The Kolmogorov–Sinai entropy in the setting of fuzzy sets for image texture analysis and classification

Abstract

The Kolmogorov–Sinai (K–S) entropy is used to quantify the average amount of uncertainty of a dynamical system through a sequence of observations. Sequence probabilities therefore play a central role for the computation of the entropy rate to determine if the dynamical system under study is deterministically non-chaotic, deterministically chaotic, or random. This paper extends the notion of the K–S entropy to measure the entropy rate of imprecise systems using sequence membership grades, in which the underlying deterministic paradigm is replaced with the degree of fuzziness. While constructing sequential probabilities for the calculation of the K–S entropy is difficult in practice, the estimate of the K–S entropy in the setting of fuzzy sets in an image is feasible and can be useful for modeling uncertainty of pixel distributions in images. The fuzzy K–S entropy is illustrated as an effective feature for image analysis and texture classification.