In mathematics, a small distance is synonymous with a small difference. Using the mathematics called shape analysis, researchers can study whether the difference between figures A and B below is smaller than the difference between B and C. The mathematically correct answer depends on which concept of distance is used in the analysis and how sensitive it is to noise. In the project “Shape analysis – geometry and computation”, Klas Modin will build upon mathematical methods in which a transformation group deforms an object. The distance between shapes A and B is then defined as the shortest transformation route that takes A to B. In shape analysis, a shape can, for example, be a surface, an image, an electrical field or a particle cloud in physics.
Need to compare non-rigid shapes
The research area arose in the early 2000s, motivated by medical imaging problems of comparing organs from different patients in magnetic resonance images (MRI). The organs are non-rigid, they are deformed as soon as the body moves and depending on posture, and a way was needed to quantify differences between shapes. By combining theories partly from the Russian mathematician Vladimir Arnold, who in the 1960s made a geometric formulation of the equations of motion in fluid mechanics, and partly from the Swedish statistician Ulf Grenander, who built a statistical theory for shapes, a useful mathematical framework was obtained.
– The research on shape analysis builds bridges between different mathematical areas, it is a sort of interdisciplinary field within mathematics with many applications that I find very interesting. My research is about developing the mathematics of shape analysis, but by extension, the mathematical techniques that are being developed can become useful tools for researchers who study the shape of proteins through cryo-electron microscopy, or for physicians who use MRIs to search for shape deviations in organs caused by tumours.
Found his field in New Zealand
Klas Modin first got in touch with shape analysis when he was a postdoctoral fellow at Massey University, New Zealand. He then received an international postdoctoral position through the Swedish Research Council at the University of Toronto, where he worked with Professor Boris Khesin, a former student of Arnold. This cooperation has continued after the return to Sweden five years ago. The grant he has now received gives a welcome opportunity to recruit PhD students and focus on the research.
The programme Wallenberg Academy Fellows is financed by Knut and Alice Wallenberg Foundation and has been established in partnership with the royal academies and 16 Swedish universities. The intention is that Sweden’s most promising young researchers should be able to concentrate on their research for a long period and have good resources. They also have the opportunity to participate in a mentoring program, which helps boost their scientific leadership.
Press release from KAW, Twenty-nine young researchers become Wallenberg Academy Fellows 2019 >>
Interview with another new Wallenberg Academy Fellow, Elin Esbjörner at Biology and Biological Engineering >>
Interview with another new Wallenberg Academy Fellow, Witlef Wieczorek at Microtechnology and Nanoscience >>
Text: Setta Aspström
Pictures: Klas Modin. The second is a shape representation using de Rham currents.
Photo: Markus Marcetic