Two-Dimensional Digitized Picture Arrays and Parikh Matrices

Abstract

Abstract: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. Response to Reviewers:The authors are again grateful to the referee for a very meticulous reading of the revised version1 and thank the referee for further suggestions on improving the work. The authors have now revised the paper and revision2 has been submitted.Details of the changes done in revision2 are listed in "Author Response to comments on revision 1" as responses to the comments. 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. Powered by Editorial Manager® and Preprint Manager® from Aries Systems Corporation

Precedence Matrices of Binary and Ternary Words and their Algebraic Properties

“…Properties of precedence matrices of binary and ternary words have been studied in this paper. It will be interesting to make a study of such matrices in the context of picture arrays taking motivation from corresponding studies of Parikh matrices of picture arrays [11].…”

Algebraic Properties of Parikh Matrices of Binary Picture Arrays

“…Picture Array e notion of the Parikh matrix of a word has been extended to a picture array in [21] by introducing a row Parikh matrix and a column Parikh matrix of an array, which we recall now again restricting to a binary alphabet.…”

“…is proves the required result. e notion of M-ambiguity of words has been extended to two-dimensional picture arrays in [21]. We now recall this.…”

“…Fair words and their properties have been studied in [24]. A weak ratio property for an array is introduced in [21]. We now extend the notion of fair words to two-dimensional arrays.…”

“…For instance, notions such as repetitions of subarrays in 2D arrays are studied in [15][16][17]19], while periodicity in arrays is dealt with in [14,18]. e notion of the Parikh matrix of a word has been extended to row and column Parikh matrices of picture arrays in [21], and their properties have been studied. e problem of reconstruction of 2D binary images has been studied [22] based on Parikh matrices.…”