# Approximation on the 3d ball

## Approximation on the 3d ball

Typical examples of applications of approximation methods on the 3d ball are tomographic problems in the geosciences and in medical imaging. Since in these contexts structures occur that have typical layers which are approximately spherical shells, the use of Euclidean approximation methods in the 3-dimensional space appears not to be the best choice. Instead, the further development of spherical methods to the inner space of the sphere promises to yield a better performance for such applications. The Geomathematics Group Siegen has developed several approximation methods for functions on the 3d ball. Among them are wavelet-based multiscale methods and spline methods (also spline multiresolution methods). The wavelet methods enable a zooming-in into parts of the ball, where details at different resolutions can be extracted and analysed. The spline methods yield interpolating or approximating functions that have certain minimal-smoothness and best approximation properties (1st and 2nd minimum property). Moreover, they can also be used for heterogeneous data, i.e. for data from different sources (e.g. gravitational and seismic data) and for data from different domains (e.g. from satellites at different orbit heights).

There is still the need for further research. One of the main reasons are the high numerical costs that are connected to such methods. Both the wavelet and the spline method require the evaluation of kernels which are represented by double series involving several Jacobi polynomials. Moreover, due to the curse of dimensionality, the calculation of solutions at a high resolution requires the use of huge data grids in numerical implementations. First results concerning an acceleration of the methods were already obtained by the group e.g. by construction special kernels that have a high localization on the ball. Further improvements - also with respect to algorithmic aspects - are currently researched.

References:

- V. Michel:
*Wavelets on the 3-dimensional ball*, Proceedings in Applied Mathematics and Mechanics (PAMM), 5 (2005), 775-776. - V. Michel:
*Tomography - problems and multiscale solutions*, in: Handbook of Geomathematics (W. Freeden, M.Z. Nashed, and T. Sonar, eds.), 2nd edition (completely revised version of the paper in the 1st edition), Springer, Berlin, Heidelberg, 2015, pp. 2087-2119. - V. Michel: Lectures on Constructive Approximation - Fourier, Spline, and Wavelet Methods on the Real Line, the Sphere, and the Ball, textbook, 326 pages, Birkhäuser Verlag, Boston, 2013.