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. Author manuscript; available in PMC: 2026 Jan 1.
Published in final edited form as: Magn Reson Med. 2024 Aug 18;93(1):8–17. doi: 10.1002/mrm.30252

Simultaneous frequency and phase corrections of single-shot MRS data using cross-correlation

Dinesh K Deelchand 1
PMCID: PMC11518653  NIHMSID: NIHMS2014857  PMID: 39155397

Abstract

Purpose:

The objective of this study was to propose a novel preprocessing approach to simultaneously correct for the frequency and phase drifts in MRS data using cross-correlation technique.

Methods:

The performance of the proposed method was first investigated at different SNR levels using simulation. Random frequency and phase offsets were added to a previously acquired STEAM human data at 7T, simulating two different noise levels with and without baseline artifacts. Alongside the proposed spectral cross-correlation (SC) method, three other simultaneous alignment approaches were evaluated. Validation was performed on human brain data at 3T and mouse brain data at 16.4T.

Results:

The results showed that the SC technique effectively corrects for both small and large frequency and phase drifts, even at low SNR levels. Furthermore, the mean square measurement error of the SC algorithm was comparable to the other three methods used, with much faster processing time. The efficacy of the proposed technique was successfully demonstrated in both human brain MRS data and in a noisy MRS dataset acquired from a small volume-of-interest in the mouse brain.

Conclusion:

The study demonstrated the availability of a fast and robust technique which accurately corrects for both small and large frequency and phase shifts in MRS.

Keywords: analysis, preprocessing, brain, MR spectroscopy

Introduction

Magnetic resonance spectroscopy (MRS) is a powerful tool for studying the biochemical composition of tissues noninvasively. With proton MRS in the brain, at least five metabolites can be reliably measured at 3T and beyond and these include total N-acetylaspartate (NAA), total creatine (tCr, creatine plus phosphocreatine), total choline (tCho), myo-inositol and glutamate (1). To have adequate signal-to-noise ratio (SNR) of these low concentration metabolites (10 μmol/g or lower) from the volume-of-interest (VOI), repeated scans or transients are constantly employed during MRS data acquisition (2). However, various imperfections are present during measurements which could affect the frequency and phase of the spectroscopic data. These artifacts may originate from physiological motion such as breathing, deliberate or unintentional subject movements, insufficient water suppression, poor MRS sequence localization (3), system instability or drift resulting from induced gradient heating from demanding sequences such as diffusion and functional MRI which preceded the MRS acquisitions (4,5). Therefore, preprocessing of MRS data is an important step to remove some of these artifacts before quantification, as recently documented in the experts’ MRS consensus papers (6,7).

For this purpose, single-shot MRS data needs to be available or raw coil-uncombined data can be utilized. The preprocessing steps may include coil-combination followed by frequency, phase, and eddy current corrections, in addition to discarding unusable spectra due to large water residual or lipid signals. These result in enhanced spectral quality, i.e. increase in SNR of the data and improvement in the spectra linewidth (8). All these lead to reliable estimation of metabolites concentrations, whether relative or absolute, and provide better quantification precision, as reflected in lower Cramer-Rao lower bounds.

Various methods are currently available for the retrospective correction of frequency and phase drifts, either individually or simultaneously, in both the time and frequency domains.

These include techniques such as constructive averaging (9), correlation (10,11), least-square optimization (12–16), principal component analysis (17), and using residual water signal information (18). Most of these approaches utilize the metabolite info to align each transient to a reference spectrum for the estimation and correction of frequency and/or phase offsets. The reference spectrum is typically either the first transient or the average of all transients. More recently, deep-learning approaches (19–22) have been introduced to correct both frequency and phase.

The purpose of the current study was to propose another approach for simultaneous frequency and phase correction using cross-correlation technique. To my knowledge, only one study (11) has reported using cross-correlation to estimate frequency only in edited MRS. The proposed technique was termed spectral cross-correlation (SC). This novel approach was first evaluated for accuracy as a function of SNR. It was then compared to current three simultaneous frequency and phase correction methods using in vivo measured STEAM 7T data, which was manipulated to model frequency and phase instabilities. The SC approach was also successfully demonstrated on human and mouse brain proton MRS data.

