Integral geometry for modeling shapes

21 Feb
Widely Applied Mathematics Seminar
Dr. Sayan Mukherjee
Thursday, February 21, 2019 - 3:00pm to 4:00pm
Pierce 209

A challenge in data science is modeling complex data such as 3-dimensional shapes such as a brain tumor or a bone. Applications of shape analysis include associating the shape of a tumor to clinical covariates and biomarkers in radio genomics to relating variation of the morphology of a bone to selection in evolutionary biology/anthropology. In this talk we will develop an approach to represent shapes using integral geometry. The representation is based on a transformation of shapes into representations that allow for analysis using standard statistical tools without requiring landmarks on the shapes and the shapes being diffeomorphic. The transformations are based on Euler integration. By using a variation of Schapira’s inversion theorem, we show that these transforms are injective on the space of shapes — each shape has a unique transform. The main theoretical result provides the first (to our knowledge) finite bound required to specify any shape in certain uncountable families of shapes, bounded below by curvature. We also show how these transformations can be used in conjunction with functional data analysis for regression problems where the shape itself is a covariate.

 

 

Sayan Mukherjee is a Professor of Statistical Science, Mathematics, Computer Science, and Biostatistics & Bioinformatics at Duke University. Sayan received his PhD from MIT and was a postdoc at the Broad Institute. He is a Fellow of the Institute of Mathematical Statistics.His research covers Bayesian methodology; computational and statistical methods in statistical genetics, quantitative genetics, cancer biology, and morphology; discrete Hodge theory, geometry and topology in statistical inference; inference in dynamical systems; machine learning; and stochastic topology

 

Speaker Bio: 

Sayan Mukherjee is a Professor of Statistical Science, Mathematics, Computer Science, and Biostatistics & Bioinformatics at Duke University. Sayan received his PhD from MIT and was a postdoc at the Broad Institute. He is a Fellow of the Institute of Mathematical Statistics.His research covers Bayesian methodology; computational and statistical methods in statistical genetics, quantitative genetics, cancer biology, and morphology; discrete Hodge theory, geometry and topology in statistical inference; inference in dynamical systems; machine learning; and stochastic topology

Host: 
Prof. Maha
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Vamsi