  In this paper, we present a residual neural network-based method for point
set registration that preserves the topological structure of the target point
set. Similar to coherent point drift (CPD), the registration (alignment)
problem is viewed as the movement of data points sampled from a target
distribution along a regularized displacement vector field. While the coherence
constraint in CPD is stated in terms of local motion coherence, the proposed
regularization term relies on a global smoothness constraint as a proxy for
preserving local topology. This makes CPD less flexible when the deformation is
locally rigid but globally non-rigid as in the case of multiple objects and
articulate pose registration. A Jacobian-based cost function and
geometric-aware statistical distances are proposed to mitigate these issues.
The latter allows for measuring misalignment between the target and the
reference. The justification for the k-Nearest Neighbour(kNN) graph
preservation of target data, when the Jacobian cost is used, is also provided.
Further, to tackle the registration of high-dimensional point sets, a constant
time stochastic approximation of the Jacobian cost is introduced. The proposed
method is illustrated on several 2-dimensional toy examples and tested on
high-dimensional flow Cytometry datasets where the task is to align two
distributions of cells whilst preserving the kNN-graph in order to preserve the
biological signal of the transformed data. The implementation of the proposed
approach is available at https://github.com/MuhammadSaeedBatikh/kNN-Res_Demo/
under the MIT license.