Methods

Spectral cross-correlation technique

The proposed simultaneous frequency and phase alignment technique was based on cross-correlation which compares the similarities between two MRS spectra. Let’s consider Sn as the complex spectrum at transient n during the MRS measurement and SRef as the reference spectrum to which all spectra will be aligned. SRef can either be the first transient or the average of all transients in the scan. The first transient will be used by default unless specified otherwise in the text. First, the cross-correlation reference signal (CRef) was calculated by cross-correlating the complex spectrum SRef with itself over a user-defined chemical shift range. Then the cross-correlation signal between SRef and Sn was computed at each transient (Cn) using the same chemical shift range used previously.

The frequency shift was determined by comparing the magnitude values of the two cross-correlation signals (CRef and Cn), and calculating the difference between their maximum amplitude. On the other hand, phase offset was determined from the phase information of the complex CRef and Cn signals. It was calculated as the difference in mean phase over a few points (i.e. 11 data points) centered on the maximum magnitude signals. Note that only one cross-correlation signal was required per transient to calculate both frequency and phase offsets. To enhance the accuracy of shifts computation, all spectra were zero-filled by a factor of 10. Figure 1 demonstrates the principle of cross-correlation approach on two singlets. The SC algorithm was implemented in MATLAB (R2019a, MathWorks, MA). It uses the cross-correlation function xcorr, where the maximum shift range (maxLag as defined in the spectralXCorr.m script) for cross-correlation covers the entire spectral width. This MATLAB script and a sample dataset (in MATLAB and NIfTI formats) are available on GitHub (https://github.com/Anesh20/spectralXCorr/).

Figure 1:

Figure 1:

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Schematic of the cross-correlation approach to simultaneously determine frequency and phase information based on two single peaks. A) Consider two transients where singlet S2 is shift by 5 Hz with a phase difference of 20° compared to the reference singlet SRef. B) Complex spectrum SRef is cross-correlated with itself to generate one reference cross-correlation signal (CRef in blue). To determine the frequency and phase, the complex spectra SRef and S2 are cross-correlated (C2 in magenta). The difference between the absolute maximum values of the two cross-correlation signals gives the frequency shift (ΔF). The phase (ΔP) is calculated from the difference in mean phase of the cross-correlation signals.

Evaluation of the SC algorithm as a function of SNR levels

The measurement error of the SC approach was first investigated with respect to different SNR levels and number of points used to determine the phase information. A noiseless spectrum, consisting of four singlets representing NAA, tCr-CH3, tCho, and tCr-CH2, was simulated at 7 T (2048 complex points, spectral width = 6 kHz, linewidth of 12 Hz, Figure S1). Random frequency shifts between 0 and 25 Hz and phase offsets between ±40 degrees were then added to the spectrum to generate 101 spectra where the first transient was the spectrum without any imposed shifts. For each spectrum, Gaussian white noise was added with a known SNR level. This procedure was repeated 100 times, each with a different noise realization at the same SNR level. Twelve different SNR levels (2.5, 7, 7.5, and from 10 to 50 in steps of 5) were chosen, yielding 1200 spectra for each shifted spectrum. Note that the first transient served as the reference spectrum, without any frequency or phase offsets.

All the spectra generated above were aligned using the SC method, with the number of points from 1 to 21 in steps of 2, to assess the impact of the point number on phase information calculation. A limited chemical shift range of 1.8 to 3.6 ppm was utilized for alignment. The measurement error was reported based on the mean absolute error (MAE) compared to the ground truth values for frequency and phase offsets.

Evaluation of SC performance using manipulated in vivo data

To validate the proposed approach, a previously acquired STEAM 7T MRS data (8 mL VOI, TE/TM=8/32 ms, 64 transients, 2048 complex points, spectral width of 6 kHz) from the occipital cortex was used (23). This dataset did not contain any baseline distortion resulting from lipids or water residual signals. First, the single-shot data was preprocessed using MRspa (24) in MATLAB to remove any frequency and phase artifacts and eddy current corrected. Then, random frequencies drifts ranging from 0 to 25 Hz and random phase shifts between ±40° were added to each transient to generate a new dataset (still with 64 transients) with known offsets, except for the first transient, which was used as the reference spectrum. The 25 Hz frequency shift was selected based on the worst-case scenario from various sources: breathing can cause shifts of up to 4 Hz (11,25), small motions up to 10 Hz (26,27), and scanner drift following an fMRI study around 11 Hz (28). A larger range of phase possibilities (12) was tested even though this kind of phase instability is rare in MRS data.

