Figure 1.
A 2d image (left top; credit: ‘Pont de Singe’, Olivier Grossetete. Photo: Thierry Bal) is recorded as a field of contrast by the retina and the LGN (left bottom). V1 neurons respond to regions of contrast changes in a direction-selective manner, performing edge detection (middle bottom). The information from edges is integrated to reconstruct long contours (middle top). In this paper, we model the visual process starting from edges in V1; sample input (bottom) and output (top) to our model are on the right.
Figure 2.
(A) Neurons send excitatory signals along approximately co-circular directions. Thus neurons in occluded gaps may get enough excitatory input along smooth contours to get excited without direct visual input. (B) The orientation at two points is said to be co-circular if they are tangential to the circle connecting the two points. If the orientation preference at the origin is along the real axis, the co-circular edge at a point has the orientation
. Multiplication by
can be written as:
.
Figure 3.
Shape of the interaction kernel.
(A)Schematic shape of the interaction kernel . Arrows represent the orientation preference and darkness and size represent the magnitude. (B) Results of dynamics with the kernel
with the current
. Here, as everywhere in this work, we use
, which optimizes the performance according to a genetic algorithm search over the parameter space, see Methods.
Figure 4.
(top and middle) Time evolution of the neural field for sample images. The magnitude (line width) and the direction of the field are plotted at every point where the strength of the field is higher than a cutoff (0.35). The parameters of the dynamics are as in Fig. 3. Dynamics removes the clutter and fills in the occlusion gaps. However, spurious activity (widening lines) appears for large simulation times, so that the best performance is obtained for intermediate times. (bottom) Performance of the model on an image used in psychophysics experiments [32]. Like human subjects, the model can identify, complete, and bind together long punctuated contours.
Figure 5.
Neural dynamics at different cutoffs.
Time evolution of a sample image at different cutoff values. At a lower cutoff the occlusions fills rapidly, but it takes longer to suppress the clutter. At higher cutoffs clutter removes quickly, while it takes longer to fill the gaps. Notice the spurious activity around the contours at longer times. This spurious activity is dominant at lower cutoffs.
Figure 6.
Precision vs Recall with an absolute cutoff.
(A) vs
averaged over 500 randomly generated images at various simulation times starting with
. The numbers indicate cutoff values for a specific data point at the corresponding simulation time. Note the weak dependence on the cutoff. The simulation lengths of
(black dots) produces the curve with the best precision and recall combination. (B)
vs
with different starting values of precision and recall averaged over 100 randomly generated images, but with the same model parameters. Legend indicates the initial
. The black dots are the same as in the top panel. Red
's correspond to a lower initial precision (more clutter), compared to the black dots. Blue
's stand for the same initial
as black, but with the target partitioned into more shorter segments (a larger number of occlusions). Pink
's correspond to higher initial precision (less clutter), but the clutter elements are longer and harder to suppress.
Figure 7.
Precision vs Recall with a relative cutoff.
vs
averaged over 500 randomly generated images at various simulation times starting with
. The numbers indicate cutoff values for a specific data point in terms of the percentage of the maximum activity of the field at the corresponding time. Note the similarity with the results in case of absolute cutoff values (Figure 6A).