An estimator for functional data with application to MRI

The authors propose a method for restoring the underlying true signal in noisy functional images. The Nadaraya-Watson (NW) estimator described in, e.g., G. S. Watson, "Smooth regression analysis," Sankhya Series A, vol. 26, p. 101-16 (1964) is a classical nonparametric estimator for this problem. Since the true scene in many applications contains abrupt changes between pixels of different types, a modification of the NW estimator is needed. In the data the authors study, the characteristics of each pixel are given as a function of time. This means that a curve of data points is observed at each pixel. Utilizing this time information, the NW weights can be modified to obtain a weighted average over pixels with the same true value. Theoretical results showing the estimator's properties are developed. Several parameters play an important role for the restoration result. Practical guidelines are given for how these parameters can be selected. Finally, the authors demonstrate how the method can be successfully applied both to artificial data and Magnetic Resonance Images.