Representations of Knowledge in Complex Systems

Abstract

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 15000x) 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 jumpdiffusion 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.

Metric Sample Spaces of Continuous Geometric Curves and Estimation of Their Centroids

“…In practice, curves without landmarks, i. e., closed curves for which no specific point is distinguished, can also be of interest (see, e. g., Grenander and Miller, 1994).…”

“…All of them generate continuous functions with probability 1, so they induce random curves in the Fréchet sense, as a consequence of Theorem 5.2. The following are typical examples: Models of random closed polygonal functions, such as the Circular Autoregressive (CAR) model for edges (Mardia et al, 1991) and the model of linearly connected arcs subject to circulant random scale-rotation transformations (Grenander et al, 1990;Grenander and Miller, 1994). Models based on spline interpolation of a random vector of landmark points (Mardia et al, 1991).…”

“…This paper is partly based on a lecture given at the Workshop on Statistics of Brain Mapping, Montreal, June [13][14] 1998. We thank Professor K. Worsley for facilitating our participation in this event, and we also express our thanks to participants that provided valuable comments on our work, in particular to M. I. Miller.…”

“…The main drawback in this approach is that information on the whole curves is lost. Also, there are situations in which the identification of landmarks is controversial (see discussion in Grenander and Miller, 1994).…”

“…Some of these representations are approximations, e. g. by means of linearly connected straight arcs (Grenander et al, 1990; Grenander and Miller, 1994). Other representations are exact but only valid for curves of particular shapes, e. g. the radius-vector function for star-shaped curves (Stoyan and Stoyan, 1994, Ch.…”