**Abstract**:

**Abstract**
**A key computation in visual cortex is the extraction of object contours, where the first stage of processing is commonly attributed to V1 simple cells. The standard model of a simple cell—an oriented linear filter followed by a divisive normalization—fits a wide variety of physiological data, but is a poor performing local edge detector when applied to natural images. The brain's ability to finely discriminate edges from nonedges therefore likely depends on information encoded by local simple cell populations. To gain insight into the corresponding decoding problem, we used Bayes's rule to calculate edge probability at a given location/orientation in an image based on a surrounding filter population. Beginning with a set of ∼ 100 filters, we culled out a subset that were maximally informative about edges, and minimally correlated to allow factorization of the joint on- and off-edge likelihood functions. Key features of our approach include a new, efficient method for ground-truth edge labeling, an emphasis on achieving filter independence, including a focus on filters in the region orthogonal rather than tangential to an edge, and the use of a customized parametric model to represent the individual filter likelihood functions. The resulting population-based edge detector has zero parameters, calculates edge probability based on a sum of surrounding filter influences, is much more sharply tuned than the underlying linear filters, and effectively captures fine-scale edge structure in natural scenes. Our findings predict nonmonotonic interactions between cells in visual cortex, wherein a cell may for certain stimuli excite and for other stimuli inhibit the same neighboring cell, depending on the two cells' relative offsets in position and orientation, and their relative activation levels.**

*r*

_{1},

*r*

_{2}, …

*r*should be combined to calculate the probability that an edge exists at a reference location and orientation follows from Bayes's rule. Bayesian inference has had significant successes in explaining behavior in sensory and motor tasks (Fiser, Berkes, Orbán, & Lengyel, 2010; Kording & Wolpert, 2004; Tenenbaum, Kemp, Griffiths, & Goodman, 2011; Weiss, Simoncelli, & Adelson, 2002; A. Yuille & Kersten, 2006; A. L. Yuille & Grzywacz, 1988). However, in the context of edge detection within a V1-like architecture, given that there are thousands of oriented filters within a small distance of a candidate edge, the need for human labeled ground truth data makes it intractable to fully populate the high-dimensional joint filter likelihood functions required to evaluate Bayes's rule.

_{N}*x*to

^{N}*x*×

*N*, where

*N*is the number of participating filters and

*x*is the number of gradations in each filter output; (b) each LL ratio term can be expressed and visualized as a function of a single filter value

*r*, making explicit the information that a filter at one location in an image carries about the presence of an edge at another; and (c) in the overall edge probability calculation, the positive and negative evidence from surrounding filters that is captured by these LL ratios can be combined linearly, a simple, neurally plausible computation. (As a caveat, the LL ratios themselves are generally nonlinear functions of the filter values, complicating the neural interpretation; see the Discussion).

_{i}*r*

_{1},

*r*

_{2}, …

*r*may be expressed in probabilistic terms via Bayes's rule, and then rewritten to make explicit the prior and likelihood ratios in the denominator:

_{N}*N*filters, the likelihoods in Equation 3 can be factored and rewritten in terms of a sum of log-likelihood ratio terms, each one a function of a single filter's value; the sum then acts as the argument to a sigmoid function

*r*

_{ref}, evaluated at the location/orientation where the edge probability is being calculated, is used in the Results section to reduce higher order statistical dependencies among the other contributing filters (see text for details and references):

*O*

_{1},

*O*

_{2},

*O*

_{3}roughly corresponded to red-green, blue-yellow, and luminance channels, respectively. In this paper we used

*O*

_{3}, the luminance channel only.

_{on}= P(

*r*|edge), P

_{off}= P(

*r*|

*y*are the filter response bins (50 bins in the range [0,1]), and

_{j}*λ*was set to 0.5 (Konishi et al., 2003).

