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Can we manipulate tree forms like numbers?

Banff International Research Station for Mathematical Innovation and Discovery

Tree-forms are ubiquitous in nature and recent observation technologies make it increasingly easy to capture their details, as well as the dynamics of their development, in 3 dimensions with unprecedented accuracy. These massive and complex structural data raise new conceptual and computational issues related to their analysis and to the quantification of their variability. Mathematical and computational techniques that usually successfully apply to traditional scalar or vectorial datasets fail to apply to such structural objects: How to define the average form of a set of tree-forms? How to compare and classify tree-forms? Can we solve efficiently optimization problems in tree-form spaces? How to approximate tree-forms? Can their intrinsic exponential computational curse be circumvented? In this talk, I will present a recent work that I have made with my colleague Romain Azais to approach these questions from a new perspective, in which tree-forms show properties similar to that of numbers or real functions: they can be decomposed, approximated, averaged, and transformed in dual spaces where specific computations can be carried out more efficiently. I will discuss how these first results can be applied to the analysis and simulation of tree-forms in developmental biology.

Mathematics; Biology and other natural sciences; Mathematical biology

video/mp4

Author affiliation: INRIA

2018-07-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/66433

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/