[Next] [Previous] [Top]

III. COMPUTATIONAL COMPONENTS OF LEGION

We use LEGION to group similar features and segregate dissimilar ones in a scene. Each oscillator will respond to a detected feature at some location in the image. Two or more oscillators that likely respond to features of the same object will group together by the process of synchronization, where these oscillators have the same phase. A synchronized group of oscillators is separated from other synchronized groups that represent different objects by desynchronization. There are four computational components of LEGION that describe its behavior: group participation, group formation, group segregation, and group suppression. Each is implemented by the following network components: input stimulus, excitatory coupling, global inhibitor, and oscillator potential, respectively.

From some initial network state, groups of oscillators will form by synchronization due to the effects of the first two network components. Two or more oscillators are said to be synchronous if they jump up and down simultaneously. Two coupled oscillators synchronize by excitation via the coupling term S. An oscillator is able to participate in a group only when it receives a positive input stimulus, i.e. when and in equation (1a). In this case, the cubic is shifted upward as in the oscillatory state shown in Fig. 3a. When an oscillator jumps up, it sends excitation to its neighbors and shifts their cubics upward. Stimulated neighbors that are close to the jumping oscillator in the silent phase will immediately jump up as well because they will lie below their LK [17]. Thus, the propagation of local excitation between coupled oscillators forms groups of oscillators, each of which is stimulated and connected. Terman and Wang [20] have shown that synchronization occurs at an exponential rate.

A group of synchronized oscillators representing an object are distinguished from other groups representing different objects by a selective gating mechanism using the global inhibitor [20]. Global inhibition works to ensure that only one group will be in the active phase at a time. Assume n oscillator groups are all travelling in the silent phase. When the leading group jumps up, it sends excitation to the global inhibitor, which in turn responds with global inhibition. The cubics of all oscillators in the network shift downward. Oscillators in other groups travelling in the silent phase enter the non-oscillatory state (Fig. 3b). The stable fixed point to the left of LK acts as a gate to prevent other groups from jumping up. On the other hand, the leading group will travel in the active phase and eventually reach RK and jump down. This removes global inhibition and the next group in the silent phase closest to LK will be able to jump up to repeat the process. Thus, each synchronous group is desynchronized from all other groups because of the global inhibitor.

The oscillator potential is added to LEGION to remove small noise fragments that could disrupt network dynamics [24]. It forces certain groups to enter and stay in the non-oscillatory state. From equation (1a) it is easy to see that the potential is able to remove the effect of I via function H. At the beginning of network dynamics, the potential is initially zero and the exponential function has a large value so that H has value 1. The exponential decay is chosen to be slow enough so that group formation occurs. The potential starts to play an important role when the exponential function becomes less than the threshold . If all oscillators in a group together could not develop a large enough potential through equation (2), then the entire group enters and stays in the non-oscillatory state. On the other hand, if at least one oscillator, called a leader [24], is coupled with enough synchronized neighbors and develops a high potential, then the entire group remains in the oscillatory state. All oscillators in the group will remain synchronous, even though not all oscillators in the group are leaders. This is because the leader is able to sustain a large potential at each cycle and its jumping up propagates local excitation to its coupled neighbors. Leaders are able to maintain high potentials because they are able to reinforce their potentials when their neighbors jump up to the active phase synchronously.

The above components have been rigorously analyzed by Terman and Wang [24][20]. To illustrate, Fig. 4 shows a computer simulation of a 20x20 LEGION using 4-neighborhood connectivity for both and (see Fig. 2). The differential equations (1)-(4) were solved using a fourth-order Runge-Kutta method. The input image is a 20x20 binary image shown in Fig. 4a, with four objects (arrow, square, cross, and rhombus) and some background noise. Pixels are mapped to oscillators in one-to-one correspondence, and network connections preserve pixel adjacency and are set to 1. An oscillator receives a positive input stimulus if its associated pixel is black. Thus, each object in the image, i.e. a spatially separate black region, is represented by a synchronized group of oscillators that desynchronizes from all other groups. Fig. 4b shows plots of the temporal activity of oscillators in each group. Each trace combines the activities for all oscillators in a group. The horizontal axis is time and the vertical axis is the normalized x activity. The trace for the global inhibitor, is the z activity. The activity of the oscillators representing the background noise is shown in the trace labeled "Background." Initially, each stimulated oscillator has random activity, but is able to quickly produce oscillatory behavior. The active phase corresponds to high activity durations and the silent phase corresponds to low activity durations. The instantaneous jumping between phases is shown by the sudden change from low to high activity or vice versa. The global inhibitor is excited whenever an oscillator enters the active phase. Synchronization is achieved quickly and desynchronization occurs after only two cycles. The near steady limit cycle behavior occurs after the vertical dotted line because all synchronized groups are separated, and oscillators representing the background have entered the non-oscillatory state.

[Next] [Previous] [Top]