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Abstract:
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With the wide availability of inexpensive digital cameras and high-quality smart phone cameras, the world is awash in images. The novelty of image data comes from the fact that it visually captures more information than numerical data does, but it also contains a myriad of different types of data sets that often don’t belong to a Euclidean space. We focus on statistical methodology that is well-informed by the topology and geometry of non-Euclidean spaces. To extend classical statistical principles to the complex settings for image data, we rely on a key feature of object spaces; namely nonlinearity. Information extracted from an image is usually represented as a collection of points on disjoint unions of non-linear spaces or manifolds. More precisely, object spaces are connected metric spaces with a topological stratification. Stratified object spaces include real algebraic varieties, Hilbert manifolds, polyhedral complexes, spaces of positive semidefinite operators, certain homogeneous spaces, etc. Here we take a nonparametric approach in Object Data Analysis with an eye toward the computational speed of algorithms required for said methodology, when applied to 3D imaging.
References: V. Patrangenaru and L. E. Ellingson. Nonparametric Statistics on Manifolds and their Applications to Object Data Analysis. CRC Press, 2015.
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