Mastering Windows: Improving Reconstruction

Abstract

Ideal reconstruction filters, for function or arbitrary derivative reconstruction, have to be bounded in order to be practicable since they are infinite in their spatial extent. This can be accomplished by multiplying them with windowing functions. In this paper we discuss and assess the quality of commonly used windows and show that most of them are unsatisfactory in terms of numerical accuracy. The best performing windows are Blackman, Kaiser and Gaussian windows. The latter two are particularly useful since both have a parameter to control their shape, which, on the other hand, requires to find appropriate values for these parameters. We show how to derive optimal parameter values for Kaiser and Gaussian windows using a Taylor series expansion of the convolution sum. Optimal values for function and first derivative reconstruction for window widths of two, three, four and five are presented explicitly.