Spatially-global integration of closed, fragmented contours by finding the shortest-path in a log-polar representation

Abstract

Finding the occluding contours of objects in real 2D retinal images of natural 3D scenes is done by determining, which contour fragments are relevant, and the order in which they should be connected. We developed a model that finds the closed contour represented in the image by solving a shortest path problem that uses a log-polar representation of the image; the kind of representation known to exist in area V1 of the primate cortex. The shortest path in a log-polar representation favors the smooth, convex and closed contours in the retinal image that have the smallest number of gaps. This approach is practical because finding a globally-optimal solution to a shortest path problem is computationally easy. Our model was tested in four psychophysical experiments. In the first two experiments, the subject was presented with a fragmented convex or concave polygon target among a large number of unrelated pieces of contour (distracters). The density of these pieces of contour was uniform all over the screen to minimize spatially-local cues. The orientation of each target contour fragment was randomly perturbed by varying the levels of jitter. Subjects drew a closed contour that represented the target’s contour on a screen. The subjects’ performance was nearly perfect when the jitter-level was low. Their performance deteriorated as jitter-levels were increased. The performance of our model was very similar to our subjects’. In two subsequent experiments, the subject was asked to discriminate a briefly-presented egg-shaped object while maintaining fixation at several different positions relative to the closed contour of the shape. The subject’s discrimination performance was affected by the fixation position in much the same way as the model’s.

Combining contour and region for closed boundary extraction of a shape

“…We followed the procedure described in Experiment 3 of Kwon et al ( 2016 ) to test boundary extraction using the fragmented boundary of an egg-like shape embedded in noise edges. The subject's task was to indicate if the pointy side of the egg was oriented to the left or to the right.…”

“…where k is the distortion coefficient such that a larger k produces an egg with a more obvious pointy side and makes the discrimination task easier. For three of our four subjects (S1-S3), we used k = 0.04, which was the same value used in the previous study (Kwon et al, 2016 ). Subject S4 was tested with k = 0.08.…”

“…Following the study by Kwon et al ( 2016 ), our model uses the log-polar representation of the retinal image. It is known that a log-polar map is a good approximation of the transformation from the retinal image to the early visual areas in the cortex (starting with V1), the first areas responsible for extraction of contours and boundaries (Schwartz, 1977 ).…”

The Role of Problem Representation in Producing Near-Optimal TSP Tours

“…The circumference of the base of the cylinder is 2π. Treating the map as the surface of a cylinder was not needed, and in fact was not desirable, when Kwon et al (2016) solved the SPP problem in a complex-log map. A discontinuity at one of the nodes (line segments) led to a map where this segment had two representations in the complex-log map.…”

“…Finally, consider an example that has been reported recently by Kwon, Agrawal, Li, and Pizlo (2016) that provided a direct motivation for the present paper. Subjects were asked to draw a closed contour that was represented in the image by short line segments embedded in a large number of similar line segments, called distracters ( Figure 2, see next page).…”