Symmetry is usually computationally expensive to encode reliably, and yet it is relatively effortless to perceive. Here, we extend F. J. A. M. Poirier and H. R. Wilson's (2006) model for shape perception to account for H. R. Wilson and F. Wilkinson's (2002) data on shape symmetry. Because the model already accounts for shape perception, only minimal neural circuitry is required to enable it to encode shape symmetry as well. The model is composed of three main parts: (1) recovery of object position using large-scale non-Fourier V4-like concentric units that respond at the center of concentric contour segments across orientations, (2) around that recovered object center, curvature mechanisms combine multiplicatively the responses of oriented filters to encode object-centric local shape information, with a preference for convexities, and (3) object-centric symmetry mechanisms. Model and human performances are comparable for symmetry perception of shapes. Moreover, with some improvement of edge recovery, the model can encode symmetry axes in natural images such as faces.

*central region*. The “receptive field” for symmetry in that region has a 2:1 size ratio elongated along the axis of symmetry it encodes (Dakin & Herbert, 1998), is less sensitive to parallel orientations (Dakin & Hess, 1997; Rainville & Kingdom, 2000), and is selective for dot density (Rainville & Kingdom, 2002; but see Dakin & Herbert, 1998). Symmetry in that region is systematically biased toward axes at given orientations (Wenderoth, 2002). Symmetry detection can be equated in peripheral vision if stimuli are increased at the same rate as would be needed for positional tasks (Sally & Gurnsey, 2001). Peripheral symmetry detection is less robust to removal of the central region of the symmetry patch, either because it has a smaller integration window (Tyler, 1999), or because it is less robust to information loss (Poirier & Gurnsey, 2005). In this region, symmetry may be encoded as the coalignment of blobs after filtering with orthogonal orientations (e.g., Dakin & Hess, 1997), or with more specialized symmetry mechanisms (see above).

*outline*. Unlike central information, symmetry information present in the outline is independent of the symmetry axis' orientation (Wenderoth, 2002). Shape symmetry can be detected peripherally (Julesz, 1971; Wilson & Wilkinson, 2002). Similarly, symmetry perception in textured patches benefit from outline information (Gurnsey et al., 1998; Labonté et al., 1995). Shape symmetry may be evaluated by point-by-point comparisons of salient features, such as corners or convexities (Wilson & Wilkinson, 2002), which have been shown to play a more important role in object perception than edges (Attneave, 1954; Bertamini, 2001, 2004; Biederman, 1987; Habak, Wilkinson, Zakher, & Wilson, 2004; Loffler, Wilson, & Wilkinson, 2003; Poirier & Wilson, 2007; Shevelev, Kamenkovich, & Sharaev, 2003; but see Hess, Wang, & Dakin, 1999; Mullen & Beaudot, 2002).

*θ*) using a sum of sinusoid functions of various amplitudes, phases, and frequencies (Wilkinson, Wilson, & Habak, 1998; Figures 1 and 3A):

*R*

_{0}is the mean radius, and

*ω*

_{n},

*A*

_{n}, and

*ϕ*

_{n}are the frequency, amplitude, and phase, respectively, for each radial modulation (

*n*of

*m*) added into the circle. RF patterns are useful to study intermediate-level shape perception because they provide controls on shape, show global shape processing properties (Hess et al., 1999; Jeffrey, Wang, & Birch, 2002; Loffler et al., 2003; Wilkinson et al., 1998), and are easily modified to create natural shapes such as faces (e.g., Wilson, Loffler, & Wilkinson, 2002; Wilson & Wilkinson, 2002). In the context of symmetry perception, phase alignment of the different components was manipulated to vary the degree of symmetry in the patterns (see Figure 1). Other stimulus parameters including the contour's luminance profile were matched to experimental conditions (see Wilson & Wilkinson, 2002).

*y*

_{o}from the filter's receptive field center (see Figures 2, left and 3D–3F):

*R*

_{center},

*θ*) is the receptive field's output), taking one sample at the receptive field's center (

*R*

_{center},

*θ*), two samples at ±Δ

*θ*of the first sample and slightly further away from the center of the object than the first sample (

*R*

_{out},

*θ*± Δ

*θ*), and the last two samples are also positioned at ±Δ

*θ*but placed slightly inward (

*R*

_{in},

*θ*± Δ

*θ*). The radii of the inward and outward samples were scaled with distance from the object center, thereby providing size constancy over a larger range of object sizes.

*S*) from center (

*R*

_{center},

*θ*) and inward samples (

*R*

_{in},

*θ*± Δ

*θ*) gives peak responses for accentuated convex curvatures (see Figures 3J and 3L), whereas combining responses from center (

*R*

_{center},

*θ*) and outward samples (

*R*

_{out},

*θ*± Δ

*θ*) gives a peak response for straighter or even concave deviations from a circle. Sampled contour responses were combined multiplicatively as

*R*

_{curv}).

*R*

_{50%}determines the point of the function where cell firing (

*R*

_{cell}) is half of its maximum (100%),

*N*determines the steepness of the function, and

*γ*is an exponential non-linearity applied to curvature responses. These 30 firing rates, one per sampled direction from the object center, give a population code for internal representation of object shape (see Figures 3J and 3L).

*R*

_{norm}equals unity ( Equation 5):

*ϕ*is a given possible axis of symmetry,

*θ*is the angle around the object center from that axis of symmetry, and

*ω*

_{inh}is a weight regulating the mutual inhibition due to deviations from symmetry.

