Xavier Pennec


Hierarchical locally affine deformations (Polyaffine)

Non-linear geometric deformation often exhibit complex patterns that are difficult to analyse. In order to provide an intuitive analysis, we aim at parametrize deformation with a very small number of interpretable elements localized at suitable places (deformation atoms). In order to be efficient, these elements should be more powerful than the traditional translation used in displacement vector fields.

Polyrigid and Polyaffine transformations

In 2003 Arsigny proposed a general framework, coined polyaffine, to parametrize deformations with a small number of locally rigid or affine components, while guaranteeing the invertibility of global deformations. The basic idea is to compute the flow of the weighted average of rigid or affine velocity fields in different regions. It was revisited in 2006 to be written as an autonomous flow (thus in the stationary velocity field framework) which could benefit from the fast algorithms to implement the group exponential.

Polyaffine log-demons registration in the SVF framework

With Christof Seiler, we merged the log-demons and polyaffine frameworks to obtain a fast diffeomorphic registration algorithm parametrized by a small number of meaningful parameters. The regions could then be estimated hierarchically from the data themselves in a multi-scale tree, or they can be determined by a template like for the AHA regions in the heart.

Locally affine and sparse multiscale deformations in the LDDMM setting

With Stefan Sommer, we derived a similar idea in the framework of right invariant metrics on diffeomorphisms (the LDDMM setting). The multi-scale encoding of momenta led us to consider a kernel bundle with different kernel sizes(metrics) along the fibres. The evolution equation for the deformation resembles the EPDiff equation with an aggregated total momentum, but it has in our case a momentum conservation per scale and an aggregated total velocity. To encode locally affine deformations, we considered deformation atoms which are derivatives of the original kernel of the RKHS. These higher order momenta give a more complex EPDiff equation, but it can be used for forward geodesic shooting and its adjoint equation for backward gradient transport for optimization.

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Xavier Pennec