This page contains explanations concerning a simulation program available on internet.
The explanations concern the stages and parameters of computational models of a center-surround cell and a dot-pattern selective cell  as proposed by Kruizinga and Petkov  and modified by Petkov and Visser . The models are used to compute the responses of such cells to a given input stimulus (image). The processing steps used to compute these responses need not follow the biological mechanisms. The models are implemented as image processing operators that deliver one or more output images for each input image that is submitted.
The first step in our model of a dot-pattern selective cell is computation of the responses of center-surround cells to an input stimulus (image). Such cells would respond strongly to intensity spots (intensity increments or decrements) of appropriate size and thus act as spot detectors. The responses of these spot detectors can then be used to detect groups (patterns) of spots or dots.
A centre-surround receptive field is circular symmetric and consists of a central region and a concentric surround region of opposite polarities. The figure shows a centre-ON, surround-OFF receptive field, briefly called ON receptive field, which means that the centre is excitatory and the surround inhibitory. Such receptive fields are typical of retinal ganglion and LGN cells and are also found in the primary visual cortex (layer 4Cβ of V1). (You can read more on this here).
|Intensity map of a two-dimensional DoG function||One-dimensional DoG profile|
The spatial summation properties of such a cell can be modelled, for instance, by a difference of Gaussians function (DoG):
The coefficients and are selected in such a way that all positive values of the DoG sum up to 1 and all negative values sum up to -1. To visualise a DoG function, select option "DoG function" under "Output image" in the interface to the simulation program.
The responses of a population of such cells that cover uniformly the visual field (represented by the frame of the input image) and have the same form but are centered at different positions are computed by convolution (*) of the input image f with the DoG receptive field function and subsequent half-wave rectification (notified by |.|+):
followed by contrast normalisation
The parameters of this operation are explained in more detail below.
This parameter specifies the radius R of the central region, i.e the zero-crossing circle, of the DoG. For optimal spot detection, it should be chosen to be equal to the radius of the targeted spots in the input image. The radius R and the standard deviation of the surround Gaussian in the DoG function are related as follows:
where is called the centre/surround size ratio (see below).
One single spot radius can be used by specifying its (real positive) value in the field "Spot radius / radii" and setting the parameter "Number of different channels (spot radii)" to 1 in the interface to the simulation program. One convolution of the input image with the corresponding DoG function will be computed in this case.
Multiple spot radii can be specified as a list of real positive values in the "Spot radius / radii" field, e.g. 2, 4, 6. An alternative way of specifying multiple radii is to enter one value in the field "Spot radius / radii", e.g. 2.0, and an integer positive number in the field "Number of different channels (spot radii)", e.g. 4. This will yield the spot radii list 2.0000,2.8284,4.0000,5.6569 that is a progressive series. The first value in this list is the value specified in the field "Spot radius / radii" and the subsequent values are obtained by multiplication with .
If multiple spot radii are specified, the corresponding number of DoG functions will be generated and the input image will be convolved with each of these DoG functions, yielding the corresponding number of center-surround cell response (convolution) results that can be visualised separately or simultaneously (see Displayed spot radii channels).
This parameter can be used to specify multiple spot radii. The value of this parameter must be a positive integer number and will only take effect when the "spot radius / radii" field contains a single value. See "Spot radius / radii" for further explanations.
This parameter specifies the standard deviation of the centre Gaussian function in the DoG as a fraction (a real value between 0 and 1) of the standard deviation of the surround Gaussian of the same DoG.
For a given radius R of the DoG centre, the extent of the surround can be changed using : small (near 0) and large (near 1) values of yield extended (large) and localized (small) surrounds, respectively. Certain experimental data suggests that biologically relevant values of lie around 0.5.
The result of convolving the input image with a DoG is normalised for local brightness by division with :
where C is a positive coefficient and is the local grey level average computed by convolving the input image with the Gaussian function with the largest standard deviation among those used. This normalisation mechanism is a simplified modification of the shunting equation at equilibrium. It can be switched off by setting the coefficient C to zero in the interface to the simulation program. In this case (C = 0) the output is equal to the convolution result and indicates local brightness differences. If C is different from 0, the local brightness differences are normalized with the average local brightness and the output indicates contrast differences.
Select "ON channel" in the interface to the simulation program to compute the responses of centre-ON, surround-OFF cells (convolution with a DoG that is positive in the centre). Select "OFF channel" to compute the responses of centre-OFF, surround-ON cells (convolution with a DoG with an opposite polarity). The ON and OFF channels enhance bright and dark intensity spots, respectively. If "ON and OFF channel" is selected, both types of response will be computed and merged in the center-surround cell response result.