*R*

_{sym}equals

*R*

_{norm}for symmetrical patterns and decreases monotonically as patterns become increasingly asymmetrical (i.e., as the absolute difference between

*R*

_{norm}· (

*ϕ*+

*θ*) and

*R*

_{norm}· (

*ϕ*−

*θ*) increases). Finally, the degree of perceived symmetry for the pattern (

*S*

_{pat}) for a potential axis of symmetry (

*ϕ*) is given by summing over orientations:

*S*

_{pat}peaks, with symmetry strength proportional to

*S*

_{pat}, and the axis orientation equal to the

*ϕ*value where the peak occurred. This not only allows the derivation of the axis of symmetry's orientation and strength but also allows for multiple peaks to occur. That is, the model recovers multiple axes of symmetry when appropriate. Because of the normalization phase (see Equation 5),

*S*

_{pat}equals unity for symmetrical patterns and decreases monotonically as symmetry is decreased.

*S*

_{pat}with a Gaussian function (

*σ*= 120°; the specific function used here or its width is not critical to the results), effectively removing these “accidental” axes of symmetry from influencing performance. It is currently unknown what the responses of participants would be if the symmetry axis' orientation was random, if participants' responses would be affected by off-axis symmetry, and whether simply removing or adjusting this bias would be sufficient to account for data from such a scenario.

*S*

_{pat}values in steps of 15° phase shift and interpolated for intermediate values. We used a threshold

*S*

_{pat}value of 0.8 to determine what participants would classify as symmetrical or not; this parameter was not considered a free variable, as it covaries with

*ω*

_{inh}without significantly improving the quality of the fit. The free parameters

*N*and

*ω*

_{inh}were adjusted to decrease the sum of squared differences between the human and model data, using the fminsearch function in Matlab (Mathworks). All other parameters are taken from the low-RF mechanism (see Poirier & Wilson, 2006), which was the mechanism responsible for detection and discrimination performance for the lower range of radial frequencies, as well as lateral masking data collected on RF5 patterns.

*σ*= 0.15°), which is better-suited to natural edges, (2) the object center was defined as the weighted average position of the second stage filters, (3) curvature responses were computed within a range of ±23° of the concentric orientation in 5 equal steps, and the maximum curvature response over that range was taken, to increase the curvature mechanisms' orientation bandwidth.

*r*(5) = 0.904,

*t*= 4.73,

*p*= 0.0052; or assuming slope = 1 and intercept = 0,

*r*(7) = 0.900,

*t*= 5.16,

*p*= 0.0013), indicating that the model replicates human performance in this task. Replacing our neural implementation of symmetry (i.e., Equations 5– 7) by the more common but biologically implausible cross-correlation operation does not alter the quality of the fit.

*N*= 1.942) and the present study (Equation 4;

*N*= 0.288). The remaining free parameter

*ω*

_{inh}was 163.7. Also plotted on Figure 8 are the relevant curvature response ranges for different data sets that were included in the fits. The new value of

*N*creates a more gradual response increase, producing a more continuous code for shape, whereas the earlier parameter produced a sharper increase, which is perhaps more important for threshold detection of the patterns. Clearly, if the previous parameters were kept as is in the present simulations, the symmetry response would be dominated by the rare cases where curvature mechanisms produced low responses. In addition, as noted in the original paper (see Poirier & Wilson, 2006), the model is robust over a 10% change in many of its parameters,

*N*included. A new parameter fit confirms that this robustness could be extended to encompass the current value of

*N*. This parameter fit used the value of

*N*used in this study, and only varied the parameters that were not critical for the present study (e.g., parameters regulating masking or percent correct in identification tasks). The resulting quality of fits was within the range of fit qualities shown in Poirier and Wilson (2006) when parameters were changed by 10%. Thus the model is robust when the value of

*N*is changed to the current value.

*effort*needed to morph a shape into its nearest symmetrical (Zabrowsky & Algom, 2002). That is, the distance is calculated between each point of the asymmetric object and its corresponding point on the symmetric object most similar to it. This computation is performed after normalization for size, thus providing size invariance. They show that for a set of random shapes, perceived “goodness” of shape (defining goodness was left up to the participants) was well correlated with a combination of reflexive symmetry and rotational symmetry. It is not clear however what would be the neural implementation of such a model, or how well thresholds derived using it would be correlated with psychophysical thresholds (e.g., Wilson & Wilkinson, 2002).

*S*/

*n,*where

*S*is the number of symmetry pairs, and

*n*is the total number of elements. As the bulk of their argument relates to this computation, it is worthwhile comparing with our own computation. Although the details of the implementation differ, our model also does a computation that is conceptually similar to

*W*=

*S*/

*n*. The division by

*n*is done in our model by normalization (Equation 5), and the sum of

*S*is done by adding (Equation 7) the normalized responses after shunting inhibition removes the responses that are asymmetrical (Equation 6). Thus, Equations 5–7 perform a computation equivalent to the WoE implementation. It follows then that any conclusion derived from the WoE will equally apply to our model. The only notable differences are (1) we use shunting inhibition to classify whether responses are random elements or pairs, and this classification varies gradually with the difference between signals of a pair, (2) the symmetry computation is done on a different internal representation of the stimulus, and (3) the two models were developed to account for performance with regards to different classes of stimuli.