Matriz de competitividad

The overall system architecture may be schematized as a feature extraction block feeding a platoon of active neural particles, as shown inFigure 8.1. The feature extraction block processes an input image and produces a vector of feature maps. Each feature map represents the spatial distribution of some local feature of the image, and associates to each pixel of the input sample a feature vector describing the local properties of the image in that point. The image may be described as a world where particles live. Each location, or pixel, of this world is associated with a local environment which is represented by the feature vector calculated from the local values of the feature maps.

The neural network is composed of a platoon of neural particles, which at rest are arranged on a square lattice. Neighboring particles are attracted toward one another by “friendship” which is modeled as an elastic force having an elastic constantc. The particles are assumed to differentiate during system adaptation, so each neural unitui j is associated to a personal ecosystem, modeled as a vector of weightswi j, which represents its ideal environment, or the environment which is best suited for the particle.

Leonardo Bocchi 161 In order to analyze an image, the platoon is superimposed on the vector of feature maps. In this way, each unitui j is located in a positionPi j =(x,y) in the image plane. The positionPi jis calledmatch point. Each unit is an autonomous entity, which is set free to move around and look for a good environment for living. In this process, the particle acts as a specialized feature detector, which tries to locate, in a neighborhood of the match point, the point in the image having a feature vector matching, as close as possible, the personal ecosystem of the unit. In other words, the particle looks for a position, in a local neighborhood, which has a local environment as close as possible to the ideal environment for the particle.

The unit is therefore attracted toward this position and starts to move in this direction, while elastic constraints expressed by the elastic forces constrain neigh- boring units to move in a coordinated way.

Without neighborhood attraction, the behavior of particles is similar to the behavior of a group of roaches dropped on the floor: each roach starts to run to- ward the best place for hiding, according to its judgement (in the model terminol- ogy, each roach looks for the place which best matches its ideal environment). The attraction forces can be represented, in this example, as a set of springs connecting together roaches and forcing them to move in a coordinated way.

It is possible to describe the attraction force and the elastic bindings in terms of energy of the unit, while the movement is associated to a process of energy minimization, which will be called relaxation.

When the process is completed, each unit is located on the point in the im- age which best corresponds to the personal environment stored in the vector of weights in the unit and which best fulfills the topological relationships between neighboring units. When this phase is completed, each particle may adapt to its location, increasing the correspondence between the local environment and its ideal environment.

Iterating this process over different images, the relaxation step and the follow- ing adaptation of the weight vector let the network units differentiate from one another, and each of them adapts to live in an environment described by a certain feature vector, which appears in the input images, also when such feature vector is located in different positions in the input samples. A particle which is often located in a certain environment (e.g., a vertical edge) gets used to living on vertical edges, wherever they appear in the input. When an unknown image is presented to the system, the particle tries to reach a position which reminds it of a vertical edge. At the beginning of the relaxation process, the unit tends, therefore, to move toward any edge having the same direction. In the same time, attraction forces which act on the units keep them from moving freely on the image. This forces neighboring units to adapt to identify feature vectors which occur in neighboring points in the input images used during the training phase. In this way, we obtain a mapping between feature vectors and weight vectors, having the property that feature vec- tors which are spatially adjacent in the image (although having different values) are mapped into units spatially adjacent in the network grid.

For instance,Figure 8.2shows a column of units (a), adapted to live on a ver- tical edge. A curved edge is then presented to this part of the network (b). The

162 Evolution of an abstract image representation

(a) (b) (c) (d)

Figure8.2. Example of network relaxation. (a) Five units have adapted to identify a vertical edge. (b) A curved edge is presented. (c) The best matching between units and image: all units are located on the vertical part of edge. (d) Elastic constraints force units to space almost evenly.

optimal match between the units and the image occurs in the point of the edge which has vertical direction, and all units are attracted to reach that point (c). Us- ing a different example, if different kinds of food are placed on a surface, a swarm of units without constraints group together in the position where the most attrac- tive spot of food is located. The elastic constraints, however, prevent an excessive crowding of units by maintaining their relative distances. The balance between the two forces distributes the units as in (d), where the elastic force due to stretching balances the attraction toward the point where the edge has a vertical direction.