Geometric Information Theory

In the context of patterns or images, information theory serves as a valuable tool for identifying the length scales and positions where information is located. In this chapter, we present an information-theoretic approach to analyse patterns described by probability densities in continuous space, which can be interpreted as the light intensity representation of an image. We then decompose the relative information between a uniform a priori distribution and the observed distribution into contributions from both varying positions within the system and different length scales. This leads to an information decomposition that is different from what we presented for discrete system, and with a focus that in some cases are more applicable to images and pattens.