Figure 1.
Model of push-pull inhibition.
(a) Model of a (top) simple cell that receives excitatory or push input from model LGN cells with appropriately aligned receptive fields, and an inhibitory or pull input from another (bottom) cortical neuron that receives input from LGN cells with receptive fields of opposite polarity. Shaded light and dark gray areas indicate ON and OFF subregions, respectively, within the receptive fields of afferent model LGN cells. The solid lines indicate excitatory synaptic connections and the dashed line indicates an inhibitory synaptic connection. (b) Preferred stimulus that evokes maximum response to the concerned model.
Figure 2.
Receptive field and orientation selectivity.
(a) The receptive field structure of a CORF model cell (of size 30×30 pixels). The solid and dashed circles represent sub-units that take as input the responses of center-on and center-off LGN model cells, respectively. (b) and (c) show a closer look at two types of sub-units. The image in (b) illustrates a sub-unit whose output is a Gaussian-weighted summation of the responses of a pool of center-on DoG functions, while the image in (c) illustrates a sub-unit that integrates center-off DoG responses. The radius of each sub-unit is a function that grows linearly with the Euclidean distance from the receptive field's center of the CORF model cell. (d) A synthetic stimulus (of size 100×100 pixels) of bright-to-dark vertical edge and (e) the corresponding response image obtained by sliding the CORF receptive field in (a) across all locations of the stimulus in (d).
Figure 3.
Automatic adjustment of a CORF receptive field for a given β value.
The black and gray dashed circles represent the original and the shifted receptive field, respectively, of a center-off sub-unit that is described by tuple i in the concerned CORF model. The new polar coordinates (), with respect to the ‘+’ marker (receptive field center of the CORF model at hand), are determined by shifting the polar coordinates (
) along the x-axis by half of the given β value.
Figure 4.
Relationship between the separation index B of the ON and OFF subregions of the receptive field of a CORF model cell (see inset) and the response to the preferred oriented edge and the orientation bandwidth at half amplitude.
For β = 0 the ON and OFF sub-regions are organized as depicted in Fig. 2a. In this case, the concerned CORF model cell achieves maximum response with an orientation bandwidth at half amplitude of 45°. The orientation bandwidth increases and the response decreases with an increasing β value. The value of β for which the response disappears depends on the size of the pool - if it does not touch the edge, there will be no response.
Figure 5.
Construction of band-limited noisy images.
(a) A test image (of size 100×100 pixels) is the sum of a (b) noiseless edge image and (c) a noise image. The noise image is a superposition of a constant value N (here N = 8) and 100 sinusoidal gratings of randomly selected orientations for the same spatial wavelength w (here w = 9 pixels). (d) The 2D spectrum of the noise image in (c). (e) Response map that is obtained by CORF model cells (with or without inhibition) to the preferred stimulus in (b). (f) A horizontal profile within the enframed region in (e). The label b (here b = 3 pixels) indicates the number of CORF responses at half amplitude along the horizontal direction, which is the direction orthogonal to the edge orientation.
Figure 6.
Experimental results of the SNR obtained with CORF model cells with no inhibition (CORF) and with push-pull inhibition (CORF+PP).
The first columns of (a–c) contain test images that are obtained by varying the spatial wavelength w and the contrast value C of band-limited noise. The second and third columns of (a–c) are the response maps obtained by the concerned CORF and CORF+PP model cells, respectively. A CORF model cell with push-pull inhibition systematically exhibits an improved SNR.
Figure 7.
Separability of spatial frequency and orientation.
(a) Response maps of a CORF model cell without inhibition (,
), to gratings of different spatial frequency and orientation, (b) with moderate push-pull inhibition (
,
) and (c) with strong inhibition (
,
) The red and green plots indicate the marginal (MARG) row- and column-wise sums, and singular value decomposition (SVD), respectively. These results are comparable to the response of biological cells (see Fig. 3 in [51]).
Figure 8.
Relationship of spatial frequency and orientation selectivity.
(a) A CORF model cell without inhibition (,
) has independent relations between the preferred spatial frequency and orientation, while (b) a CORF model cell with push-pull inhibition (
,
) shows a dependent relationship. This is similar to what is observed in biological simple cells (see Fig. 1 in [45]).
Figure 9.
Spatial frequency sensitive to contrast.
Spatial frequency tuning curves as a function of contrast obtained by two CORF model cells; (a) with no inhibition (,
) and (b) with push-pull inhibition (
,
). The dependence of spatial frequency tuning and contrast changes is achieved only when the model LGN cells are processed by a sigmoid function.
Figure 10.
Examples of RuG images, their ground truth and the respective contour maps obtained by five operators.
(First row) Images of objects in natural scenes taken from the RuG data set. (Second row) The corresponding contour maps hand drawn by a person. Best contour maps obtained by (third row) the proposed push-pull CORF model, (fourth row) the basic CORF model without inhibition, (fifth row) the Gabor filter model with isotropic surround inhibition, (sixth row) the Gabor energy model with isotropic surround inhibition and by (seventh row) the classical Canny edge detector.
Figure 11.
Examples of Berkeley images, their ground truth and the respective contour maps obtained by five operators.
(First row) Images of objects in natural scenes taken from the Berkeley data set. (Second row) The corresponding collection of superimposed contour maps hand drawn by five persons. Best contour maps obtained by (third row) the proposed push-pull CORF model, (fourth row) the basic CORF model without inhibition, (fifth row) the Gabor filter model with isotropic surround inhibition, (sixth row) the Gabor energy model with isotropic surround inhibition and by (seventh row) the classical Canny edge detector.
Table 1.
The best parameters for the five evaluated operators.
Figure 12.
Comparison of contour detecton results to the images of the RuG data set.
The proposed push-pull CORF model outperforms CORF (without inhibition), Gabor function with isotropic inhibition (GF+II), Gabor energy function with isotropic inhibition (GEF+II) and the Canny edge operators in the majority of the cases.