Binary images, M-vectors, and ambiguity

TLDR: This work introduces the notion of M-vector of a binary word which allows us to have a linear notation in the form of a unique vector representation of the Parikh matrix of the binary word and extends this notion to a binary image treating it as a binary array over a two-symbol alphabet.

Abstract: Mateescu et al (2001) introduced the notion of Parikh matrix of a word as an extension of the well-known concept of Parikh vector of a word. The Parikh matrix provides more numerical information about a word than given by the Parikh vector. Here we introduce the notion of M-vector of a binary word which allows us to have a linear notation in the form of a unique vector representation of the Parikh matrix of the binary word. We then extend this notion of M-vector to a binary image treating it as a binary array over a two-symbol alphabet. This is done by considering the M-vectors of the words in the rows and columns of the array. Among the properties associated with a Parikh matrix, M-ambiguity or simply ambiguity of a word is one which has been investigated extensively in the literature. Here M-ambiguity of a binary array is defined in terms of its M-vector and we obtain conditions for M-ambiguity of a binary array.