Geometric Discretization in Shape analysis

Licentiatavhandling, 2022

Discretizations in shape analysis is the main theme of this licentiate thesis, which comprises two papers. 
The first paper considers  the problem of finding a parameterized time-dependent vector field that warps an initial set of points to a target set of points.
 The parametrization introduces a restriction on the number of available vector fields.
  It is shown that this changes the geometric setting of the matching problem and  equations of motion in this new setting are derived. 
  Computational algorithms are provided, together with numerical examples that emphasize the practical importance of regularization. 
Further, the modified problem is shown to have connections with residual neural networks, meaning that it is possible to study neural networks in terms of shape analysis. 
The second paper concerns a class of spherical partial differential equations, commonly found in mathematical physics, that describe the evolution of a time-dependent vector field. 
The flow of the vector field generates a diffeomorphism, for which a discretization method based on quantization theory is derived. 
The discretization method is geometric in the sense that it preserves the underlying Lie--Poisson structure of the original equations. 
Numerical examples are provided and potential use cases of the discretization method are discussed, ranging from compressible flows to shape matching.