Datos, información y conocimiento

The network, in order to build a knowledge about the imaged objects, needs to be trained with a set of images which represent the same class of objects. During the training process, each particle differentiates from the others, progressively adapt- ing to live in a particular environment. The training procedure is similar, apart from the relaxation step, to the training of an unsupervised neural network, which can be described as follows. After a suitable initialization of the weights, the pat- terns, or images, in the training set are presented to the network inputs, usually in a random order. For each input pattern, the output of the network is evaluated, and some, or all, of the weights of the network are adapted. The process is then iteratevely repeated until convergence.

Once the training process has been completed, the weight of the units em- bed an abstract representation of the imaged objects. Indeed, each unit represents a feature which is present in the images, while topological relations between fea- tures in the image are translated into the topological relation between the units representing them. A simple example of the obtained representation is described

inFigure 8.4. The figure represents a small ENM network trained with a set of

digitized hand-drawn capital H (top row). In this case, the selected feature maps include the gray level of the image, and a lowpass filtered version of the same im- age. After the training, the first component (which corresponds to the gray-level feature map) of the unit weights contains a “mean” shape of the letters used dur- ing the training (bottom row). The network has stored in its weights an abstract representation of the image set, by discovering that all the input images contain the same basic shape, although stretched and deformed.

The adaptation process is performed iteratively. After an initialization phase, an image from the data set is presented to the system, which relaxes, allowing units to find a suitable position in the image. Then, each unit adapts to live in that po- sition, by increasing the match between its personal ecosystem and the local envi- ronment present in that point. At this point, it is possible to present a new image and repeat the process until the system reaches the desired adaptation level.

168 Evolution of an abstract image representation

Image 1 Image 2 Image 3 Image 4 Image 5

Figure8.4. Abstract representation extracted by a network trained with five images of a capital H (top row). The network weights (bottom row) shown that the network has detected that a single shape is present in all the images.

8.5.1. Initialization

The initialization of the network weights can be done either by a random selection of the weights in a suitable range of values, or by direct assignment to some known value. To enhance stability and convergence speed, it is usually advisable to assign some “reasonable” value to the weights of the units. In this case, a good initializa- tion value can be obtained by sampling the images in the training set. The network grid may be used to assign to each unit a weight vector equal to the feature vector evaluated in the match point before relaxation.

8.5.2. Presentation of input patterns

The main loop of the training phase starts by presenting the input patterns, one at a time, to the network. In order to improve the learning and the generaliza- tion properties of the network, the order of presentation of the input patterns is randomized before each iteration. When each pattern has been presented to the network, the relaxation process starts as described above. After the relaxation pro- cess has completed, it is possible to let the units adapt to the new environment by changing the associated weight vector. It is worth noting that in this kind of ar- chitecture, the activation of the units does not relate to the output of the network. The output, indeed, is described by the final position of each unit in the image plane. For instance,Figure 8.5reports the deformation patterns of the network grid caused by the inputs reported inFigure 8.4to the trained network.

8.5.3. Learning rule

The training phase, after the relaxation process has completed, has been designed to increase the fitness of the unit to the environment, by increasing the match

Leonardo Bocchi 169 0 20 40 600 10 20 30 40 50 60 (a) 0 20 40 600 10 20 30 40 50 60 (b) 0 20 40 600 10 20 30 40 50 60 (c) 0 20 40 600 10 20 30 40 50 60 (d) 0 20 40 600 10 20 30 40 50 60 (e)

Figure8.5. Deformation of the network grid representing the mapping between input images and network units for the data reported inFigure 8.4.

between the weights of each unit and the image features. This has been done by modifying the weights of the unit according to the following rule:

Δwi j=IPi j−wi j, (8.10) whereis the learning ratio, bounded between 0 and 1, andPi jis the position of the unit at the end of the relaxation process.

The proposed learning rule, similar to the one introduced by Kohonen [17], improves the fitness of the unit by increasing the correspondence between each unit and the image features. In the following relaxation steps, the unit will be, therefore, attracted toward a point with the same feature vector. In this way, the unit learns to act as a feature detector specialized to identify points having a feature vector similar toI(Pi j).

8.5.4. Iteration

The process described above for a single image needs to be repeated for all the images in the training set, which are presented to the network in a random order. The entire sequence is then repeated iteratively, for a fixed number of iterations. As happens for the parameters which drive the relaxation process, the best results can be obtained by using a linearly decreasing learning rate.