*x*axis of its likelihood function with a Poisson “kernel.” This was equivalent to considering a filter response to be the average firing rate of a noisy neuron, and then repeatedly measuring the virtual neuron's actual spike count over a short time window, and histogramming the results. Greater smoothing was achieved by using a shorter time window for spike counting. Let

*r*be the measured filter value (ranging from zero to one),

_{i}*f*

_{max}the virtual noisy neuron's maximum firing rate,

*τ*time window for measuring spikes, and

*λ*=

*r*×

_{i}*f*

_{max}×

*τ*the expected spike count within the time window. The Poisson distribution

*P*(

*k*) = (

*λ*/

^{k}*k*!)

*e*

^{−λ}then gives the probability of reading out

*k*spikes for that filter value, which is then accumulated in a histogram indexed by

*k*. After all of a filter's measured values are processed in this way, the histogram is normalized both horizontally (from zero to one) and vertically (to convert it to a probability density). Example likelihood functions processed in this way are shown in Figure 3A using

*f*

_{max}= 100 Hz, and

*τ*= 500 ms.

*r*[0,1]. When the filter was applied at noncardinal orientations, pixel values off the grid were determined by bilinear interpolation. The probability distribution function (pdf) of the filter's response at all locations and orientations in the database is shown in Figure 1B. The filter had a mean response of 0.012 (out of a maximum of one), and a roughly exponential fall off over most of the range so that, for example, the probability of measuring a filter value near 0.6 was 100,000 times lower than the probability of measuring a value near zero.

_{i}∈*r*

_{ref}). To compute the edge prior, 1,000 image patches were drawn at random from the database, and a randomly oriented reference location was marked by a red box corresponding to the 5 × 2 pixel filter profile shown in Figure 1A. Human labelers were asked to judge whether an edge was present in each image that (a) spanned the length of the red box (i.e., entered and exited through opposite ends of the box; (b) remained entirely within the box; and (c) was unoccluded at the center of the box adjacent to the shaded pixel. Labelers were instructed to score edges as shown in Table 1.

Score given | Interpretation | Assigned edge probability |

1 | Certainly no edge | 0 |

2 | Probably no edge | 0.25 |

3 | Can't tell—around 50/50 | 0.5 |

4 | Probably an edge | 0.75 |

5 | Certainly an edge | 1 |

*r*

_{ref}, the filter value computed at the reference location itself (i.e., in the red box). Image patches were again drawn at random from the database, this time collected in narrow bins centered at five values of

*r*

_{ref}= {0.1, 0.3, 0.5, 0.7, 0.9}. Bin width was 0.02. Image patches with

*r*

_{ref}values outside the bin ranges were discarded. The collection process continued until each bin contained 500 exemplars. Example patches are shown in Figure 2C for three of the five values of

*r*

_{ref}, showing the clear tendency towards higher edge probability as the value of

*r*

_{ref}increased. Using the same labeling scheme as above, edges were scored and scores were averaged within each bin. The result is plotted in Figure 1D (red data points) along with a sigmoidal fit (black solid curve).

*r*

_{ref}, to calculate the edge probability at the reference location (Equation 3). Multiple strategies, described in the following, were used to narrow down the large population of filters surrounding a reference location to a subset that is as CCI as possible. As it was also our goal to include only the most informative filters in the chosen filter set, but we wished to avoid measuring the informativeness of large numbers of filters that would later be rejected based their failure to meet the CCI criteria, the steps taken to minimize filter dependencies and maximize filter informativeness were interleaved so as to reduce overall computational effort.

*variance*in other filters (Karklin & Lewicki, 2003; Parra, Spence, & Sajda, 2001; Schwartz & Simoncelli, 2001; Zetzsche & Röhrbein, 2001). It has been previously pointed out that such dependencies can be suppressed through divisive normalization (Carandini & Heeger, 2012; Karklin & Lewicki, 2003, 2005; Liang, Simoncelli, & Lei, 2000; Parra et al., 2001; Schwartz & Simoncelli, 2001; Wainwright & Simoncelli, 2000; Zetzsche & Röhrbein, 2001; Zhou & Mel, 2008). Adopting a different but related strategy (with secondary benefits as discussed below), we tabulated the1-D likelihood distributions for each candidate filter conditioned both on the edge/no edge distinction, and on the value of

*r*

_{ref}, in order to obtain the likelihood functions P(

*r*|edge,

_{i}*r*

_{ref}=

*C*) and P(

*r*|no

_{i}*r*

_{ref}=

*C*) (Figure 3A). Fixing the contrast at the reference location served a similar decorrelating function among surrounding filters as would a divisive normalization, in the sense that image patches within any given

*r*=

_{ref}*C*bin exhibit far less variation in regional power than image patches in general (data not shown).