The mean signal-to-noise ratio (SNR) of each transient was 35.4±3.5 where SNR is defined as the ratio of the NAA singlet amplitude to the root-mean-square noise measured between −2 to −4 ppm (without any apodization), unless otherwise specified. The SNR of this high SNR dataset was intentionally reduced by adding random Gaussian noise, resulting in a mean single-shot SNR of NAA of 7.4±1.2, denoted as low SNR dataset.

Baseline distortion artifacts, typically resulting from residual unsuppressed water signal, were also simulated. Briefly, a singlet peak was simulated at 4.2 ppm with a linewidth of 14 Hz to represent the residual water signal and this peak was scaled to be 10 times higher than the NAA singlet. For each transient, random phase variation was added to the water peak to reflect poor water suppression. This baseline distortion was added to both the low and high SNR STEAM datasets.

All the above STEAM datasets underwent simultaneous frequency and phase estimation using the proposed SC approach. A chemical shift range of 1.8 to 3.6 ppm was utilized for the spectra without baseline issue, while the frequency range was centered on NAA singlet peak, i.e. between 1.8 to 2.2 ppm for dataset with baseline issues. No other apodization was applied to the data.

Evaluation against alternative methodologies

The effectiveness of the SC method was assessed by comparing it to three alternative simultaneous frequency and phase correction approaches: an iterative correlation (IC) method (10), spectral registration (SR) (13) based on least-square optimization and Robust Alignment to a Target Spectrum (RATS) (12) which employs variable-projection in the frequency domain.

The IC method was implemented in MATLAB where the normalized scalar product between two spectra were maximized. Frequency shifts were tested within a range of ±50 Hz with a step of 2 Hz, while the phase offsets ranged between ±50° with a step of 2.5°. To maintain consistency, the frequency and phase estimation were conducted in the limited frequency domain, utilizing a shift range of 1.8 to 3.6 ppm for STEAM dataset without baseline artifacts and between 1.8 to 2.2 ppm for dataset with baseline issues, consistent with the approach used for SC.

The SR algorithm available in the FID-A package (29) in MATLAB was also utilized for comparison. Specifically, the frequency domain implementation of the SR algorithm was employed with a chemical shift range of 1.8 to 3.6 ppm for STEAM data without baseline issue and a range of 1.8 to 2.2 ppm for spectra with baseline issues. The script was called using op_alignAverages_fd(input,1.8, 3.6, 0.3, ‘r’, ref) where input represents all the transients in the scan, ref is the first transient in the scan and 0.3 is the time used for alignment.

RATS, freely available in the spant package in R, was also used to correct the STEAM data, which was initially converting into NIFTI format. RATS script was called using mrs_corr <- rats(input, ret_corr_only = FALSE, xlim = c(3.6, 1.8), ref = ref, max_shift = 50) where mrs_corr is the corrected spectra, input represents all the transients in the scan, ret_corr_only indicated whether to output the estimated frequency and phase offsets, xlim is the chemical shift range to perform the alignment, ref is the first transient in the scan used as reference and max_shift represents the maximum allowed frequency shift.

These three simultaneous alignment methods were tested on the low and high SNR STEAM spectra, both with and without baseline issues. The performance of IC, SC, SR, and RATS was assessed based on the mean absolute error (MAE) between frequency and phase estimation and the ground truth values across datasets. Additionally, one-way ANOVA with post hoc Tukey HSD pairwise comparison (P < 0.01) was employed to compare the MAE between the methods.

