Discussion
The main advantage of polyharmonic spline interpolation is that usually very good interpolation results are obtained for scattered data without performing any "tuning", so automatic interpolation is feasible. This is not the case for other radial basis functions. For example, the Gaussian function needs to be tuned, so that k is selected according to the underlying grid of the independent variables. If this grid is non-uniform, a proper selection of k to achieve a good interpolation result is difficult or impossible.
Main disadvantages are:
- To determine the weights, a linear system of equations must be solved, which is non-sparse. The solution of a non-sparse linear system becomes no longer practical if the dimension n is larger as about 1000 (since the storage requirements are O(n2) and the number of operations to solve the linear system is O(n3). For example n=10000 requires about 100 Mbyte of storage and 1000 Gflops of operations).
- To perform the interpolation of M data points requires operations in the order of O(M*N). In many applications, like image processing, M is much larger than N, and if both numbers are large, this is no longer practical.
Recently, methods have been developed to overcome the aforementioned difficulties. For example Beatson et.al. present a method to interpolate polyharmonic splines at one point in 3 dimensions in O(log(N)) instead of O(N).
Read more about this topic: Polyharmonic Spline
Famous quotes containing the word discussion:
“Americans, unhappily, have the most remarkable ability to alchemize all bitter truths into an innocuous but piquant confection and to transform their moral contradictions, or public discussion of such contradictions, into a proud decoration, such as are given for heroism on the battle field.”
—James Baldwin (1924–1987)
“It was heady stuff, recognizing ourselves as an oppressed class, but the level of discussion was poor. We explained systemic discrimination, and men looked prettily confused and said: “But, I like women.””
—Jane O’Reilly, U.S. feminist and humorist. The Girl I Left Behind, ch. 2 (1980)
“This is certainly not the place for a discourse about what festivals are for. Discussions on this theme were plentiful during that phase of preparation and on the whole were fruitless. My experience is that discussion is fruitless. What sets forth and demonstrates is the sight of events in action, is living through these events and understanding them.”
—Doris Lessing (b. 1919)