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

Shapes

Objects on which the diffeomorphism group acts

$d=2,3$

• landmarks in $\mathbb{R}^d$: $\mathbf{q}=(q_1,\ldots,q_n)$
• curves: $\gamma:\mathbb{S}^1\to\mathbb{R}^d$
• surfaces: $\gamma:\mathbb{S}^2\to\mathbb{R}^d$
• images: $I:\Omega\to\mathbb{R}$, $\Omega\subset\mathbb{R}^d$
• tensor fields: $\Omega\to\mathcal{T}^k_l$, $\Omega\subset\mathbb{R}^d$
• ...

One framework - multiple shape types

Image Matching

matching problem, image registration:

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

Dissimilarity measure:

     $S(I^0,I^1)=\int_\Omega|I^0(x)-I^1(x)|^2dx$

     or mutual information or similar

$I^0$ moving image: $\phi_t.I^0$            $I^1$ fixed image

landmarks can be matched precisely, images generally cannot: lack of transitivity of $G$-action

Image Matching

matching problem, image registration:

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

Optimal flows $\phi_t$ for $E$ with

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

are geodesics on $G=\mathrm{Diff}(\Omega)$

Geodesic equation:

Euler-Poincaré reduction

General form of geodesic equation:

$$\frac{d}{dt} \frac{\delta l}{\delta v}+\mathrm{ad}^*_v \frac{\delta l}{\delta v}=0$$

Valid for all shape space (only shape action varies)

Uses right invariance of metric on $G$: 

$\|\partial_t\phi_t\|_{\phi_t}=\|v_t\|$, $v_t=\partial_t\phi_t\circ\phi_t^{-1}$

The (infinite dimensional) variable $\phi_t$ is reduced

Lagragian to Eulerian coordinate change in fluid systems

$$\frac{\partial m}{\partial t} + v \cdot \nabla m + \nabla v^T \cdot m + m \nabla \cdot v = 0$$

Computation

Path straigthening: Gradient descent on $v_t$

$$v_t^{k+1}=v_t^k-\epsilon\nabla_{v_t}E(v_t)|_{v_t=v_t^k}$$

swipe back and forth between t=0 and t=T

Shooting: use that geodesic path is determined by $v_0$

$$v_0^{k+1}=v_t^k-\epsilon\nabla_{v_0}E(v_0)|_{v_0=v_0^k}$$

use adjoint equations for transporting

$\nabla_{\phi_T} S(\phi_T.s^0,s^1)$ to $\nabla_{v_0}S(\phi_T.s^0,s^1)$

(backpropagation)

Both schemes are numerically challenging

Curves and Surfaces

curves: $\gamma:\mathbb{S}^1\to\mathbb{R}^d$,           surfaces: $\gamma:\mathbb{S}^2\to\mathbb{R}^d$

Similarity $S(\gamma^0,\gamma^1)=\int_{\mathbb{S}^1}\|\gamma^0(\theta)-\gamma^1(\theta)\|^2d\theta$ is dependent on parametrizations of $\gamma^0$ and $\gamma^1$

Current metric provides invariant metric on curves and surfaces:

    RKHS on $\Omega$ with kernel $G$

    curve $\gamma$ determines a covector in $V^*$:

            $v\mapsto\int_\Omega v(x)^T\gamma(x)dx$

    currents norm is $\|\gamma\|$ using the RKHS norm on $V^*$

discretized: $l\approx \sum_{i=1}^n\delta_{l(t_i)}^{\dot{l}(t_i)}$ gives 

Deformetrica http://www.deformetrica.org/

Inner metrics on curves

outer metrics: measure deformations of $\Omega$

inner metrics: measure deformations of $\gamma$ itself

• L2: $g(v,w)=\int_{\mathbb{S}^1}v(\theta)^Tw(\theta)ds$
• Sobolev-type: $g(v,w)=\int_{\mathbb{S}^1}(L_\gamma v(\theta))^Tw(\theta)ds$

e.g. $L_\gamma=I_d-D_s$ or higher order

Elastic: $g^{a,b}(h,k)=\int_{\mathbb{S}^1}a^2|(D_s h)^{\perp}|\:|(D_s k)^{\perp}|+b^2|(D_s h)^{\top}|\:|(D_s k)^{\top}|ds$

Parametrization invariance:

    $L_{\gamma\circ\phi}h\circ\phi=(L_\gamma h)\circ \phi$ implies metric on $\mathrm{Emb}(\mathbb{S}^1,\mathbb{R}^2)/\mathrm{Diff}(\mathbb{S}^1)$

h2metrics: https://github.com/h2metrics/h2metrics

Other shape models

• Kendall's shape space
       points modulo rotation, translation, scale
• Point distribution models (PDMs): $s\in\mathbb{R}^{dn}$
       linear
• Elastic shape models
       inner metric
• Level set based models
• Hausdorff distances of sets
• ... and many more

Curve and surface matching

Exercise session:

• curve matching, Fréchet mean and PCA in h2metrics (MATLAB)
  • run 'example.m' from h2metrics
  • compute Fréchet mean of a subset (or all) of the OASIS corpus callosum data (folder 'examples/data')
  • make PCA plot
• shape analysis in Deformetrica
  • run some examples from https://gitlab.com/icm-institute/aramislab/deformetrica/wikis/home e.g.
    • first example from deformetrica/wikis/2_tutorials/2.1_registration
    • registration example in examples/registration/landmark/3d/bundles_cortico/
  • visualize results in e.g. ParaView https://www.paraview.org/