From Scattered Data Interpolation to Locally Regular Grid Approximation
Prof. Christophe Rabut, Centre de Mathématiques, INSA, Toulouse Cedex, France
On the opposite, the use of polyharmonic splines to approximate data on regular grids is quite easy, thanks to a particularly efficient basis (so-called polyharmonic B-splines), which is an extension of the polynomial B-splines in one dimension. But the regularity of the centres of the so-obtained spline is then an important drawback since in that case we need to use dense centres, even where the shape of the obtained function does not need it.
This is why we propose to use locally regular grids, i.e. grids which are regular on various parts, with a step depending on the considered parts. We do so in a progressive way, refining the grid in the places where the obtained function is too far from the data (and only in these parts) , obtaining so a fine (regular) grid where necessary, and a coarse (regular) grid where sufficient. Furthermore in such grids, we use a hierarchical B-spline basis, which means that B-splines with a small step live together with B-splines with larger step. Some examples show the efficiency of the method, which can also be used with any B-spline-type function, such as refinable functions or tensor product polynomial B-splines.
Prof. Christophe Rabut, Centre de Mathématiques, INSA, Toulouse Cedex, France
Seminar • 29 August 2005 • BM 5.202
AbstractWe briefly explain in this talk why polyharmonic splines are particularly well suited for multivariate scattered data interpolation. However these functions have an important drawback : the linear system associated to them is full and poorly conditioned, and the computation itself of a given function may be heavy and unstable; this is, probably, the main reason why polyharmonic splines are not more widely used.On the opposite, the use of polyharmonic splines to approximate data on regular grids is quite easy, thanks to a particularly efficient basis (so-called polyharmonic B-splines), which is an extension of the polynomial B-splines in one dimension. But the regularity of the centres of the so-obtained spline is then an important drawback since in that case we need to use dense centres, even where the shape of the obtained function does not need it.
This is why we propose to use locally regular grids, i.e. grids which are regular on various parts, with a step depending on the considered parts. We do so in a progressive way, refining the grid in the places where the obtained function is too far from the data (and only in these parts) , obtaining so a fine (regular) grid where necessary, and a coarse (regular) grid where sufficient. Furthermore in such grids, we use a hierarchical B-spline basis, which means that B-splines with a small step live together with B-splines with larger step. Some examples show the efficiency of the method, which can also be used with any B-spline-type function, such as refinable functions or tensor product polynomial B-splines.