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 parameterize deformation with a very small number of interpretable elements localized at suitable places (deformation atoms). In order to be efficients, 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 parameterize deformations with a small number of rigid or affine components, while guaranteeing the invertibility of global deformations. The basic idea is to compute the flow the 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 stationnary 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 parameterized by a small number of meaningful parameters. The regions could then be estimated hierarchically from the data themselves in a multiscale tree, or be fixed to a template for instance to matche the AHA regions in the heart.
- C Seiler, X Pennec, M Reyes. Capturing the multiscale anatomical shape variability with polyaffine transformation trees. Medical image analysis 16 (7), 1371-1384, 2012. [PDF].
A preliminary version (Geometry-aware multiscale image registration via OBBTree-based polyaffine log-demons) presented at MICCAI 2011 got the Young investigated award.
- K McLeod, M Sermesant, P Beerbaum, X Pennec. Spatio-temporal tensor decomposition of a polyaffine motion model for a better analysis of pathological left ventricular dynamics. IEEE transactions on medical imaging 34 (7), 1562-1575, 2015 [PDF].
Related preliminary works were presented at
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 multiscale encoding of momenta led us to consider a kernel bundle with different kernel sizes(metrics) along the fibers. The evolution equation for the deformation is similar to the EPDiff equation with an agregated total mo, but with a momentum conservation per scale and an agregated 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.
- S Sommer, F Lauze, M Nielsen, X Pennec.
Sparse multi-scale diffeomorphic registration: the kernel bundle framework.
Journal of mathematical imaging and vision 46 (3), 292-308, 2013. [PDF]
Preliminary versions of this work were presented at
IPMI 2011 and
Keywords: Sparse multiscale momentum in a bundle of metric kernels for LDDMM.
- S.Sommer, M. Nielsen, S. Darkner, X. Pennec Higher-order momentum distributions and locally affine LDDMM registration. SIAM Journal on Imaging Sciences 6 (1), 341-367, 2013. [PDF].
Keywords:Locally affine LDDMM, first order deformation atoms.