The Shannon sampling theorem tells us that 2-D band-limited signals can be reconstructed from their values on a scaled integer lattice, and it provides a reconstruction formula in terms of the sinc function. In practice, one can not compute an infinite amount of data, so Shannon's formula must be truncated. The apodized sinc function has poor decay and it generalizes to 2-D in a separable manner. These properties yield, in turn, unsatisfactory image modeling and interpolation results.
As an alternative, one can define a reconstruction algorithm by using 2-D functions that are non-separable and have fast decay, for example the collection of Wendland functions. These functions have the following properties: 1)They are defined radially, taking a single value along circles about the origin; 2) Along a radial line, they are defined by piecewise polynomials; 3)They have compact support.
The goals of this project are to implement an image interpolation algorithm that utilizes the Wendland functions, to demonstrate their approximation properties, and to compare it with the separable polynomial B-spline model.
Prerequisites: a course on image processing, some coding experience in Matlab or ImageJ