5.4 Bose-lertrattanapanich pixel interpolation

In data processing applications, it often occurs that certain points on a surface are known and are all samples of a common function (ex: an image). However, if these samples are too few for a particular purpose (ex: viewing an image in high resolution), then the question arises: how can the value of this function at other points be found? If there is no formula that, given (x,y) will return the function's value, then such a formula must be approximated. Interpolation is the generation of such an approximation.

The task of creating a high-resolution (HR) image from a set of low-resolution (LR) images requires interpolation. One method, Bose-Lertrattanapanich interpolation, is described below.

• Given a set of LR images and a registration algorithm, apply the registration algorithm to obtain a set of non-uniformly distributed points. These points represent the relative locations of all LR pixel values.
• Construct a Delaunay triangulation of the points.

  Delaunay triangulation example

• Estimate the gradient vector of image intensity (dz/dx,dz/dy) at each triangle vertex from the normal vectors of surrounding regions.

  Normal vector at a vertex

• Approximate the image intensity values (z) for each triangle patch by a continuous surface.

• For each point (x,y) on the HR grid, apply the appropriate polynomial to calculate the pixel value.

There are many simpler implementations of the last three steps of this process. The algorithm our code implements generates a constant function for each triangle patch, rather than a bivariate polynomial. In our implementation, the interpolated pixel value in each triangle patch (pT) is the average of the pixel values at the three vertices of each triangle (pA,pB,pC): pT = (pA + pB + pC)/3. This function is less accurate than the bivariate polynomial, but is more intuitive and cost-efficient to implement.