Denoising scheme based on singular-value decomposition for one-dimensional spectra and its application in precision storage-ring mass spectrometry

This work concerns noise reduction for one-dimensional spectra in the case that the signal is corrupted by an additive white noise. The proposed method starts with mapping the noisy spectrum to a partial circulant matrix. In virtue of singular-value decomposition of the matrix, components belonging to the signal are determined by inspecting the total variations of left singular vectors. Afterwards, a smoothed spectrum is reconstructed from the low-rank approximation of the matrix consisting of the signal components only. The denoising effect of the proposed method is shown to be highly competitive among other existing nonparametric methods, including moving average, wavelet shrinkage, and total variation. Furthermore, its applicable scenarios in precision storage-ring mass spectrometry are demonstrated to be rather diverse and appealing.

Figure 1

Illustration of different denoising effects with two matrix-embedding approaches. (a) The green line shows the denoised result with a Hankel matrix, while the blue line corresponds to a partial circulant matrix. The orange line is the ground truth. (b) Deviations of the two results with respect to the ground truth.

Figure 2

Typical examples of SVD components in the case of noise-corrupted signal (left), sole signal (center), and sole noise (right). (a1) Full spectra of singular values in a nonincreasing order, where the blue crosses belong to the signal and the green dots belong to the noise. (a2) similar to (a1), but for sole signal; (a3) similar to (a1), but for sole noise. (b1) Zoomed spectra focusing on the largest six singular values. (b2) similar to (b1), but for sole signal; (b3) similar to (b1), but for sole noise. (c1) The corresponding left singular vectors. (c2) similar to (c1), but for sole signal; (c3) similar to (c1), but for sole noise.

Figure 3

Illustration of the ringing artifact in the smoothed data. The blue line is obtained with linear detrending, whereas the green line without.

Figure 4

Illustration of the smoothing effect on the noise-corrupted periodic signals with various SNRs. (a) The noise obeys a uniform distribution. The orange line is the ground truth, while the blue line is the denoised approximation. (b) Same as (a), except the noise is normally distributed. (c) The resultant NRMSDs obtained in different cases.

Figure 5

Same as described in the caption of Fig. 4, but for an asymmetric peak.

Figure 6

Same as described in the caption of Fig. 4, but for a Gaussian peak (left) and a Lorentzian peak (right) on a ramp.

Figure 7

Same as described in the caption of Fig. 4, but for the amplitude of a Duffing oscillator.

Figure 8

A zoomed pulse recorded by a time-of-flight detector, which is denoised by (a) the proposed method in this work, (b) moving average, (c) wavelet shrinkage, and (d) total variation. The crosses on the leading edge mark the $90\%$ to $10\%$ interval used to calculate the fall time. See Sec. 4a for the detail.

Figure 9

Background estimation of a Schottky spectrum with a sharp peak on top of it. The blue line is a direct fitting of a Lorentzian function, whereas the red line is obtained by the proposed method in this work. The green line acts like a reference.

Figure 10

Weak peak detection in a Schottky spectrum by the proposed method in this work. (a) The red line is the denoised spectrum, and the green line, which is obtained by averaging 100 similar spectra, provides a reference. All local maxima in the red spectrum are marked with blue crosses, which also respectively indicate their prominences (vertical extent) and widths (horizontal extent). The detected peaks are labeled, in a descending order of PWR, as 1, 2, and 3. (b) The PWR of all eight peak candidates, among which the largest three are considered as real peaks associated with particles.