A Metric for Data Set Comparison Via QR-factorization

Abstract

A new metric is introduced for comparing matrix data sets based on QR-factorizations. The metric measures the degree of similarity between the bases of any two matrices, with measure values in the interval [0, 1]. The metric is initially developed in a general framework, for any m×n matrix. The proposed measure is then discussed specifically in reference to matrices representing theoretical and physical data sets. The measure has direct applications for projects involving particle travel, and image recognition, comparison, and registration, particularly in instances involving intermodal data acquisition. The measure is shown to satisfy the requirements for a metric and is demonstrated to calculate rotation angle between two data sets that differ by angle, θ. The results show that the metric works well in assessing differences between data sets that differ by small perturbations, assuming proper alignment prior to comparison. It is also shown to be an effective tool for 1) determining optimal rotational alignment of test 2 − D images and 2) identifying an unknown sample in a data base in a single blind test.