Publication8

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. © 2011 Springer-Verlag Berlin Heidelberg.

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained. © 2013 World Scientific Publishing Company.