Reconstructions from incomplete projections

As for the reconstruction algorithms, we can divide them in two major groups iterative and non-iterative techniques. A non-iterative method produces the final result with a formula which is applied only 

Iterative algorithms are clearly less precise than single-application techniques, but their great advantage is that they introduce the time dimension in the computation, and this makes them particularly suitable to simulate biological processes. Even more important is the fact that the temporal dimension allows us to reconsider the problem of the minimum number of projections that are required for a complete reconstruction. During an iterative procedure, we could discover properties of the original structures that were not recorded in the projections, and in this case we could obtain a reconstruction even if 

The interesting point is that this is a mathematical version of the problem that we face in embryonic development. The fertilised egg contains far less information than the adult organism (whatever criterion is used to measure information in biological systems), and embryonic development can be described therefore as a process that is reconstructing a structure from incomplete information. This is another way of saying that embryonic development is a process that increases the complexity of a living system. The reconstruction of structures from incomplete information, in short, is a model that could help us understand how it is possible for a system to obtain a convergent increase of complexity. 

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Figure 3.2 A reconstruction from incomplete projections is a method where structure matrices and memory matrices are reconstructed in parallel.

As we can see, the problem can be given a precise formulation, but what really counts is that it can also be given a solution. I have demonstrated that structures can indeed be reconstructed by using only 10% of the minimum number of projections (Barbieri, 1974a, 1974b, 1987), and an iterative algorithm which exploits memory matrices. More precisely, a reconstruction from incomplete projections is possible if two conditions are met (1) if the reconstruction method employs memory matrices where new information appears, and (2) if the reconstruction method employs codes, or conventions, which transfer information from the memory space to the real space. 

A picture and its projections are both structures of the real space, and, when projections are incomplete, there is no possibility of perfoming exact reconstructions if information comes only from structures of the real space (or from equivalent structures of the Fourier space). Only in a related but autonomous space we can find genuinely new information, and the memory space is precisely that type of independent world. It is in fact the only space where a system can get the extra information that allows it to increase its own complexity. The MRM model, in other words, leads to a universal concept to the principle that there cannot be a convergent increase of complexity without memory. 

Before the DH lab was officially created, its director initiated collaboration with a Swiss newspaper focusing on their database of 200 years of digitalised articles (4 million articles). This base is currently accessible in image and text mode. Users can access it freely and select articles using a search-engine. The project is about transforming this large number of disparate, incomplete and (to a certain extent) unreliable archives into a coherent and searchable reconstruction of the past. The idea is to extract from this textual and visual database semantic information that could permit users to browse and visualise differently the information contained in the corpus. 

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