Abstract
Data Rectification by univariate filtering is popular for processes
that lack an accurate model. Linear filters are most popular for
on-line filtering. Unfortunately, these methods are single scale
in nature and are best for rectifying data containing features and noise
that are at the same resolution in time and frequency. Consequently,
for multiscale data, linear filters are forced to trade-off the extent
of noise removal with the accuracy of the features retained. In contrast,
nonlinear filtering methods such as, finite impulse response median
hybrid filters, and wavelet thresholding are multiscale in nature, but
they can not be used for on-line rectification. This paper presents a technique
for on-line nonlinear filtering based on wavelet thresholding. On-line
multiscale (OLMS) rectification applies wavelet thresholding to data in
a moving window of dyadic length to remove random errors. Gross errors
are removed by combining wavelet
thresholding with multiscale median filtering. Theoretical analysis
shows that OLMS rectification using Haar wavelets subsumes mean filters
of dyadic length, while rectification with smoother boundary corrected
wavelets is analogous to adaptive exponential smoothing. If the rectified
measurements are not needed on-line, the quality of rectification can be
further improved by averaging the rectified signals in each window. The
resulting approach overcomes the boundary effects encountered in translation
invariant (TI) rectification of Coifman and Donoho (1995), and is called
boundary corrected translation invariant (BCTI) rectification. Examples
based on synthetic and industrial data demonstrate the benefits of the
on-line multiscale and boundary corrected translation invariant rectification
methods.