Education

Abstract

This issue contains two papers in the Education section. The first paper, “Computed Origami Tomography,” is presented by Axel Kittenberger, Leonidas Mindrinos, and Otmar Scherzer. The paper is motivated by advances in scanner technology and the increased sophistication of the accompanying 3D image reconstruction methods. The authors have implemented the idea of constructing a system that eliminates dangerous X-ray radiation but preserves the main features of CT scanners. Such a system is safe to use in training. The proposal is to replace the X-rays with optical illumination and detection of thin objects that are translucent in the visible spectrum. The origami figures, which are made out of thin sheets of cellulose acetate, become an attractive surrogate model for the objects in real CT scans. The first part of the paper contains an explanation of the physics assumptions and laws on which the X-ray tomography is based. The theory of light-ray propagation is similar to the theory of X-rays and provides a geometric optics approximation. The second part of the paper contains instructions for building a crafting device that can record digital images of small 3D objects at different rotation directions. Open source software for reconstructing the 3D object from the collected images is provided. An introduction to cryo-imaging and cryo-EM data provides additional insight into new directions in computerized tomography (``cryo” means “cold” in Greek). In this context, “cryo” refers to the process of immobilizing a sample specimen (a virus or a molecular cluster in a cell) by freezing it. The authors maintain that the described setup allows for conducting tomographic imaging experiments even at home. They have offered a training course along the lines of this paper multiple times and have produced a YouTube video explaining the basic ideas (the link to the video is included in the reference list). For those who wish to experiment with computational tomographic reconstructions without assembling the device, the authors have a database of several objects together with their tomographic measurements. The paper is written in an engaging style, focusing on the main ideas and pointing to further references. The second paper, authored by Bernardo Gouveia and Howard A. Stone, is “Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods.” The paper highlights a general, very flexible, yet relatively simple method to construct a resonant or repeated root solution to ordinary differential equations (ODEs). The idea of the method is to introduce a small parameter $\epsilon$ in the forcing function and then to construct a suitable Taylor expansion of a known homogeneous solution in that parameter. Next to the natural desire to analyze any situation mathematically, unwanted oscillations are highly relevant in civil engineering and other areas. Therefore, software systems provide solutions addressing resonance, but the results might be cumbersome or not insightful at times. The paper starts with background material on methods to obtain resonant solutions or repeated root solutions to ODEs. Then the authors present seven examples of increasing complexity, on which the ideas are developed and the advocated approach is explained. In the context of some of those examples, the advantages of the method are illustrated by comparison of its solutions to the solutions obtained by Mathematica. The concluding section of the paper contains a more general derivation, which encompasses all of the examples discussed before. The authors assert that the method is beneficial to students as well as to practitioners who come across this type of problem in their work. The paper is written in a clear and deductive manner.