Statistical Shape models

Session 1: (L 9-9:45) Shape analysis and actions of the  diffeomorphism group

Session 2: (E 10-10:45) Landmark analysis in Theano Geometry

Session 3: (L 11-11:45) Linear representations and random orbit model

Session 4: (E 12:30-13:15) Landmarks statistics in Theano Geometry

Session 5: (L 13:30-14:15) Shape spaces of images, curves and surfaces

Session 6: (E 14:30-15:15) Analysis of continuous shapes

Shape Matching

$\phi\in\mathrm{Diff}(\Omega)$ diffeomorphism of domain $\Omega$

$\phi$

Variational problem to find optimal $\phi\in G$:

Group and quotient

Riemannian View

$\phi_t:[0,T]\to\mathrm{Diff}(\Omega)$ path of diffeomorphisms (parameter t)

Flow Equations

Constructing diffeomorphisms:

Forward flow:

$$\frac{d}{dt}\phi(x,t)= v(\phi(x,t),t) ,\ \forall x\in\Omega$$

$\phi(x,0)=x$

$\phi_t\in G=\mathrm{Diff}(\Omega)$

$v_t$ is a vector field on $\Omega$: $v\in V\subset \mathcal{X}$

It is the Eulerian velocity that controls the flow

Variational Problem

The family $v_t$, $t\in [0,T]$ determines the flow

Variational problem:

     $E_{s_0,s_1}(\phi)=\mathrm{min}_{\phi_t}R(\phi_t)+\frac1\lambda S(\phi_T.s_0,s_1)$

     $R(\phi_t):=\int_0^T\|\partial_t \phi_t\|_{\phi_t}^2dt$, $v_t=\partial_t\phi_t\circ\phi_t^{-1}$

So we need a norm on $V$ for defining $\|\partial_t\phi_t\|_{\phi_t}=\|v_t\|$

Velocity Fields

Norm on $V$ is defined via a Reproducible Kernel Hilbert Space (RKHS) structure

Velocity $v$

Momentum $m=Lv$ (recall: $p$ for landmarks)

$L$ is the momentum operator $L:V\to V^*$

Kernel $K$ is the inverse momentum: $K=L^{-1}$

$K$ is as well a matrix-valued function on $\Omega\times\Omega$: $$K: \Omega\times\Omega\to\mathbb{R}^{d\times d}$$

e.g. $K(x,y)=\alpha e^{-\frac{\|x-y\|^2}{2\sigma^2}}$

RKHS

Kernel $K$ is the inverse momentum: $K=L^{-1}$

$K$ is as well a matrix-valued function on $\Omega\times\Omega$: $$K: \Omega\times\Omega\to\mathbb{R}^{d\times d}$$

e.g. $K(x,y)=\alpha e^{-\frac{\|x-y\|^2}{2\sigma^2}}$

For $a\in\mathbb{R}^d$, $K(\cdot,x)a\in V$

$$\langle K(\cdot,x)a,K(\cdot,y)b \rangle= a^TK(x,y)b$$

$V$ is the completion of finite linear combination of $K(\cdot,x)a$

$V$ is embedded in $L^2(\Omega,\mathbb{R}^d)$

RKHS

$V$ is the completion of finite linear combination of $K(\cdot,x)a$

$V$ is embedded in $L^2(\Omega,\mathbb{R}^d)$

vector $w\in L^2(\Omega,\mathbb{R}^d)$ gives covector

$v\mapsto\int_\Omega w(x)^Tv(x)dx$

velocity vectors $v_t$ are often smooth

momenta $m_t=Lv_t$ are often rough

e.g. Dirac deltas:

    $v_t=\sum_{i=1}^NK(\cdot,x_i)a_i$

    $m_t=\sum_{i=1}^Na_i\otimes\delta_{x_i}(\cdot)$  i.e. landmark momenta $p_i=a_i\otimes\delta_{x_i}(\cdot)$

Smooth vs. Rough

Geodesic Equation

The norm $\|\partial_t\phi_t\|_{\phi_t}^2$ determines a Riemannian metric (that's why we draw $G$ curved)

Riemannian geodesics, length/energy minimizing:

$$\phi_t=\mathrm{argmin}_{\phi_t}\int_0^T\|\partial_t\phi_t\|_{\phi_t}^2dt$$

$v_0=\partial_t\phi_t|_{t=0}$ determines evolution

$\mathrm{Exp}:V\to G$ endpoint map

$\mathrm{Log}:G\to V$ initial velocity map

Shape Geodesics

$\pi$ is a Riemannian submersion

it induces a Riemannian metric on $\mathcal{S}$

Geodesics on $\mathcal{S}$ lift to

    horizontal geodesics on $G$

Horizontal geodesics on $G$

    project to geodesics on $\mathcal{S}$

Those were the landmark geodesics

But we can do this for any shape space

Exp/Log maps are defined on $\mathcal{S}$ as well

Random Orbit

$V$ is a vector space (infinite dimensional)

We can perform regular statistics on $V$

Normal distribution $N(0,\Sigma)$ maps to distribution $\mathrm{Exp}_*N(0,\Sigma)$ on $G$ - random orbits

Data $s_1,\ldots,s_N$ can be analysed

in $V$: $v_i=\mathrm{Log}(s_i)$

Statistics

Template estimation (mean shape/image):

$$s_0=\mathrm{argmin}_{s\in\mathcal{S}}\sum_{i=1}^nd(s,s_i)^2$$

Geodesic distance on $\mathcal{S}$:

$$d(s,s_i)=\argmin_{\phi_t,\phi_T.s=s_i}\int_0^T\|v_t\|^2dt$$

Tangent space Principal Component Analysis:

$$\mathrm{PCA}(\mathrm{Log}(s_0,s_1),\ldots,\mathrm{Log}(s_0,s_n))$$

Stochastic Flows

e.g. Riemannian Brownian motion

Landmark Statistics in Theano

Exercise session:

• open notebook courses/shape19-E2.ipynb
• sample random initial momenta and forward integrate to create a distribution of shapes (replace example code that samples from a Riemannian Brownian motion)
• ​​find Frechet mean of the distribution
• perform PCA on landmark tangent space

hint: use a low number of landmarks to reduce computation time