Validation of SC method

To demonstrate the effectiveness of the proposed technique, previously measured in vivo datasets were processed without any manipulation using SC, IC, SR and RATS techniques. Two MRS data were utilized: semi-LASER spectra (8 mL, TE/TR=28/3000ms, 64 transients) acquired from the posterior-cingulate cortex (PCC) at 3T (30), and LASER spectra (6 μL, TE/TR=12.3/5000ms, 128 transients) obtained from the mouse striatum at 16.4 T (31). Both datasets were acquired following FASTMAP B0 shimming in the MRS voxel and B1-field calibration. Water suppression was achieved using VAPOR with interleaved OVS pulses in semi-LASER and VAPOR in LASER.

Results

Figure 2 shows the frequency and phase MAE at various SNR levels ranging from 2.5 to 50, where the number of points used for phase calculation was 11. At low SNR (≤ 5), the estimated values were less reliable, with a standard deviation greater than 1.5 Hz in frequency offsets and greater than 5 degrees in phase offsets. As SNR increased, the measurement errors decreased. At an SNR level of 50, the error was 0.09 Hz in frequency and 0.5 degrees in phase.

Figure 2:

Figure 2:

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Frequency (left) and phase (right) estimation MAE of the SC approach at different SNR levels where 11 points were used for phase calculation. Each point represents the mean error from 10,100 spectra with different shifted and noise realizations. Error bars represent the standard deviation. For SNR > 5, the frequency error is smaller than 1 Hz while the phase measurement error was smaller than 5 degrees. No apodization was applied when running the SC algorithm.

Minimal effect on the phase estimate error was observed when using different number of points, ranging from 1 to 21, to calculate the phase information in the SC algorithm (Figure S2). This suggests that the algorithm is robust, maintaining consistency and accuracy regardless of the number of points employed, whether small or large.

Figure 3 shows the in vivo STEAM 7 T spectra with imposed offsets before and after correcting for simultaneous frequency and phase shifts using the SC approach at two different SNR levels without (A) and with baseline (B) artifacts. In the uncorrected cases (i.e. no frequency and phase correction), broad spectral pattern with unrecognizable metabolite signals was observed. After simultaneous frequency and phase correction, high-quality MRS spectra with narrower linewidth were observed. In the low SNR dataset for STEAM data, the SNR of NAA in the summed spectrum increased by ~97% from 32 (uncorrected) to 63 (corrected), both with and without baseline artifacts. Similarly, in the high SNR dataset, the SNR improved by ~91% from 155 to 296 after correction.

Figure 3:

Figure 3:

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Demonstration of the proposed spectral cross-correlation technique for simultaneously estimation of frequency and phase offsets using single-shot STEAM data (TE/TM=8/32 ms, 64 transients, 7T) with imposed frequency and phase offsets at each transient except for the first transient before and after correction estimated using the SC approach. Data without (A) and with baseline artifacts due to insufficient water suppression (C) were considered. Additionally, data are shown for both low and high SNR levels, using identical offsets for each transient. Bold black spectrum in each case represents the mean spectrum. No apodization were applied to the spectra. Comparable estimated frequency and phase values were measured from both low and high SNR data (B, D), with smaller variation in frequency at high SNR.

Good agreement in the estimated frequency and phase values was measured by SC (Figure 3B, D). For the low SNR dataset, the mean absolute error between the estimated frequency and phase values and the ground truth was 1.5±0.9 Hz and 4.2±2.8°, respectively, for spectra without baseline artifacts and 1.5±1.1 Hz and 10.1±7.1°, respectively, for spectra with baseline distortion (Table 1). Conversely, for high SNR data, the corresponding errors were 0.2±0.2 Hz and 0.5±0.5° for spectra without distortion and, 0.9±0.4 Hz and 10.1±5.4° for spectra with baseline distortion. These results suggest good frequency and phase alignments achieved with the SC approach, consistent with the simulation data from Figure 2.

Table 1:

Mean absolute errors ± standard deviation in frequency and phase for the four alignment methods with low and high SNR STEAM datasets, with and without baseline artifacts.

