MDSLAB: A toolbox for the analysis of point sets using multi-dimensional scaling, hartigan dip test and α-stable distributions

Abstract

Background and objective: Mapping different point sets onto each other is a frequent problem in many fields of science. In radiation therapy, for example, electromagnetic tracking allows to measure the spatial location of radiation sources inside the treatment volume without external registration. The various source loci form point sets, with concomitant subsets, which refer to different coordinate systems when data is collected during different treatment sessions. We present a toolbox, called MDSLAB, which allows to unify a given point set and its reference point set. Methods: The toolbox relies on multi-dimensional scaling and the estimation of principal coordinates only from observed distances between pairs of points in each set. Deviations between the principal coordinates, i.e. the data projections, are quantified and their histograms analyzed employing a Hartigan dip test. Observed uni- or bimodal, asymmetric distributions of distance deviations are approximated by α-stable distributions. Results: We illustrate the working of the toolbox with an application to data collected during high dose rate brachytherapy. Sensor dwell positions inside catheters, implanted into a female breast, are collected, their distance matrices estimated and their underlying principal coordinates computed by diagonalizing the related kernel matrix. Considering the projections of the data onto their principal axes, distance deviations are quantified, their underlying distributions determined and approximated by heavy-tailed distributions. Tools, either images or whole animations, are developed to visualize the spatial dwell positions of the sensors, which map out the catheter shapes before any radiation treatment is started. Distance deviation histograms are also visualized and fit to α-stable distributions. Conclusion: MDSLAB provides a convenient tool to register point sets which are collected in different coordinate systems but whose relative distances should be identical, ideally. In practice, however, they mostly differ and MDSLAB allows to quantify such deviations and analyze their statistics as well as conveniently visualize them in images or movies.