Quantitative assessment of linear noise-reduction filters for spectroscopy.

Long V Le,Young D Kim,David E Aspnes

doi:10.1364/oe.411768

Abstract

Linear noise-reduction filters used in spectroscopy must strike a balance between reducing noise and preserving lineshapes, the two conflicting requirements of interest. Here, we quantify this tradeoff by capitalizing on Parseval's Theorem to cast two measures of performance, mean-square error (MSE) and noise, into reciprocal- (Fourier-) space (RS). The resulting expressions are simpler and more informative than those based in direct- (spectral-) space (DS). These results provide quantitative insight not only into the effectiveness of different linear filters, but also information as to how they can be improved. Surprisingly, the rectangular ("ideal" or "brick wall") filter is found to be nearly optimal, a consequence of eliminating distortion in low-order Fourier coefficients where the major fraction of spectral information is contained. Using the information provided by the RS version of MSE, we develop a version that is demonstrably superior to the brick-wall and also the Gauss-Hermite filter, its former nearest competitor.

Highlights

Reducing noise in spectra, optical or otherwise, is easy; reducing noise without compromising information is not
We find that similar advantages result if the measures that assess the filters – mean – square error and noise – are considered in reciprocal- (Fourier-) space (RS)
The resulting perspective leads to insights that cannot be achieved from direct- (spectral) space (DS) considerations alone, and opens up additional possibilities in linear filtering

Summary

Introduction

Optical or otherwise, is easy; reducing noise without compromising information is not. Efforts have been directed nearly universally to compromises such as the binary [12], Savitzky-Golay (SG) [15], and above-mentioned GH [7] filters, to name a few These approximate the ideal filter via the Butterworth approach [25], i.e., eliminate as many derivatives as practical in a Taylor-series expansion of the transfer function about the lowest-index coefficient C0 , followed by a rolloff to zero near the white-noise cutoff Cnc. The results presented here illustrate the problem with these filters. The transfer functions mentioned above drop below 1 well before nc is reached, thereby compromising information This is damaging because data coefficients tend to decrease either exponentially or as Gaussians up to the white-noise cutoff. Our purposes here are to develop the theory and to provide examples

Theory

Gauss-Hermite and cosine-terminated filters

Findings

Discussion

Conclusion