Low SNR High SNR
Baseline Artifacts Methods Frequency (Hz) Phase (degrees) Frequency (Hz) Phase (degrees)
No SC 1.5 ± 0.9 4.2 ± 2.8 0.2 ± 0.2 0.5 ± 0.5
IC 1.4 ± 0.9 4.2 ± 3.7 0.5 ± 0.3 1.5 ± 0.9
RATS 1.8 ± 1.3 10.3± 7.7 0.1 ± 0.1 1.0 ± 0.8
SR 1.9 ± 1.3 6.8 ± 3.8 0.2 ± 0.2 1.1 ± 0.7
Yes SC 1.5 ± 1.1 10.1 ± 7.1 0.9 ± 0.4 10.1 ± 5.4
IC 1.7 ± 1.2 11.1 ± 7.2 1.0 ± 0.6 10.3 ± 6.1
RATS 1.9 ± 1.3 11.0 ± 7.8 0.2 ± 0.1 1.2 ± 0.9
SR 1.6 ± 1.0 9.7 ± 5.4 0.8 ± 0.4 9.1 ± 4.8
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SC: spectral cross-correlation; IC: iterative correlation; RATS: robust alignment to a target spectrum; SR: spectral registration.

To demonstrate that the SC algorithm works for shorter acquisition times, especially used during fast MRSI scans, the number of points in the FID was reduced to achieve acquisition time of 113 ms and 57 ms. The SC approach successfully aligned the spectra without any issues (Figure S3).

A comparison of measurement error between SC and the three other simultaneous alignment methods for the manipulated STEAM data (low and high SNR with/without baseline artifacts) is shown in Figure 4. At low SNR, the estimated frequencies between techniques were comparable with no significant statistical differences. Interestingly, without baseline artifacts, RATS exhibited the largest variability and was statistically different compared to the SC, IC, and SR methods. No such difference was observed with baseline distortion data. At high SNR, consistent measurement errors of less than 0.5 Hz for frequency and less than 1° for phase were observed for spectra without baseline artifacts across all four methods (Table 1), except the frequency estimated from IC, which was statistically different. On the other hand, RATS performed better with data containing baseline artifacts (P<0.01), while SC, IC, and SR errors were higher and comparable.

Figure 4:

Figure 4:

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Comparison of estimated frequency and phase MAE using the proposed spectral cross-correlation (SC) method versus iterative correlation (IC), RATS and spectral registration (SR) approaches under low and high SNR conditions, without (A) and with baseline artifacts (B). Comparable results were observed among methods without baseline artifacts, whereas with baseline artifacts, RATS exhibited superior performance particularly at high SNR. * represents P < 0.01 (one way ANOVA with Tukey’s post-hoc test).

The mean preprocessing time taken to estimate simultaneous frequency and phase values for the four STEAM datasets with 64 transients was 109 ± 21 ms with SC, 9.2 ± 0.6 sec with IC, 883 ± 56 ms with RATS and 840 ± 374 ms with SR, running on Windows 10 operating system with an i7–1065G7 CPU and 32 GB of RAM.

The effectiveness of the SC technique was initially assessed using high SNR human 3T data obtained from PCC, as illustrated in Figure 4. Minor phase variation and frequency shifts were observed in the single-shot data. Following SC processing, these shifts were confirmed, with minor frequency drifts of up to 2.5 Hz observed over the 3-minute data acquisition period, along with noticeable phase offsets of up to 40 degrees. These offsets are likely related to physiological motion. As a result, a 16% improvement in SNR was noted when comparing processed data to measured data lacking frequency and phase correction. Similar SNR increase was observed with RATS and SR techniques while this gain was modest with IC (~6%).

Figure 5 also demonstrates the SC approach when applied to MRS data obtained from a small VOI of 6 μL in the mouse striatum at 16.4T. This dataset was also aligned with IC, RATS and SR approaches. Single-shot data exhibited remarkably low SNR (mean SNR of tCr-CH3 was 5±1 where noise was measured from −0.5 to 0.5 ppm), hindering the ability to effectively distinguish metabolites signals. As such, the mean transient was used as the reference in this dataset for all correction techniques. Upon processing the raw measured data with the SC technique, there was a notable enhancement in SNR, with improvements of 56% observed compared to no frequency and phase correction, shifts of up to 40 Hz and significant phase offsets. Comparable gain in SNR of tCr-CH3 was observed with IC (75%), RATS (80%) and SR (59%). To further investigate potential SNR enhancements, the raw data underwent pre-processing through apodization (line-broadening of 5 Hz and Gaussian filtering of 0.12 s) during SC approach (denoted as SC+). This step resulted in an additional SNR improvement of 20% when compared to using the SC approach and 86% compared to measured data without frequency and phase correction. Particularly noteworthy is the discernible separation between creatine (Cr) and phosphocreatine (PCr) peaks at 3.93 ppm when processed with SC+.

