## Image processing: Active Connection Matrix

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**Introduction **The procedures of treatment of the image processed in Semeion are based on an innovative conceptualisation: the image is not processed according to criteria previously established; it is not interpreted in the light of specific methodological bounds. We do not ask the image to conform to some expectation of ours. In reality, the image is put into condition to "show itself as it is”, in the limits of what it is possible, without epistemological naivety that disregard the problem of the relationship subject-object of the research. In more specific terms, we do not apply a neural network to an image, but in a sort of a Copernican revolution, we transform the image into a network, each pixel in a node, in relation weighted with the pixels/nodes that surround it.

On this base, it is useful to distinguish three levels of complexity, each corresponding to the phases of the image processing:

- Syntactic Level
- Semantic Level
- Pragmatic Level

The Syntactic level corresponds to the development of the system for the processing of an image that is as neutral as possible, in accordance with the guiding principle. The result is an analysis of the image that shows particulars that no other technique succeeds in obtaining, which shows “something" that other systems do not see, without bringing it to pre existing classifications. This “something” can be, as we will show further on, a particular area individuated by a border, by a particular colour, what in general we can conceive as a simple syntactic relationship of juxtaposition among colours and pixels, in practice the structure of the internal relationships among the elements of the image. Something of which we do not suspect minimally the presence but that is visible only after the processing, something that we still do not know and that we do not want to define at this level of investigation.

The semantic level corresponds to the phase of identification of what it appeared after the processing. In this phase the system, through a series of passages and of comparisons with traditional techniques of diagnosis, reaches a correct interpretation of the image. As we will see, the results obtained through the algorithms elaborated at Semeion are superior than the others obtained with the techniques obtained in literature and, in particular, in the medical field, allow a precocious recognition of cancerous tissues and masses.

The pragmatic level corresponds to a further development of the semantic level: we have to put into relationship the interpretations of the images, of the particular emerged with the processing, with the variable context in which the same particulars can be. Then, as a same linguistic expression can have different meaning dependent on the communicative context, or on the tone, as the "something" emerged from the image can be interpreted in different ways depending on the space-time context. This implies a comparison even closer with the traditional diagnostic techniques.

**Basic notions**

The relationship that develops, in space and time, between the entity and the context in which it interacts and which enables one to define the phenomenon, is expressed through strains and stresses; the existing forces, between the minimal elements of the phenomenon.

These forces are called:

- finite when they acquire true finite values in any space and time neighbourhood of a considered initial point;
- continue when there is no point of a space and time neighbourhood where the value of the force depends on the direction reached in the considered point, and
- local when the propagation of the effects of these forces continues through every single space and time point subsequent to the initial one being considered.

The ability of a phenomenon to keep the forces finite, local and continue among its minimal elements, is the space- time cohesion the phenomenon itself. The relevant topology of a phenomenon is its intrinsic geometry which expresses the form which can be taken from the frequency spectra where the phenomenon is analysed and from the respective variations of energy. These variations of energy become relevant information because they are an expression of the relevant topology of the phenomenon, that is to say of its form or intrinsic geometry. In this case the phenomenon is called a visual phenomenon.

Every relevant topological phenomenon manifests an identity and unity of the phenomenon itself which is determined by its space-time cohesion. This means that each minimal element of the phenomenon is directly or indirectly contiguous and connected by specific forces to the others. Therefore the quantitative value of each minimal element of the analysed phenomenon results from the action of these forces. It can be demonstrated that in a relevant topological phenomenon the forces which connect each of the minimal elements to each other in its local neighbourhood are enough to explain the space-time cohesion of the whole phenomenon.

This allows one to say that each visual phenomenon can be expressed as a matrix of values, locally connected each other, by other values (weights), representing the local cohesion forces of the minimal units of that phenomenon.

The phenomenon to we refer, in general, in our research concerns the image which is perceived by our senses when a light-shrouded subject appears to us precisely as a phenomenon. The image of the subject caught and made available as a phenomenon, can be represented for its analytic treatment by a matrix of points corresponding to the pixels the assumed initial image. Trying to extract from this image - from this phenomenon- other information about the subject producing it, which is not visible in the initial image which is being considered, allows us to consider the initial image's matrix of pixels as a dynamic system which develops in its phase space until it creates a final configuration matrix of the pixels. It is important not to mistake this space phase with the two-dimensional or three-dimensional space of the initial image. In fact, it is the other dimension which is derived by the intensity of the connection forces of the pixels to each other which is added to the latter space's dimension. This occurs when the point matrix is considered and when its active meaning is considered that the initial matrix will results in a final matrix precisely because of a dynamic evolution of these connections.

The functioning of ACM systems is based on local, deterministic and iterative operations:

- Local, because in each elaboration cycle the operations involve a central pixel and its relations with the very contiguous pixels (neighbourhood of the central pixel);
- Deterministic, because the static state towards which the dynamic system tends is represented by the matrix of pixels with the new image, is based on deterministic equations: therefore the elaboration can be repeated always resulting in the same outcome;
- Iterative, because the operations of the dynamic system repeat themselves, iteratively, until the evolution in the space of phases, reaches its attractor.

ACM families differ according to how they let units and connections evolve. They are divided, more specifically, into 3 families, shown in the figure below, according to the following evolution rules:

a. Fixed Connections: Units ux are allowed to evolve until they reach a static state showing the presence of an attractor. The static state, and therefore the attractor, changes according to the evolution rule which is used. A connection matrix is initially determined by an equation called Automata Rule which uses the pixel's matrix of the assigned image.

b. Fixed Units: Connections are allowed to evolve until they reach a static state showing the presence of an attractor. The static state, and therefore the attractor, changes according to the evolution rule that is used. It uses the pixel's matrix of the assigned image, which remains constant in every elaboration cycle;

c. Dynamic Connections and Units: Connections reaches their attractor according to the matrix of the units ux updating in every elaboration cycle. In this case, the brightness of the matrix pixel of the units ux, is updated during the evolution of the system, participating in the correction of the matrix of the connections.