Approximation of manifold data

 
 


Consider a mapping on a manifold, that is a function defined on Euclidean domain (typically the real line) and with manifold values. For example, a trajectory on the sphere, as shown below.




Our problem is given samples of the mapping, to construct an approximation, such that

1 - The resulted approximation (which is itself a mapping) is, at least, continuous.

2 - The construction of the approximation is feasible.

3 - The approximation error can be bounded, based on the manifold properties and the function class which the mapping belong to.

4 - Additional properties of the function or its samples can be inherited by the approximation. For example, if the samples are matrices with common determinant, we may require to preserve this determinant on the entire approximation function.




 

The problem

Related papers


SPD interpolation

  1. •Subdivision Schemes for Positive Definite Matrices. In FOCM.


SPD approximation

  1. •Approximation schemes for functions of positive-definite matrix values.

In IMA numerical analysis.


Refinement of Manifold data

  1. •A global approach to the refinement of manifold data, in Mathematics of Computation.

  2. •Manifold-valued subdivision schemes based on geodesic inductive averaging, in Journal of Computational and Applied Mathematics.

Manifold APPROXIMATION ORDER

  1. •On preparation

MATRIX DATA APPROXIMATION based on decomposition theorems

  1. •Submitted


 

Our solution

The input data consists of SPD matrices, modeled as centered ellipsoids. These matrices are sampled from a smooth function, and given in the figure. The resulted interpolation, a smooth curve of SPD matrices, is presented in the animation.


Contrary to the development of classical approximation methods and numerical analysis methods for real-valued functions, the development in the case of manifold-valued functions, which is rather recent, was mainly concerned in its first stages with advanced numerical and approximation processes, such as geometric integration of ODE on manifolds, subdivision schemes on manifolds and wavelets-type approximation on manifolds.


Our main prototype (but definitely not the only one) is the class of subdivision schemes. These schemes are based on repeated refinement of the data, where the limit convergences to a continuous function. The main advantages of these schemes are their simple evaluation and their local nature (the result depends only on a local neighborhood). We use intrinsic averages of the manifold in order to adapt subdivision schemes and lift them from numbers to manifold data.

One interesting case is the manifold (or cone) of symmetric positive definite (SPD) matrices. This is a special case of Hadamard manifolds (or CAT-0 metric space). We use the geometric average for matrices, which describes the Riemannian geodesic curve on this manifold, to lift interpolatory schemes, such as the famous 4-point scheme and the Bernstein polynomials, to SPD matrices. Furthermore, we derive many unique properties. For example, preserving the Lowvner partial ordering of the data matrices to the approximation.

Some illustrations

In the animation: traveling over interpolation based on geodesics in the world of SPD matrices!

The univariate subdivision schemes can be applied in tensor fashion for bivariate samples over grids:


BIVARIATE EXAMPLE 1 - a smooth (synthetic) function. On the left, the data. On the right, one up-sampling to form “super-resolution”. The result is generated by tensor product 4-point scheme for SPD matrices.

BIVARIATE EXAMPLE 2 - the tensors field is taken from a real world application of D-MRI scan of a human brain. Samples are on the left, the up-sample (calculated higher resolution) appears on the right.