independent sources of information, added to prior orientation log-odds to produce the odds of grouping percepts. Distance likelihood is well captured by a power law, and discollinearity likelihoods by generalized Laplace distributions, with higher kurtosis for stochastic zigzag. While observers prefer vertical over horizontal orientations, the exact prior distribution is idiosyncratic. Perceptual grouping along cardinal axes is less affected by distance, but more by discollinearity, than along oblique orientations. Results are qualitatively and quantitatively compared to ecological statistics of contours (J. H. Elder & R. M. Goldberg, 2002). The potential of hierarchically extended Bayes models for a better understanding of principles in cue integration is discussed. Keywords: perceptual organization, grouping, Bayesian inference, cue integration, proximity, collinearity, good continuation, contour integration Citation: Claessens, P. M. E., & Wagemans, J. (2008). A Bayesian framework for cue integration in multistable grouping: Proximity, collinearity, and orientation priors in zigzag lattices. Journal of Vision, 8(7):33, 1–23, http://journalofvision.org/8/7/ 33/, doi:10.1167/8.7.33. Introduction The current paper focuses on combination of informa- tion sources and prior knowledge relevant for perceptual grouping, more specifically applied to the integration of proximity and collinearity to a global orientation percept. We analyze grouping of local elements into lines as a problem of contour integration. Historical and experimental paradigms in grouping Two experimental paradigms have been popular in the psychophysical investigation of grouping mechanisms. In the contour detection paradigm, dots or Gabor patches which together define a virtual line of a certain length are embedded in a background of otherwise unrelated similar elements (Beck, Rosenfeld, & Ivry, 1989; Chinnis & Uttal, 1974; Field, Hayes, & Hess, 1993; Hansen & Hess, 2006; Kova ´cs & Julesz, 1993; Uttal, 1975; Uttal, Bunnell, & Corwin, 1970). Participants in this line of experiments are to indicate whether or not they were able to discern a contour, for example, in a yes/no or two-alternative forced choice design. Other research groups made use of stimuli in which two or more candidate groupings were simulta- neously available (Ben-Av & Sagi, 1995; Claessens & Wagemans, 2005; Gepshtein & Kubovy, 2000, 2005; Kubovy, Holcombe, & Wagemans, 1998; Kubovy & Wagemans, 1995; Kurylo, 1997; Oyama, 1961; Rock & Brosgole, 1964; Zucker, Stevens, & Sander, 1983). Grouping under these conditions turns out to be a multi- stable phenomenon: prolonged or repeated presentation of one and the same stimulus potentially evokes different percepts. Note the difference with the contour detection paradigm: while contour detection measures detectability of a single contour in background noise, the multistable grouping paradigm measures relative preference among various candidates that are all valid perceptual solutions. Foundations of Bayesian integration of grouping cues The Bayesian framework has been valuable in explaining how information, from one or several sources, is combined with prior knowledge in perceptual Journal of Vision (2008) 8(7):33, 1–23 http://journalofvision.org/8/7/33/ 1 doi: 10.1167/8.7.33 Received October 2, 2007; published December 24, 2008 ISSN 1534-7362 * ARVO