*C*took on the same five values as were used previously to measure P(

*r*

_{ref}), all the image patches needed to construct the likelihood functions for the 112 filter candidates had already been collected and labeled.

*r*

_{ref}values, beyond its effect of decorrelating surrounding filters, is that the approach greatly increases the amount of on-edge data from which the on-edge likelihood functions are constructed. To see this, consider a labeling strategy in which reference locations are selected at random: this yields on-edge cases only 2% of the time, so that populating the on-edge likelihood distributions requires a very large amount of data to be labeled. In contrast, when image patches are automatically preselected at specific values of

*r*

_{ref}, at the higher values of

*r*

_{ref}, which are very rarely encountered by random selection, the proportion of on-edge data is dramatically increased – constituting more than 30% of cases when

*r*

_{ref}= 0.3, and 77% of cases when

*r*

_{ref}= 0.5 (Figure 2D). More data in the on-edge likelihood distributions leads to more accurate estimates of the log-likelihood ratios that underlie the local edge probability calculation.

*r*data was smoothed by replacing each measured

_{i}*r*value with a horizontally scaled Poisson distribution with the same mean value. This reflected the “biological” assumption that the true filter value from the image would not be directly accessible for computation, since it would need to be communicated through a noisy spike train within a limited time window (see Methods for details).

_{i}*r*

_{ref}at which labeled data was actually collected. Each Poisson-smoothed distribution was divided into three sections: (a) a delta function at the origin, which collected all negative values of the filter due to rectification, in addition to bona fide zero values; (b) the left section of the density from zero through the peak; and (c) the right section of the density from the peak to one. The left and right parts of the distribution were fit by separate Gaussian functions that met at the peak. Example fits of the on-edge and off-edge distributions are shown in Figure 3A at two reference contrast levels (upper and lower panels, respectively) for the filter at the same center but rotated 45° relative to the reference filter.

*σ*

_{left}and

*σ*

_{right}for the two Gaussians), were plotted at the five reference contrast levels for which labeled data was collected. A piecewise cubic Hermite interpolating polynomial was then fit through each of the five parameter plots for both the on-edge and off-edge distributions for each filter. Plots of the five spline-fit functions are shown in Figure 3B for a different filter, this one rotated 22.5° relative to the reference location. The spline fits allowed the parameters of the on-edge and off-edge likelihood distributions to be generated for any value of reference filter contrast; the fits (red curves) to the collected likelihood functions (black curves) are shown in Figure 3C, along with several likelihood functions at interpolated values of

*r*

_{ref}(green dashed curves). For purposes of cross validation, a new set of labeled data was collected for

*r*

_{ref}= 0.2. The resulting likelihood function (green solid curve) corresponded closely to its prediction based on interpolation of the fit parameters.

*C*(0.3, 0.5, 0.7; Figure 4A). The Chernoff information was calculated based on the Poisson-smoothed likelihood distributions. Filters were ranked within each

*C*level (from 1 = best to 112 = worst) and the ranks for each filter were averaged across

*C*levels, weighted by the log probability of encountering that

*C*level in the database (see Figure 4B). For viewing convenience, the weighted ranks were inverted (newrank = 112 − oldrank) so that the best filters had the highest scores (Figure 4B). The top 30% of the filters ( = 34) were kept for further evaluation (Figure 4C).

