A K-iterated scheme for the first-order Gaussian recursive filter with boundary conditions

Abstract

Abstract-Recursive Filters (RFs) are a well known way to approximate the Gaussian convolution and are intensively used in several research fields. When applied to signals with support in a finite domain, RFs can generate distortions and artifacts, mostly localized at the boundaries of the computed solution. To deal with this issue, heuristic and theoretical end conditions have been proposed in literature. However, these end conditions strategies do not consider the case in which a Gaussian RF is applied more than once, as often happens in several realistic applications. In this paper, we suggest a way to use the end conditions for such a K-iterated Gaussian RF and propose an algorithm that implements the described approach. Tests and numerical experiments show the benefit of the proposed scheme.

Error analysis for the first-order Gaussian recursive filter operator

“…For K = 1, the filter merely becomes an n-order Gaussian RF filter. Equations in (5) and (6) are conveniently referred to as the advancing and backing filters, respectively: when a Gaussian RF is implemented as an algorithm, the index j must be treated in increasing order in the former and decreasing in the latter [8]. The values α t and β are called smoothing coefficients and satisfy the constraint:…”

“…In the next section we will provide the structure of D and F . (7) and (8). If β = 1 − α and |α| < 1, then the output entries s j are given by the series:…”

“…More in particular, among RFs, the Gaussian RFs are very efficient implementations that approximate Gaussian-based convolutions. Gaussian RFs can be constructed in several ways but, in this work, we deal just with the kind derived by Deriche [7] and Young and van Vliet (see [8], and the references therein). As is well known, Gaussian RF based algorithms, applied to a signal with compact support, generate unbounded distortions in the output signal boundary entries (a detailed explanation is in [8]).…”

“…Gaussian RFs can be constructed in several ways but, in this work, we deal just with the kind derived by Deriche [7] and Young and van Vliet (see [8], and the references therein). As is well known, Gaussian RF based algorithms, applied to a signal with compact support, generate unbounded distortions in the output signal boundary entries (a detailed explanation is in [8]). This is known as edge effect and, to the aim of removing it, theoretical tools (named boundary conditions) and implementative improvements have been proposed in literature [8], [9].…”

“…As is well known, Gaussian RF based algorithms, applied to a signal with compact support, generate unbounded distortions in the output signal boundary entries (a detailed explanation is in [8]). This is known as edge effect and, to the aim of removing it, theoretical tools (named boundary conditions) and implementative improvements have been proposed in literature [8], [9]. Here we are not interested in edge effects and only focus on the error due to the the transition from the starting filter operator to the RF approximating it.…”