Remarks on M-type Smoothers with Edge Preserving Properties

(1998) have formulated an important and practical problem: How to smooth the noise out of image data while at the same time preserving unsmooth features such as jumps, spikes and edges? They formalize the problem as the estimation of a mostly smooth regression function function, m(x), based on observations Y i that decompose as follows: By assuming m satisses standard nonparametric smoothness assumptions except at an unspeciied number of jump discontinuities (edges), they provide a technical formulation of the problem that points the way toward possible solutions. The standard Nadarya-Watson estimator blurs the edges, so the authors instead use redescending M smoothers to resolve jumps and spikes that exceed the noise level. It is an appealing direct attack on the problem. Moreover, their asymptotic analysis provides an excellent start on a general theory for edge preserving smoothers. We expect this paper to stimulate more research in the area. It raises many issues and provides many practical insights. M estimation has a long history in the statistical literature, with the traditional focus being on the accommodation of outliers. Translated to the present context outliers appear as spikes and jumps in the data. If the spikes and jumps are large relative to the noise level indicated by the bulk of the data, then the M smoother \passes" them provided it is tuned properly. The M-smoother treats any outlying observation as a jump, even if it is an isolated point. This seems reasonable, because in the absence of modeling assumptions, outliers and spikes are indistinguishable. In this regard the present paper diiers from earlier work on robust smoothing by HH ardle and Gasser (1984), which focused on smoothers that limit the innuence of spikes by smoothing them over. The redescending property allows the estimator to resolve jumps that exceed the noise level. It also leads to computational challenges, which the authors address in considerable detail using a monitored quasi-newton algorithm. As an alternative we have implemented an iterative weighted least squares (IWLS) algorithm, which is straightforward to program. Although the IWLS algorithm could potentially nd the wrong zeroes of the estimating equation, in practice we have not found this to be a problem if one uses the data as starting values, and if one follows the authors' general recommendations concerning the smoothing parameters g and h.