Pixel Interpolation

Summary: One method for interpolating the pixel values of several low-resolution (LR) images to create a single high-resolution (HR) image is to approximate the patches of a Delaunay triangulation with bivariate polynomials.

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.

1. 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.
2. Construct a Delaunay triangulation of the points.

   Figure 1: MATLAB Delaunay triangulation of registered points from a set of LR images.

   Delaunay Triangulation Example

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

   Figure 2: The gradient vector at a vertex (dz/dx,dz/dy) can be estimated from the normal vector at the vertex: n = [nx,ny,nz]. n is calculated by summing the normal vectors of the surrounding triangle patches weighted by their areas, then dividing this sum by the total area of these triangles. The result is dz/dx = -nx/nz, dz/dy = -ny/nz. (Source: 2)

   Normal Vector at a Vertex

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

   Figure 3: Bivariate polynomial to model surface; c values are based on the gradient vectors. (Source: 2)

5. 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.