Organization:
Funded by: | UT/EWI |
PhD: |  |
Supervisor: Daily supervisor: |   |
Collaboration: | - |
Description:
In many applications, geometric data is the main quantity of interest, but it is far from obvious how to represent shapes in a way that is both accurate and computationally efficient. In recent years, new representations are still being proposed to mitigate existing approaches’ shortcomings.
One such idea is that of level-set representations with neural networks, which has recently been proposed for shape reconstruction and matching tasks in computer vision and graphics. This is an extrinsic approach with implicit regularisation, with the tradeoff that its accuracy in representing a surface and its associated geometric quantities is obscured. To counter this disadvantage, in this project we plan a theoretical study of these representations and their accuracy. The natural approach is from a point of view of nonlinear approximation theory, which in this case needs to be strongly influenced by differential geometry, since the quantities of interest are curves and surfaces.
On the other end of the spectrum, parametric representations also offer complementary advantages, like the ability to rigorously define metrics on spaces of geometric objects. In this second category, we plan to explore generalizations of these notions for curves and surfaces of low regularity, with the ultimate goal of solving inverse problems formulated directly in terms of them.
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