Figure 5:

Figure 5:

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A) In vivo 1H semi-LASER spectra (TE/TR=28/3000 ms, 64 transients) measured in the human posterior cingulate cortex (VOI of 8 mL) at 3 T and aligned using SC, IC, RATS and SR methods. The first transient was used as the reference to align the data in all methods. Small improvement in SNR was observed after processing the data; the measured SNR of NAA was 153 with the measured technique without any alignment applied, 178 with SC, 161 with IC, 178 with RATS, and 176 with SR. For display purposes, all spectra were processed with Gaussian factor of 0.12 s. The estimated frequency and phase values using the SC algorithm (B).

Discussion

The current study shows a novel approach to perform simultaneous frequency and phase estimation using spectral cross-correlation in the frequency domain based on a user-defined frequency range. The technique demonstrates accurate correction for both small and large frequency and phase drifts, even in noisy MRS data, emphasizing its robustness across various SNR levels. Additionally, SC shows remarkably fast processing times compared to previous simultaneous alignment techniques. The successful application of the SC approach was also demonstrated in human and mouse brain proton MRS data.

The SC algorithm showed comparable frequency and phase measurement errors to the IC, RATS, and SR techniques in MRS data, both with and without baseline distortion resulting from insufficient water suppression (Figure 3). As expected, at low SNR, all four methods exhibited higher measurement errors, which decreased at high SNR levels. Conversely, the RATS algorithm, designed to handle large baseline distortion, demonstrated the lowest measurement errors in MRS data with baseline distortion, particularly at high SNR levels. Adhering to recent MRS consensus guidelines (7,32,33) that recommend using advanced sequences like SPECIAL, semi-LASER, or LASER which provide good MRS localization and effective water suppression, can minimize chemical shift displacement errors and reduce artifacts such as baseline or lipid artifacts.

Improving spectral quality in challenging experimental conditions, particularly in low SNR data scenarios where single-shot data tends to be noisy (34,35), can benefit from this technique. Applying apodization to the single-shot spectra to estimate the frequency and phase offsets may enhance the estimation of frequency and phase shifts. As shown in Figure 6, with optimal B0 shim at ultra-high field strengths, the separation of the two peaks of total creatine at 3.93 ppm is expected, as previously demonstrated (36). Similar outcomes are anticipated with other simultaneous alignment techniques. This further underscores the efficacy of the SC approach.

Figure 6:

Figure 6:

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A) In vivo 1H LASER spectra (TE/TR=12.3/5000 ms, 128 transients) measured in the mouse striatum (VOI of 6 μL) at 16.4 T and aligned using SC, IC, RATS, SR and SC+ methods. The mean transient was used as the reference to align the data in all methods. Improvement in SNR and spectral linewidth were apparent after processing the data with all techniques. When single-shot measured data were apodized during SC processing (denoted as SC+ approach), further spectral SNR improvement was observed. For display purposes, all spectra were processed with Gaussian factor of 0.12 s. B) The measured frequency and phase offsets with SC and SC+ approaches.

The SC algorithm demonstrated superior performance compared to similar simultaneous methods, particularly in terms of processing speed. RATS and SR algorithms (12,13) rely on least squares fitting, making them iterative processes that can be computationally intensive. In contrast, the IC approach (10) processed the MRS data in a few seconds. This disparity is anticipated since this approach relies on user-defined frequency and phase shifts ranges that are typically expected in in vivo data. If the actual shift exceeds the specified range, there is a risk of algorithm failure. Therefore, the SC approach is therefore well-suited for simultaneous frequency and phase correction.

