AU mathematics and statistics professor Michael Robinson sees his recently completed book as a beginning, not an end. “This book is a jumping off point,” says Robinson. “I’m looking to take the ideas in the book and use them as a springboard to further address the problems presented. It’s very much a foundation to build off of.”
Published by Springer, Topological Signal Processing examines the intersection between signal processing, the art of collecting and analyzing measurements, and topology, the study of abstract notions of space. Exploring challenges in the signal processing community through the lens of topology gives mathematicians and engineers new tools for problem-solving, enabling them to develop approaches outside of traditional methodology. “Typically topology has been a pure math subject, but in the past few years applied topology has become a hot area,” says Robinson. “However, no one has really applied it to signal processing, so this book is the first unified treatment of that.”
Robinson is specifically interested in using topology to retrieve geometric data from sensors—objects that convert physical measurements into electrical signals—that are not cameras. This data can be used to create pictures, allowing people to see non-visual information, such as sounds and echoes. These sensors include complex equipment such as radar or sonar systems, as well as more simple devices such as an audio recorder. Applying topology to these sensors’ geometric data helps engineers bypass uncertainties that traditional statistical methods struggle with, enabling them to create clearer, more accurate pictures.
Following up on a curiosity that developed while working as an engineer, Robinson was able to investigate the intersection between topology, mathematics, and engineering through his PhD research at Cornell University. Robinson studied differential equations through the point of view of topology, allowing him to apply concepts developed from an abstract notion of space to concrete, mathematical equations. This research led him to pursue a post-doctorate at the University of Pennsylvania with Robert Ghrist, a person known for applying topology to engineering problems. Robinson credits Ghrist for his motivation to write the book, acknowledging that it was through Ghrist’s urging that he committed to the project. “He basically said, ‘There is no one else who can write this and this information needs to be accessible to the public; therefore, you will write it,’” Robinson says.
The concepts in Topological Signal Processing are not just applicable for seasoned mathematicians and engineers, though; Robinson’s students are using his topological methods in their projects here at AU. One such project is an attempt to more accurately measure and form pictures of the structure of the wind over the ocean. Typically, satellites are used to measure wind structure by assessing how rough the surface of the ocean is. They direct radar beams on the ocean’s surface and measure how much signal comes back. If it is very windy, the signals reflect a turbulent surface; if the wind is calm, the signals indicate the ocean is more tranquil. These signals are processed and form a picture of the wind’s structure. However, these pictures are taken at a low resolution, resulting in an inaccurate presentation of the data. By applying topology to the satellite’s signal processing data, Robinson and his students are able to form clearer pictures. “We’re taking ideas developed through this topological signal processing framework, turning them into algorithms that can process the satellite data, and then putting the data through these algorithms,” says Robinson. “This approach allows us to see structural detail you wouldn’t be able to see using any other system.”
This breakthrough technique has huge implications for weather and climate prediction, connecting to industries such as air navigation. If these models can inform airline pilots the wind is turbulent in a certain area, they can avoid this region, ensuring a safe flight for passengers. And if the data and pictures reveal the wind is frequently turbulent in an area, perhaps these are regions that should be avoided altogether.
Robinson hopes the concepts in Topological Signal Processing will be points of entry for solving specific problems like wind structure measurement and believes bringing topology and engineering together is key in providing new solutions. “I want to increase visibility for signal processing problems in the applied topology community, and I want to increase visibility of topological methods in the engineering community,” Robinson says. “Engineers have very interesting problems and topologists—mathematicians—have techniques that can be applied to solve them. Both of these communities have a lot to learn from each other.”
You can read more about Topological Signal Processing by visiting the publisher’s website.