Representations of Knowledge in Complex Systems

Modern sensor technologies, especially in biomedicine, produce increasingly detailed and informative image ensembles, many extremely complex. It will be argued that pattern theory can supply mathematical representations of subject-matter knowledge that can be used as a basis for algorithmic `understanding' of such pictures. After a brief survey of the basic principles of pattern theory we shall illustrate them by an application to a concrete situation: high magnification (greater than 15 000 ×) electron micrographs of cardiac muscle cells. The aim is to build algorithms for automatic hypothesis formation concerning the number, location, orientation and shape of mitochondria and membranes. For this we construct a pattern theoretic model in the form of a prior probability measure on the space of configurations describing these hypotheses. This measure is synthesized by solving sequentially a jump-diffusion equation of generalized Langevin form. The jumps occur for the creation-annihilation of hypotheses, corresponding to a jump from one continuum to another in configuration (hypothesis) space. These continua (subhypotheses) are expressed in terms of products of low dimensional Lie groups acting on the generators of a template. We use a modified Bayes approach to obtain the hypothesis formation, also organized by solving a generalized Langevin equation. To justify this it is shown that the resulting jump-diffusion process is ergodic so that the solution converges to the desired probability measure. To speed up the convergence we reduce the computation of the drift term in the stochastic differential equation analytically to a curvilinear integral, with the random term computed almost instantaneously. The algorithms thus obtained are implemented, both for mitochondria and membranes, on a 4000 processor parallel machine. Photographs of the graphics illustrate how automatic hypothesis formation is achieved. This approach is applied to deformable neuroanatomical atlases and tracking recognition from narrow band and high resolution sensor arrays.