The tilt illusion is a paradigmatic example of contextual influences on perception. We analyze it in terms of a neural population model for the perceptual organization of visual orientation. In turn, this is based on a well-found treatment of natural scene statistics, known as the Gaussian Scale Mixture model. This model is closely related to divisive gain control in neural processing and has been extensively applied in the image processing and statistical learning communities; however, its implications for contextual effects in biological vision have not been studied. In our model, oriented neural units associated with surround tilt stimuli participate in divisively normalizing the activities of the units representing a center stimulus, thereby changing their tuning curves. We show that through standard population decoding, these changes lead to the forms of repulsion and attraction observed in the tilt illusion. The issues in our model readily generalize to other visual attributes and contextual phenomena, and should lead to more rigorous treatments of contextual effects based on natural scene statistics.  At its most basic, perceptual organization concerns which sensory stimuli are treated collectively, and what consequences there are for each stimulus in being either aggregated with, or separated from, the others (e.g., Behrmann, Kimchi, & Olson, 2003; Wertheimer, 1923) . One central assumption is that the logic of this organization depends on the statistics of natural images, through phylogenetic and/or ontogenetic programming (e.g., Attneave, 1954; Barlow, 1961; Brunswik & Kamiya, 1953; Elder & Goldberg, 2002; Geisler, Perry, Super, & Gallogly, 2001; Krüger, 1998; Sigman, Cecchi, Gilbert, & Magnasco, 2001) . In this paper, we consider how the tilt illusion may arise as a downstream consequence of perceptual organization. In the standard (direct) form of the tilt illusion, the perceived orientation of a central bar of light is tilted away from the orientation of a surround grating that forms its visual context (Figure 1a, left) . Under particular circumstances, one can also observe a weaker but consistently attractive effect (called the indirect tilt illusion) toward the surround orientation (Figure 1a , right). The tilt illusion is an appealing target for elucidating perceptual organization since its psychological (e.g., Goddard, Clifford, & Solomon, 2008; Wenderoth & Johnstone, 1988; Westheimer, 1990 , and see Figures 1a and 1b) and neural (e.g., Cavanaugh, Bair, & Movshon, 2002; Felsen, Touryan, & Dan, 2005; Gilbert & Wiesel, 1990; Li, Thier, & Wehrhahn, 2000; Sengpiel, Sen, & Blakemore, 1997 , and see Figure 1c ) bases have been extensively probed. The illusion has also been a focus of theoretical interest (e.g., Clifford, Wenderoth, & Spehar, 2000; Gibson & Radner, 1937; Schwartz, Hsu, & Dayan, 2007; Series, Lorenceau, & Frégnac, 2003; Solomon & Morgan, 2006) . Indeed, it has long been known that simple, mechanistic, changes to tuning curves at the neural level might underlie observed contextual biases in the tilt illusion (and also in aftereffects, a related phenomenon associated with adaptation, e.g., Jin, Dragoi, Sur, & Seung, 2005; Kohn, 2007; Kohn & Movshon, 2004; Teich & Qian, 2003) . However, why the surround stimulus might lead to these tuning curve changes, and the Journal of Vision (2009) 9(4) Wegmann, & Barth, 1993) . In particular, it has been shown that filters with similar orientation preferences representing nearby spatial locations have statistically correlated magnitudes. However, only one main computational implication of this fact has been drawn, namely the redundancy reduction conclusion (Barlow, 1961) that the correlations can be reduced (Schwartz & Simoncelli, 2001b ) by a nonlinear model of primary visual cortical neurons, known as divisive gain control (Carandini, Heeger, & Movshon, 1997; Geisler & Albrecht, 1992; Heeger, 1992; Wilson & Humanski, 1993) , by dividing out the common magnitude of the correlation. Further, the implications of the coordinated statistics have only been considered for contextual effects at the single neural unit level; not at the level of populations of neurons, nor perception. Moreover, modern versions of the statistical model known as the Gaussian Scale Mixture model (GSM, described in Background to modeling approach section; Andrews & Mallows, 1974; Portilla, Strela, Wainwright, & Simoncelli, 2003; Schwartz et al., 2006b; Wainwright & Simoncelli, 2000; Wainwright et al., 2001) , have only been applied in domains outside neuroscience, such as image processing and statistical learning. In this paper we study the implications of the GSM model for orientation surround effects both at the neural population and perceptual levels, with an emphasis on the tilt illusion. A key question for the model is which filters are statistically coordinated and thereby subject to common divisive gain control. We argue that the direct tilt illusion arises when filter activations in the center and surround locations are coordinated, intuitively because natural visual objects and textures are spatially extended. We demonstrate that the resulting contextual gain control leads to tilt repulsion, through population decoding. However, along with continuous regions that are part of the same object or texture, images also contain regions of segmentation (Blake & Zisserman, 1987; Geisler, 2008; Geman & Geman, 1984; Mumford & Shah, 1989; Shi & Malik, 2000) . We suggest that when segmentation is a more likely explanation of the center and surround stimuli, then the gain control pool of the center filter should place less weight on the surround filter activations. We show that this leads to the indirect tilt illusion. There has been a number of previous theoretical treatments of the tilt illusion or aftereffect (Bednar & Miikkulainen, 2000; Clifford et al., 2000; Dayan, Sahani, & Deback, 2003; Schwartz, Sejnowski, & Dayan, 2006a; Stocker & Simoncelli, 2006; Wainwright, 1999) . However, previous approaches have not incorporated findings about the nature of statistical coordination in natural scenes, and in most cases have not encompassed both the neural population and perceptual levels. Our main goal in the paper is not to fit the illusion data better than other modeling approaches, but rather to provide a framework with rich links to modern treatments of natural scene statistics. The key innovation in our approach is combining a recent formal model of natural scene statistics widely used in the image processing and statistical learning literature, with a pre-existing model of inference about (i.e., decoding of) orientation from the activity of a population of neurons. We emphasize that throughout the paper, we are not learning the parameters of the GSM model (e.g., as in Schwartz et al., 2006b ). The new goal here is rather to study the implications of the GSM framework, for both the neural and perceptual levels. This work sets the basis for future studies incorporating learning (and adaptation) of the model parameters from natural scenes. We expect this framework to be applicable to studying a wealth of other neural and perceptual contextual data (e.g., Clifford & Rhodes, 2005; Li, 1999; Schwartz et al., 2007) in a more rigorous manner based on scene statistics. We first discuss the background for our approach (Background to modeling approach section). In the Methods section we describe the formal methods and parameters for the neural unit and population decoding model. We next apply the methods to changes in tuning curves and repulsion and attraction in the tilt illusion (Model simulations: Tuning curve changes and the direct tilt illusion section; and Segmentation model and the indirect tilt illusion section). Finally, in the Discussion, we discuss the wider implications of our model and its relationship to the other accounts of the tilt illusion.
Matching to the scene statistics in the texture tilt illusion has been done through parametric image models such as Gaussian Scale Mixture that lead to divisive normalization-like responses (CoenCagli et al., 2009; REF .
Perceptual organization in the tilt illusion
Journal of vision
Journal of vision
Medicine Psychology