In the previous article we considered how noise reduction filters may be useful in practical applications. In this article we walk through the mathematical and technical aspects of implementing the noise reduction filters used there. We conclude with final remarks comparing our regularization operators to the broader class of moving averages. The reader is assumed to be conversant in elementary calculus.
We present a family of discrete regularization operators, suitably adapted to remove noise from time series data. The operators are discrete analogues of their continuous mollifier counterparts, which are well-known in the literature on partial differential equations for producing smooth approximations of irregular functions. In this article we demonstrate the utility of the noise reduction filters on electrocardiographic data, after which a technical discussion is presented in part two of this two-part series.