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{Diff}(M)

\mathrm{P}(M)

\mathrm{P}(M)

\mathrm{Id}

\mathrm{Id}

\mu

\mu

\nu

\nu

\pi(\varphi)=\varphi_*\mu

\pi(\varphi)=\varphi_*\mu

Riemannian metric

\displaystyle\mathcal{G}_\varphi(\dot\varphi,\dot\varphi) = \int_{M}\left\vert \dot\varphi \right\vert^2 \mu

\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)

\overline{\mathcal{G}}_\mu(\dot\mu,\dot\mu) \Rightarrow d_W^2(\mu_0,\mu_1)

\mathrm{Hor}

\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)

\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{Diff}_\mu(M)

\mathrm{Ver}

\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 $T^*(\mathrm{Diff}/\mathrm{Diff}_\mu) \simeq T^*\mathrm{P}$
$\dot S + \frac{1}{2}|\nabla S| = 0 \quad \dot\rho + \mathrm{div}(\rho\nabla S) = 0$

$v=\nabla S$

Klas Modin

What is shape analysis?

Illustrating example

Basic idea

History of shape analysis

Fluids and optimal transport

What is shape analysis?

What is shape analysis?

Klas Modin PRO

Klas Modin

What is shape analysis?

What is shape analysis?

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