Interpolation can also be carried out in 2-D space. Given a set of sample points at 2-D points in either a regular grid or an irregular grid (scattered data points), we can construct an interpolating function that passes through all these sample points. Here we will first consider methods based only on regular grids and then those that also work for irregular grids.
Methods based on sample points in regular grid
Given a set of 2-D sample points in a regular grid, we can use the methods of bilinear and bicubic 2-D interpolation to obtain the value of the interpolating function at any point inside each of the rectangles in a 2-D grid with the four corners at , , , and . In the following, for convenience and without loss of generality, we only consider one of such rectangles with , and define , , , and .
First recall that at any point in the 1-D interval can be approximated by linear interpolation based on and :
This method of 1-D linear interpolation can be extended to the bilinear interpolation method to calculate the function value at any 2-D point with and based on the known sample values , , , and at the four corners of the rectangle in a 2-D grid. This is carried out in the following two steps:
As the final expression for the bilinear interpolation is symmetric with respect to and , the order of the two steps is irrelevant, i.e., if the interpolation is first carried out in y-direction and then in x-direction, the result is exactly the same.
The same set of 2-D sample points can be more smoothly approximated by a bicubic function in the following form:
Methods based on sample points in irregular grid
The methods discussed above require the data points to be available on a regular (rectangular) grid. They do not work if the data points are hileramdomly scattered in the 2-D space (irregular grid). We now discuss methods that work for both regular and irregular grids.
A radial basis function (RBF) is any function that is centrally symmetric with respect to a specific point , i.e., the value of the RBF at any 2-D point can be simply represented by , as a function of the distance between the point and . Typical RBFs include the Gaussian and Butterworth functions:
Based on a given RBF , we can construct an interpolating function as the weighted sum of such RBFs each centered around one of the given sample points :
In this method, the value of the interpolating function at any 2-D point is calculated as the weighted average of all available sample points:
In particular, at any sample point , we have
The weight function can be generalized to any RBF function centrally symmetric with respect to , such as the Gaussian or Butterworth fucntions considered above: