The second stage implements an anisotropic filtering by means of even Gabor filters. This level has its physiological correspondence in the primate primary visual cortex (Daugman 1985; Glezer 1985; Palmer et al. 1985). Output of this stage is "full-wave" rectified with the effect of converting negative responses into positive responses (absolute value), thus rendering this stage independent of contrast polarity. It is described by the following equations:
where
6
is the rectification of the activity of cellyi
at position (i, j ) , X are inputs, k is the filter orientation, f and cp are respectively the frequency and phase of the side bands, el and e , are respectively the normalized receptive field elongation and width. Note the definition of the aspect ratio (AR). The range of value p, q is contained in the lirnit of the receptive field size (elliptic accordingly to the aspect ratio).Asynchronous model. I. Feedforward 103
5.2.4 Cooperation
In the context of this thesis, cooperation has not been used to propose a solution to contour completion, treated theoretically for instance by Grossberg (e.g. Grossberg 1987), but rather to propose ad hoc methods which could enhance the quality of edges. The principal idea is that spatially distant features, such as the responses of various cells of the anisotropic stage for a given orientation, are supposed to be interconnected (physiologically plausible according to
$ 2.4.4). On the other hand, and to remain in the spirit of asynchrony, cooperation will be based on the notion of events temporally distributed (Figure 5.9A). Outcome of these two constraints is that cooperation is viewed as a nonlinear process which links spatially distant features shar- ing similar characteristics (in this case, orientation).
Events temporally distributed
andcoherency
The dependence between the frequency of a train of spikes impinging on a cell and the la- tency of the appearance of the first spike from this cell was established in § 4.1.3. To measure the similarity between various cell responses, also called "coherency", the latency of their first spike, referred to as the phase, is thus significant. If the phase difference between cells is small, then it can be deduced that the activities of those cells are similar. When designing an operator measuring the similarity between signals, it is important to take into account the signal ampli- tudes and not only their phase. It is wished, for instance, that the output of this operator for two signals in phases of small amplitude gives a weaker response than for two signals in phases of large amplitude. To express these two factors (phase and spike frequency), cell responses are represented by vectors whose length is related to spike frequency and whose angle is related to spike latency (thus both, length and angle are dependent on signal amplitude). In consequence, the transformation of a train of spikes into a vector, called vector conversion (illustrated in Fig- ure 5.9B), involves two functions: (i) one for converting frequencies into vector magnitudes;
(ii) another for converting latencies into vector angles (in the range [0, .n] to avoid cancella- tion of vectors pointing in opposite directions; this situation would correspond to inhibitory connections, which are not wanted). These two functions are chosen linear (G and H in Figure 5.5).
Measuring coherency
Measuring the coherency among responses for a specific image column1 of a specific ori- entation is made in three stages: (i) convert responses into vectors according to the previous rules; (ii) calculate the vectorial sum of the vectors determined in the previous step. The result- ing vector gives a measure of strength of the responses along an image column with a weak measure of the extent of spreading of the vectors; (iii) make a measurement of the dispersion of all the vectors with respect to the resulting vector calculated in the previous step. The dis- persion of n vectors of length R i and angle O i with respect to a resulting vector of length R, and angle OR,
,
is measured by applying the following definition2:1. An image column refers to the lattice of points which is supposed to form columns according to the horizontal and vertical orientations. Extension to n orientations is not considered (see text).
2. A similar definition is used by Rao and Schunck (1991) to measure the flow orientation coherence of vectors corresponding to spatial gradients.
CHAPTER 5. Asynchronous Visual Processing I. Foundations
An alternative to this vectorial method for measuring the coherency among neuronal re- sponses is the coincidence detection which favors simultaneous spike arrivals by implementing a multiplicative operator, recently formalized by Bugmann (1992).
For practical reasons, the technique of cooperation (or coherency detection) just described is applicable to an image for only two orientations: horizontal and vertical. Extension to more orientations would lead to a combinatorial explosion. A second limitation is that the whole col- umn contributes to the response. Thus, gaps existing along a column are not considered. A project aimed at finding solutions to these limitations is being carried out (Durante and Burgi 1992). This project involves the analysis of nonlinear activity of neuronal structures, and par- ticularly, a study of the oscillatory mode.
Example:
Figure 5.9 : Illustration of the correspondence between a cell response and vector. (A): Cooperation based on a measure of coherency between a set of neurons whose activity is characterized by spike frequency and phase. The
? operator is described in the text. (B): Illustration of the vector conversion. There is a linear relation between spike frequency and vector magnitude as well as between latency and vector angle, described respectively by the func- tion G(x) = a - x + b , where b is representative of a maintained discharge, and H(x) = - a'x + b' , where b' = 7t (a and a' are chosen accordingly to the considered range of signal amplitude). The example on the right shows how the vectors corresponding to three resAponses with different frequencies are disposed with the resultant of the vectorial addition indicated by the vector R,.
5.2.5 Temporal integration
In accordance with the relationship between intensity and latency, luminance differences yield a dynamic data flow of the visual information. The outputs of the last stage of processing of the feedforward architecture will thus evolve dynamically along time (the time unit is arbi- trary). According to the principle that dynamic transinfomation of a source perturbed by white noise has, along time, an optimum, one goal is to make use of it. As it has been pointed out, the main function of the feedforward architecture is to implement a high pass filtering. Thus, it de-
tects boundaries separating regions which are differentiates by their luminance values. Also, an increase in the confidence for classifying a point of the lattice as belonging either to one region or to another may result in an increase in the confidence of boundary locations. The idea of tem- porally integrating outputs of the last stage of processing aspires at conserving responses which correspond to periods where dynamic transinformation was particularly high during the pro- cessing (these optimum responses are particularly visible in Figure 5.13). Temporal integration is defined as follows:
z$t) = %(t - I)
+
$(t)~ $ 0 ) = 0 b'i, b'j, b'k
where is defined in equations (5.1 1) and 2; is the output of the integrative stage (orienta- tion k and position (i, j) ). Time t - l indicates the time of the previous processing (defined with respect to the sampling time).