Default: ON channel
An intensity spot in the input image will result in a spot of intensity in the center-surround cell response (convolution) image. In the spot detection step such a spot of activity is reduced to a single point that is positioned approximately in the centre of the spot. Furthermore, the activity in the centre-surround cell response image that is not due to intensity spots but to other features, such as edges, is eliminated. A schematic overview of the spot detection process is given below.
|Schematic overview of the spot detection process.|
The non-maxima suppression (NMS) module produces a binary mask that has the value 1 in each point in which the center-surround cell response has a local maximum and the value 0 elsewhere. In this way contiguous activity regions are reduced to single pixels.
The low response removal (LRR) module produces a binary mask which has the value 1 at each position for which holds , where is a threshold coefficient and is the maximum center-surround response for the considered channel. The binary mask has the value 0 in all points in which the above condition is not satisfied.
The lateral inhibition (LI) module produces a binary mask as follows: the response in a position is compared with the responses in a number of neighbouring positions at a distance R from . If the condition , where is a positive parameter value smaller than 1 to be specified by the user, is fulfilled for any of these neighbouring positions, the value of the LI binary mask at is set to 0, otherwise to 1. In practice, a fixed number of positions that lie on a circle of radius R centred at are probed.
The winner-takes-all competition (WTA) module is used in case that multiple channels (radii) are employed. It produces a binary mask for each channel. The value of the binary mask of a given channel at position is set to 1 if the center-surround cell response at that point is larger than the center-surround cell responses at the same point for all other channels used. Otherwise, the value of the mask is set to 0.
The binary masks produced for a given channel by the NMS, LRR, LI and WTA modules are combined by a pixel-wise AND function and the resulting binary mask specifies the positions at which spots (of a radius corresponding to that channel) are centered. The output of the spot detection step at such image positions is equal to the corresponding center-surround cell response . In all other positions the output is 0.
The lateral inhibition (LI) response ratio parameter in the interface to the simulation program is used in the lateral inhibition condition where is the center-surround cell response in a point and is the response in another point at a distance R from . If this condition is fulfilled for any point at a distance R from , the output of the spot detection step at point is set to 0.
The integer parameter in the interface to the simulation program specifies the number of positions on a circle of radius R centered on a point that are probed in checking the lateral inhibition condition. and are related - larger values of should be taken for larger values of .
The low response removal (LRR) threshold in the interface to the simulation program is used in the condition where is the center-surround response in position and is the maximum center-surround response for the considered channel. If this condition is not satisfied, the output of the spot detection step at that position is set to 0.
In this step groups of spots are detected. The input to this step is the output of the previous, spot detection step. The output in a given point will be different from 0 only if a certain minimum number of spots responses of a given minimum strength are found in a certain surroundings of the point. The output of this step can chosen to be either of type binary or to specify the relative spot density.
Only those spot detection responses will be taken into account that are larger than θ. The value of θ should be a positive real number.
The number of points with spot responses larger than θ in a given surroundings of a point must be larger than the value of the parameter m to invoke response from the spot-group detection step at that point. The value of m should be a positive integer number.
The surroundings of a point that is examined for presence of spot responses is a square window with a side 2ζR;. The window side is thus specified in units of the radius 2R of the spots that are counted in the window. The value of ζ should be a positive integer number.
Two different output types of the spot-group detection step can be selected:
Density. Density is defined as the number of spots detected in the considered surroundings divided by ζ2 (the area of the square surroundings window in units of (2R)2). This number should be larger than the specified minimum number of spots m; otherwise the output is set to 0.
The response of a dot-pattern selective neuron is computed by a weighted summation of the responses of spot-group subunits by means of convolution with a Gaussian function with a standard deviation :
The parameter specifies, roughly speaking, the receptive field size of a dot-pattern selective neuron in terms of a number of center-surround fields.
One can view a DoG function, specific intermediate results or the final result of the dot-pattern selective filter. The output image and its corresponding parameter file can be downloaded.
When the spot-detecting subunit has been selected as output of the algorithm, non-zero pixel responses can be replaced by disks of appropriate spot sizes. This facilitates response inspection as the responses in the default output are hard to distinguish: they are 1 pixel in size.
When this field is checked the polarity of the output will be inverted.
When multiple radii channels have been selected, the output is a pixel-wise maximum value superposition of all channels.
K. Tanaka, H. Saito, Y. Fukada and M. Moriya: Coding visual images of objects in the inferotemporal cortex of the macaque monkey, Journal of Neurophysiology, 66, 1991, 170-180
P. Kruizinga and N. Petkov: Computational model of dot-pattern selective cells, Biological Cybernetics, 83 (4), 2000, 313-325.
[bibTex], [pdf 470 KB] © Springer
N. Petkov and W. T. Visser: Modifications of center-surround, spot detection and dot-pattern selective operators. Technical Report 2005-9-01, Institute of Mathematics and Computing Science, University of Groningen, The Netherlands, 20 January 2005, 4 pages.
[bibTex], [pdf 183 KB]