Shape analysis with applications
Shapes exist everywhere: in arts, science, and engineering; in everyday life, during day and night. Our research group, consisting of PhD students Erik Jansson and Carl-Joar Karlsson supervised by Prof. Klas Modin, studies shapes as mathematical objects. That is, we investigate how shapes can be described and what is possible with such descriptions.
Medical image analysis is one of the applications of our methods. A medical doctor could for instance ask how the shape of a region of brain tissue is changed on patients suffering from Alzheimer’s disease. Imagine having an image of the original brain and later taking a new image of the brain. The original “template’’ can be deformed into the new image, and the goal of the so-called registration problem is to identify how the template transforms into the target image, see the figure. Finding a suitable mathematical description of such problems is a challenge.
The approach, often called shape analysis, utilizes fundamental concepts of differential geometry to enable more stable and reliable computer algorithms. It is also crucial that algorithms are efficient. To meet these requirements, we use Riemannian gradient flows as our main tool. These are geometric generalizations of the familiar steepest descent method, an iterative method that steps along the negative of the gradient. Spaces of shapes are typically infinite dimensional (for example, the set of closed curves cannot be described by a finite number of parameters). Although this complicates the analysis, it is among these infinitely many transformations that we find the tools needed to guarantee, for instance, that image objects (representing a part of the brain or certain tissue, etc.) will not disappear or tear apart.
Another interesting application of the registration problem in shape analysis is that it could open up new pathways to understand certain deep learning architectures, allowing us to understand the dynamics of neural networks in a new, inherently geometric, perspective. Another application of shape analysis in AI could be to suggest ways to increase the capability of AI systems to learn shapes.
Mathematically, our task is to identify the algorithms that the computers should use and to prove that they work as intended. Therefore, we combine many aspects of infinite dimensional geometry and analysis. Similar approaches have become state-of-the-art solutions to the template matching problem in image analysis, most notably the so-called Large Deformation Diffeomorphic Metric Mapping (LDDMM).
Grenander, U. & Miller, M., Pattern theory: from representation to inference. Oxford University Press, Oxford, 2007. ISBN: 978-0-19-929706-1
Younes, L., Shapes and diffeomorphisms. Springer, Berlin, 2010. ISBN: 978-3-642-12055-8
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