*N*filters that had low mean absolute pairwise correlations (MAPC) between their responses: where

*N*= 6 and

*ρ*(

*r*,

_{i}*r*) is the correlation between two filters

_{j}*i*and

*j*over all pixel locations and orientations in the image database. The distribution of MAPC values is shown in Figure 5 for the

*r*

_{ref}-conditioned version of Bayes's rule (Equation 6), and measure the position and orientation tuning curves of the resulting probabilistic edge detector. The filter set with the sharpest tuning in both position and orientation would be selected.

*r*

_{ref}were similar in form, but were pushed towards higher or lower ends of the

*r*range, respectively.

_{i}*r*

_{ref}axis. The image patches are shown in Figure 8B, along with their corresponding LEP scores. Inspection of the patches confirm that edge probability within the reference boxes (according to the scoring rubric of Table 1) was much higher for the 90th percentile cases (top row) than the 10th percentile cases (bottom row). To extend this type of comparison to a more global perspective, we located all sites in the image where the linear score was between 0.12 and 0.38 (corresponding to all gray dots in Figure 8A). All cases above the 80th percentile in the LEP score distribution (i.e., above the red line in Figure 8A) were presumptive “good edges” and were labeled with red line segments in Figure 8C (left frame). Similarly, all cases below the 20th percentile of the LEP distribution for the corresponding linear score (below the blue line in Figure 8A) were presumptive “bad edges” and were labeled with blue line segments in Figure 8C (right frame). The upper cutoff of 0.38 on the linear axis was chosen because at that linear score, the edge probability reached 50% (Figure 2D), so that the visual distinction between “good” and “bad” edges within any given linear bin above that value would necessarily begin to fade. Consistent with the examples of Figure 8B, red-labeled edges were much more likely to be properly aligned and positioned relative to actual object edges, whereas blue edges were typically misaligned by a pixel or two, and/or misoriented.

*****riented filters”). Example images are shown in Figure 10, in comparison to a graded Canny-like algorithm (PbCanny) developed at UC Berkeley (D. R. Martin et al., 2004). We found that the rm* algorithm, with no free parameters, does a good job extracting bona fide local edge structure at the five-pixel length scale.

*r*

_{ref}, as previously mentioned this approach is closely related to divisive normalization, which is considered to be a canonical cortical operation (Carandini & Heeger, 2012). The third strategy, identifying mutually decorrelated subsets of nearby filters, was accomplished here using an exhaustive search, but a similar result could likely be achieved using a biologically plausible learning rule (e.g., Gerhard, Savin, & Triesch, 2009).

*r*

_{1}= 100%,

*r*

_{2}= 60%), Cell 2 provides positive evidence for Cell 1's hypothesis. If on the other hand Cell 2 is firing either too weakly (

*r*

_{2}< < 60%) or too strongly (

*r*

_{2}> > 60%) compared to Cell 1, Cell 2 provides negative evidence for Cell 1's hypothesis. This relative-levels effect is the source of the variable excitatory-inhibitory interaction within the Bayesian probability calculation, and the nonmonotonic LLR functions that occur in some cases as shown in Figure 9. Interestingly, the fact that nearby cells in the cortex do generally connect to each other through both excitatory and inhibitory pathways (Anderson, Carandini, & Ferster, 2000; Bonds, 1989; Isaacson & Scanziani, 2011; Priebe & Ferster, 2012) indicates that nearby cells are in principle capable of providing net positive or net negative evidence to each other depending on their relative firing rates, but determining whether this actually occurs along the lines proposed here will require further experiments.

_{1}). Likewise, inputs to the reference location from tangentially aligned filters (which we excluded from consideration given their statistical dependencies) may in some cases be nearly purely excitatory (Polat & Sagi, 1993), though interestingly, Polat et al. (1998) reported nonmonotonic modulation effects in visual cortical neurons for tangentially aligned cues of increasing contrast in a contour detection task. In general, it is most revealing to characterize the pairwise interactions centered on a reference location/orientation not as excitatory or inhibitory per se, but in terms of log likelihood ratios as in Figure 9, which are non-trivial functions of the filter responses at both locations.

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