Statistical Altas : [Roux, 2019] Frequence of apparition by voxels.

Anatomical Atlas : [Beg and Khan, 2005] Average shape.

Topological Altas : Collection of homeomorphisms covering a topological space (ex: manifold)

Definition

Normalisation of medical images is a important step for data acquisition

Before

linear registration :

After

Non Linear registration

Diffeomorphic matching  => Keeps the Topology

Constant Vector fields

• Dartel [Ashburner 2007]
• Demons [Vercauteren et al 2009] [Lorenzi et al 2013]

Temporal vector field

• LDDMM [Trouvé 1995] [Christensen et al. 1995] [Beg et al. 2005]

LDDMM

Comparing the amount of deformation in between two images.

LDDMM

Image $I$ deformed by a temporal vector field $v = (v_t)_{t\in [0,1]}.$

$$\dot I_t = v_t \cdot I_t$$

LDDMM

Energy of the deformation generated by $v$

$$\mathcal E(v) \doteq \frac12 \int_0 ^1 \|v_t \|_V^2dt = \frac12 \int_0 ^1 \langle v_t , K_\sigma \star v_t \rangle_2 dt$$

$K_\sigma \star$ being the convolution with a Gaussian kernel.

Distance between two images

$$d(I,J) = \mathrm{min}_{v} \mathcal E(v), \qquad \text{s.t. }I_0 = I;  I_1 = \int_0^1 v_t \cdot\nabla I_t dt = J$$

Geodesic := shortest path on a Manifold

To find the minimum of this cost :
$$E(v,I) = \frac12 \left\| I_1 - T \right\|_2^2 + \frac12\int_0^1 \|v_t\|_V^2  dt$$
We have to integrate over this set of geodesic equations :

$$\left\{\begin{array}{rl}v_t &= -\frac{\rho}{\mu} K_\sigma \star  (z_t \nabla I_t)\\ \partial_t z_t &= -\quad \nabla \cdot (z_t v_t)  \\ \partial_t I_t &= -\langle v_t , \nabla I_t\rangle\end{array}\right.$$

Advection equation

Continuity equation

LDDMM

Karcher mean : The average Atlas builder.

Given a group of image $I^1,\dots,I^n$ and the Riemannian metric $d$ we compute the template $I$ by minimising:

$$\mathcal M(I) = \frac{1}{2n} \sum_{k=1}^{n} d(I,I_1^k)^2$$