Application to Glioblatomas
Anton François
suppervised by
Joan Glaunès & Pietro Gori
About Me :
2012-2015
2014-2016
2016-2019
2019-2022
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)
Normalisation of medical images is a important step for data acquisition
Before
linear registration :
After
Diffeomorphic matching => Keeps the Topology
Constant Vector fields
Temporal vector field
LDDMM
Comparing the amount of deformation in between two images.
Image III deformed by a temporal vector field v=(vt)t∈[0,1].v = (v_t)_{t\in [0,1]}.v=(vt)t∈[0,1].
I˙t=vt⋅It \dot I_t = v_t \cdot I_t I˙t=vt⋅It
Energy of the deformation generated by vvv
E(v)≐12∫01∥vt∥V2dt=12∫01⟨vt,Kσ⋆vt⟩2dt\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 E(v)≐21∫01∥vt∥V2dt=21∫01⟨vt,Kσ⋆vt⟩2dt
Kσ⋆K_\sigma \star Kσ⋆ being the convolution with a Gaussian kernel.
Distance between two images
d(I,J)=minvE(v),s.t. I0=I; I1=∫01vt⋅∇Itdt=Jd(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 = Jd(I,J)=minvE(v),s.t. I0=I; I1=∫01vt⋅∇Itdt=J
Geodesic := shortest path on a Manifold
To find the minimum of this cost : E(v,I)=12∥I1−T∥22+12∫01∥vt∥V2 dt E(v,I) = \frac12 \left\| I_1 - T \right\|_2^2 + \frac12\int_0^1 \|v_t\|_V^2 dt E(v,I)=21∥I1−T∥22+21∫01∥vt∥V2 dt We have to integrate over this set of geodesic equations :
{vt=−ρμKσ⋆ (zt∇It)∂tzt=−∇⋅(ztvt) ∂tIt=−⟨vt,∇It⟩\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.⎩⎨⎧vt∂tzt∂tIt=−μρKσ⋆ (zt∇It)=−∇⋅(ztvt) =−⟨vt,∇It⟩
Advection equation
Continuity equation
Given a group of image I1,…,InI^1,\dots,I^nI1,…,In and the Riemannian metric ddd we compute the template III by minimising:
M(I)=12n∑k=1nd(I,I1k)2\mathcal M(I) = \frac{1}{2n} \sum_{k=1}^{n} d(I,I_1^k)^2M(I)=2n1k=1∑nd(I,I1k)2
Metamorphosis := Diffeomorphic registration + Intensity changes
[Trouvé, 2005]
By Anton FRANCOIS
