Klas Modin PRO
Mathematician at Chalmers University of Technology and the University of Gothenburg
Klas Modin
What is shape analysis?
Illustrating example
Source
Target
Q: What is a 'shape' ?
A: Anything diffeomorphisms act on
Basic idea
- Riemannian geometry on
manifold of diffeomorphisms
- Variational problem \(\Rightarrow\) PDEs
Distance between shapes =
shortest diffeomorphic warp
from source to target
History of shape analysis
Shape
analysis
Optimal transport
Information theory
Hydrodynamics
Non-linear Schrödinger eq.
Probability theory
Shallow water equations
Numerical analysis of PDE
Fluids and optimal transport
\mathrm{Diff}(M)
\mathrm{P}(M)
\mathrm{Id}
\mu
\nu
\pi(\varphi)=\varphi_*\mu
Riemannian metric
\displaystyle\mathcal{G}_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu
Induces metric
\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)
\mathrm{Hor}
[Arnold (1966), Benamou & Brenier (2000), Otto (2001)]
Invariance: \(\eta\in\mathrm{Diff}_{\mu}(M)\)
\displaystyle\mathcal{G}_\varphi(\dot\varphi,\dot\varphi) = \mathcal{G}_{\varphi\circ\eta}(\dot\varphi\circ\eta,\dot\varphi\circ\eta)
Exactly \(L^2\)-Wasserstein distance
\mathrm{Diff}_\mu(M)
\mathrm{Ver}
Geodesic equation on \(T\mathrm{Diff}_\mu/\mathrm{Diff}_\mu \simeq \mathcal{X}_{\mu}\)
\(\dot v + \nabla_v v = -\nabla p\)
Geodesic equation on \(T^*(\mathrm{Diff}/\mathrm{Diff}_\mu) \simeq T^*\mathrm{P}\)
\(\dot S + \frac{1}{2}|\nabla S| = 0 \quad \dot\rho + \mathrm{div}(\rho\nabla S) = 0\)
\(v=\nabla S\)
What is shape analysis?
By Klas Modin
What is shape analysis?
A brief overview of shape analysis, aimed at a general mathematics crowd.
