The Levenshtein distance as a measure of mirror symmetry and homogeneity for binary digital patterns

Godfried T Toussaint

doi:10.5194/ica-abs-1-367-2019

Abstract

Abstract. The complexity of a digital pattern, image, map, or sequence of symbols is a salient feature that finds numerous applications in a variety of domains of knowledge [1], [7], [10], [11]. Two features of patterns that form inherent components of pattern complexity, are mirror (reflection) symmetry and homogeneity [8], [9]. In the raster graphics representation mode, a pattern consists of a two-dimensional array (matrix) of elements (pixels, symbols). It is assumed here that the elements are binary-valued (black-white). With such a representation it is common to compute properties of 2-dimensional patterns, such as complexity, mirror-symmetry, and homogeneity, along the 1-dimensional rows, columns, and diagonals of the array [4]. In addition, within each row, mirror symmetries may be analysed either globally or locally [3]. A pattern that does not exhibit global mirror symmetry may still possess an abundant number of local mirror symmetries. Local symmetries permit graded measures of symmetry rather than all-or-nothing decisions. One powerful type of local symmetry is the sub-symmetry, a contiguous subset of elements of the pattern that is palindromic (has mirror symmetry). It has been shown empirically that the total number of sub-symmetries present in a pattern may serve as an excellent predictor of the perception of both visual pattern complexity [5], and auditory pattern complexity [6]. The present research project explores how two well-known measures of the distance between binary patterns and their inversions, correlate with sub-symmetries, as well as other measures of symmetry and homogeneity.

Highlights

Given two equal-length binary symbol sequences A and B, the Hamming distance between A and B is the total number of substitutions of individual symbols required to transform A into B
The fact that similarity between a pattern and its mirror image constitutes a measure of mirror symmetry provides the motivation for the exploration of the Levenshtein distance in this context
We analyze the 19 sequences with the new proposed measure of symmetry based on the Levenshtein distance, and compare the results with those obtained with sub-symmetries and Krüger’s measures of symmetry and homogeneity for several values of k

Summary

Introduction

Given two equal-length binary symbol sequences (patterns) A and B, the Hamming distance between A and B is the total number of substitutions of individual symbols required to transform A into B. Given two binary symbol sequences (patterns) A and B, the Levenshtein ( edit) distance between A and B is defined as the minimum number of insertions, deletions, and substitutions of individual symbols required to transform A into B. On the other hand the Hamming distance is 9, and fails to capture the similarity.

Results

Conclusion