Iterative reconstruction using a pyramid-shaped basis function

In iterative reconstruction, it is common to represent the continuous image as a finite linear combination of basis functions. Popular basis functions include the B-splines and the blobs. It turns out that the B-spline of order 0 corresponds to nearest-neighbour interpolation, and its parallel-beam projections are piecewise linear functions. Also, the B-spline of order 1 corresponds to bilinear interpolation and its parallel-beam projections are piecewise cubic polynomials. This observation brings the question of whether there exists an intermediate basis function with parallel-beam projections taking the form of a piecewise quadractic polynomial. Here, we discuss image reconstruction using a pyramid-shaped basis function that satisfies this condition. It turns out that this function performs as well as the B-splines of order 1 in terms of bias and noise. Unfortunately, reconstruction with this basis function requires a correction method to remove an inconvenient grid pattern.