We present the Julia package Manifolds.jl , providing a fast and easy-to-use library of Riemannian manifolds and Lie groups. This package enables working with data defined on a Riemannian manifold, such as the circle, the sphere, symmetric positive definite matrices, or one of the models for hyperbolic spaces. We introduce a common interface, available in ManifoldsBase.jl , with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using Bézier splines, an optimization task on manifolds, and principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms all comparable packages for low-dimensional manifolds in speed; over Python and Matlab packages, the improvement is often several orders of magnitude, while over C/C++ packages, the improvement is two-fold. For high-dimensional manifolds, it outperforms all packages except for Tensorflow-Riemopt, which is specifically tailored for high-dimensional manifolds.

Laser-induced breakdown spectroscopy (LIBS) is an important analysis technique with applications in many industrial branches and fields of scientific research. Nowadays, the advantages of LIBS are impaired by the main drawback in the interpretation of obtained spectra and identification of observed spectral lines. This procedure is highly time-consuming since it is essentially based on the comparison of lines present in the spectrum with the literature database. This paper proposes the use of various computational intelligence methods to develop a reliable and fast classification of quasi-destructively acquired LIBS spectra into a set of predefined classes. We focus on a specific problem of classification of paper-ink samples into 30 separate, predefined classes. For each of 30 classes (10 pens of each of 5 ink types combined with 10 sheets of 5 paper types plus empty pages), 100 LIBS spectra are collected. Four variants of preprocessing, seven classifiers (decision trees, random forest, k-nearest neighbor, support vector machine, probabilistic neural network, multi-layer perceptron, and generalized regression neural network), 5-fold stratified cross-validation, and a test on an independent set (for methods evaluation) scenarios are employed. Our developed system yielded an accuracy of 99.08%, obtained using the random forest classifier. Our results clearly demonstrates that machine learning methods can be used to identify the paper-ink samples based on LIBS reliably at a faster rate.

This work extends the dynamic programming approach to calculation of an elastic metric between two curves to finding paths in pairs of graph drawings that are closest under this metric. The new algorithm effectively solves this problem when all paths between two given nodes in one of these graphs have the same length. It is then applied to the problem of pattern recognition constrained by a superpixel segmentation. Segmentations of test images, obtained without statistical modeling given two shape endpoints, have good accuracy.

The gamma index is a measure used routinely for the quality control of dose delivery in radiotherapy, implemented in commercial systems for the verification of treatment plans. It involves comparison of the difference between planned and delivered doses to a single reference. The same reference value is selected for all points in the plan that can potentially hide dose delivery errors, especially in medium and low dose areas. In this study, a receiver operating characteristic analysis is used to demonstrate the limits of the performance of the global gamma index as a method for detecting dose delivery errors. The performance of a global gamma index is compared with two approaches based on statistical tests for outlier detection. Two statistical approaches are considered: according to the first, the distribution of the delivered doses is estimated based on an appropriate calibration procedure. According to the second, the distribution of the delivered doses is estimated based on the detection of relatively homogeneous regions of a plan and analyzing the distributions of planned doses within these regions. The performance of the three approaches is compared based on analytical considerations and in simulations in which errors are intentionally introduced to the plan delivery and noise related to dose delivery is modeled. We have shown that a statistics-based approach to gamma analysis generally leads to better detection of true delivery errors. The results of analytical consideration coincide with the simulations. In simulations, we observe that both statistical approaches are better detectors of true delivery errors than the global method for the gamma-index passing rate in the range from 0.9–1.0. It is shown that the global gamma index is a weak detector of dose delivery errors, which in some circumstances behaves only slightly better than a purely random classifier.