Plato's Cave: Directed Diffusion

The Directed Diffusion Model

The Directed Diffusion model [Lehar 1994], which is an extension of the Boundary Contour System, provides a computational implementation of this Gestalt concept in a neural network architecture. In this model, oriented edge detectors respond to the presence of oriented edges in the input by spatial receptive fields in the manner of cortical simple cells reported by Hubel and Wiesel. The response of such a cell computed as an image convolution is proportional to the contrast across the edge. The figure below illustrates two horizontal oriented cells responding to two horizontal edges in the Kanizsa figure by way of oriented receptive fields. Similar edge sensitive cells representing edges of all orientations (not shown) are also present at every spatial location in the system.

The spatial interactions responsible for illusory boundary formation occur in a higher level cooperative cell layer, which also contains cells representing all orientations at every spatial location, and these cooperative cells receive input from the corresponding oriented cells in the oriented layer. In the figure above, for example the activation of the horizontal oriented cells stimulates horizontal cooperative cells at those same locations. Neural activation diffuses in the cooperative layer in an orientation specific manner by way of bipolar receptive fields, whereby cooperative cells receive input from other like-oriented cells in the cooperative layer through the cooperative receptive fields. For example in the figure, each horizontal cell in the cooperative layer has a pair of horizontally oriented receptive fields which receive input from other horizontal cooperative cells which are horizontally adjacent to them.

The figure below illustrates a possible form for the cooperative receptive field, which can be defined by a Gaussian function of radial distance from the cell, as well as a Gaussian function of angular deviation from the line of colinearity. Cooperative cells which are horizontally adjacent to the active cells depicted above will receive input from the active cells. This secondary activation will continue to propagate to other cooperative cells still further from the input in a colinear direction. A passive decay term in the differential equation governing the cooperative cell prevents runaway positive feedback so that the pattern due to an isolated input will define at equilibrium a spatially decaying trail of activation colinear with the original edge, and extending outward to a distance which is a function of the magnitude or contrast of the original edge. The equilibrium pattern of diffusion from an isolated edge signal would therefore appear similar in form to the Gaussian receptive field shown below except that the final range of diffusion would be much greater than the size of any individual receptive field, as suggested in the figure above, and the range would be greater still at locations between colinear oriented inducers where the cooperative cells would receive activation from both sides simultaneously.

The subjective appearance of the illusory contour in the Kanizsa figure however is not a broad diffuse region as shown above, but looks more like a sharp well defined edge. In order to achieve this result, an element of spatial sharpening was incorporated in the Directed Diffusion model by the addition of inhibitory sidelobes to the cooperative receptive fields. This was achieved by defining a difference of Gaussians profile as a function of angular deviation as shown below, rather than the straight Gaussian profile shown above. This has the effect of boosting the strength of the illusory contour along the crest of greatest magnitude, and suppressing it to either side of that crest.

At every spatial location a certain cross-talk between adjacent orientations was also defined in the model, resulting in an additional diffusion of activation across orientations at each spatial location, as suggested below (left). This orientational cross-talk results in a fanning-out of the diffusing signal, as suggested below (right). This feature allows boundary completion to occur around smooth curves, as seen in the curved Kanizsa illusion, since the cooperative cells along the curve receive activation from both sides simultaneously, and thus become more active than cells along either line of colinearity. The strength of the resulting illusory contour will however be somewhat diminished, as is seen in the psychophysical studies.

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