The fast processing time of SC makes it suitable for real-time application directly on the scanner. For instance, following MRS data acquisition, frequency and phase correction can be carried out using the proposed SC method, alongside eddy current correction to generate the summed spectrum. This streamlining of the workflow should help to reduce the burden of offline preprocessing. Moreover, the feasibility of quantifying data directly on the scanner is enhanced, enabling the extraction of metabolite concentrations from the measured data. For example, the recently freely available LCModel software (37) can be utilized for this purpose. Overall, the ability to preprocess data rapidly offers significant advantages for real-time applications, facilitating efficient data analysis and decision-making.

One limitation of the proposed approach is the lack of X-nuclei demonstration. It is important to highlight that SC is not confined to proton MRS, as demonstrated in this study; it can also be adapted for X-nuclei, such as carbon-13 and phosphorus-31. Moreover, the SC technique is not limited to single-voxel MRS data but can also be applied to MRSI data. If lipids are present in the spectrum, these signals can be excluded by adjusting the chemical shift region during the cross-correlation process within the SC algorithm.

In conclusion, SC is a robust and fast method which is insensitive to large frequency and phase offsets. It can be easily incorporated in the preprocessing pipeline to improve spectral quality across diverse datasets.

Supplementary Material

Supinfo

Supporting Figure S1: Schematic of the steps used to generate spectra with different noise realizations at 12 different SNR levels to evaluate the proposed SC algorithm. A) The simulated spectrum contains the four major singlets: tNAA, tCr, and tCho. B) 100 random frequency shifts (0 – 25 Hz) and phase offsets (±40 degrees) were generated, with the first transient being without any shifts. C) These shifts were applied to the simulated spectrum to generate 101 noiseless shifted spectra. D) For each spectrum, Gaussian noise was added at a known SNR level with 101 different noise realizations. Twelve SNR levels were used. E) All generated spectra were analyzed using the proposed SC algorithm.

Supporting Figure S2: The absolute mean phase error in degrees (ground truth value minus estimated phase using SC) computed when using different number of points to calculate the phase of the spectrum at different SNR levels. No significant effect is observed when using different number of points.

Supporting Figure S3: Effect of using truncated STEAM data with the proposed SC approach, where the number of points was reduced to have an acquisition time of approximately 113 ms (left) and 57 ms (right). The SC algorithm successfully aligned the spectra in both cases, providing comparable estimations of the frequency and phase shifts.

Acknowledgements

This work was supported by funding from the National Institutes of Health (NIH) R01 EB030000, P41 EB027061, 1S10OD017974–01 and S10 RR025031.

Data Availability Statement

This proposed spectral cross-correlation code written in MATLAB is available from https://github.com/Anesh20/spectralXCorr/.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supinfo

Supporting Figure S1: Schematic of the steps used to generate spectra with different noise realizations at 12 different SNR levels to evaluate the proposed SC algorithm. A) The simulated spectrum contains the four major singlets: tNAA, tCr, and tCho. B) 100 random frequency shifts (0 – 25 Hz) and phase offsets (±40 degrees) were generated, with the first transient being without any shifts. C) These shifts were applied to the simulated spectrum to generate 101 noiseless shifted spectra. D) For each spectrum, Gaussian noise was added at a known SNR level with 101 different noise realizations. Twelve SNR levels were used. E) All generated spectra were analyzed using the proposed SC algorithm.

Supporting Figure S2: The absolute mean phase error in degrees (ground truth value minus estimated phase using SC) computed when using different number of points to calculate the phase of the spectrum at different SNR levels. No significant effect is observed when using different number of points.

Supporting Figure S3: Effect of using truncated STEAM data with the proposed SC approach, where the number of points was reduced to have an acquisition time of approximately 113 ms (left) and 57 ms (right). The SC algorithm successfully aligned the spectra in both cases, providing comparable estimations of the frequency and phase shifts.

NIHMS2014857-supplement-Supinfo.docx (850.8KB, docx)

Data Availability Statement

This proposed spectral cross-correlation code written in MATLAB is available from https://github.com/Anesh20/spectralXCorr/.

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