Noise reduction for NMR FID signals via Gabor expansion

A denoising method for multidimensional magnetic resonance spectroscopy and imaging based on compressed sensing

Both in spectroscopy and imaging, t₁-noise arising from instabilities such as temperature alterations, field-related frequency drifts, electronic and sample-spinning instabilities, or motions in in vivo experiments, affects many 2D Magnetic Resonance experiments. This work introduces a post-processing method that aims to attenuate t₁-noise, by suitably averaging multiple signals/representations that have been reconstructed from the sampled data. The ensuing Compressed Sensing Multiplicative (CoSeM) denoising starts from a fully sampled 2D MR data set, discards random indirect-domain points, and makes up for these missing, masked data, by a compressed sensing reconstruction of the now incompletely sampled 2D data set. This procedure is repeated for multiple renditions of the masked data -some of which will have been more strongly affected by t₁-noise than others. This leads to a large set of 2D NMR spectra/images compatible with the collected data; CoSeM chooses out of these those renditions that reduce the noise according to a suitable criterion, and then sums up their spectra/images leading to a reduction in t₁-noise. The performance of the method was assessed in synthetic data, as well as in numerous different experiments: 2D solid and solution state NMR, 2D localized MRS of live brains, and 2D abdominal MRI. Throughout all these data, CoSeM processing evidenced 2-3 fold increases in SNR, without introducing biases, false peaks, or spectral/image blurring. CoSeM also retains a quantitative linearity in the information -allowing, for instance, reliable T₁ inversion-recovery MRI mapping experiments.

Systematic and statistical uncertainties of the hilbert-transform based high-precision FID frequency extraction method

Pulsed nuclear magnetic resonance (NMR) is widely used in high-precision magnetic field measurements. The absolute value of the magnetic field is determined from the precession frequency of nuclear magnetic moments. The Hilbert transform is one of the methods that have been used to extract the phase function from the observed free induction decay (FID) signal and then its frequency. In this paper, a detailed implementation of a Hilbert-transform based FID frequency extraction method is described, and it is briefly compared with other commonly used frequency extraction methods. How artifacts and noise level in the FID signal affect the extracted phase function are derived analytically. A method of mitigating the artifacts in the extracted phase function of an FID is discussed. Correlations between noises of the phase function samples are studied for different noise spectra. We discovered that the error covariance matrix for the extracted phase function is nearly singular and improper for constructing the χ² used in the fitting routine. A down-sampling method for fixing the singular covariance matrix has been developed, so that the minimum χ²-fit yields properly the statistical uncertainty of the extracted frequency. Other practical methods of obtaining the statistical uncertainty are also discussed.

Sensor Configuration and Algorithms for Power-Line Interference Suppression in Low Field Nuclear Magnetic Resonance

Low field (LF) nuclear magnetic resonance (NMR) shows potential advantages to study pure heteronuclear J-coupling and observe the fine structure of matter. Power-line harmonics interferences and fixed-frequency noise peaks might introduce discrete noise peaks into the LF-NMR spectrum in an open environment or in a conductively shielded room, which might disturb J-coupling spectra of matter recorded at LF. In this paper, we describe a multi-channel sensor configuration of superconducting quantum interference devices, and measure the multiple peaks of the 2,2,2-trifluoroethanol J-coupling spectrum. For the case of low signal to noise ratio (SNR) < 1, we suggest two noise suppression algorithms using discrete wavelet analysis (DWA), combined with either least squares method (LSM) or gradient descent (GD). The de-noising methods are based on spatial correlation of the interferences among the superconducting sensors, and are experimentally demonstrated. The DWA-LSM algorithm shows a significant effect in the noise reduction and recovers SNR > 1 for most of the signal peaks. The DWA-GD algorithm improves the SNR further, but takes more computational time. Depending on whether the accuracy or the speed of the de-noising process is more important in LF-NMR applications, the choice of algorithm should be made.

Spectral data de-noising using semi-classical signal analysis: application to localized MRS

In this paper, we propose a new post-processing technique called semi-classical signal analysis (SCSA) for MRS data de-noising. Similar to Fourier transformation, SCSA decomposes the input real positive MR spectrum into a set of linear combinations of squared eigenfunctions equivalently represented by localized functions with shape derived from the potential function of the Schrödinger operator. In this manner, the MRS spectral peaks represented as a sum of these 'shaped like' functions are efficiently separated from noise and accurately analyzed. The performance of the method is tested by analyzing simulated and real MRS data. The results obtained demonstrate that the SCSA method is highly efficient in localized MRS data de-noising and allows for an accurate data quantification.

Probing the influential factors of NMR T1-T2 spectra in the characterization of the kerogen by numerical simulation

The low field nuclear magnetic resonance (NMR) spectroscopy has been widely used to characterize the longitudinal and transversal relaxation (T1-T2) spectrum of unconventional resources such as shale gas and tight oil containing significant proportions of kerogen and bitumen. However, it requires exquisite design of the acquisition model and the inversion algorithm due to the fast relaxation nature of the kerogen and bitumen. A new direct two dimensional (2D) inversion algorithm combined the iterative truncated singular value decomposition (TSVD) and the Akaiake Information Criterion (AIC) is presented to perform the data inversion efficiently. The fluid component decomposition (FCD) is applied to construct the forward T1-T2 model of the kerogen, and numerical simulations are conducted to investigate factors which may influence inversion results including echo spacing, recovery time series, signal to noise ratio (SNR), and the maximal iteration time. Results show that the T2 component is heavily impaired by the echo spacing, whereas the T1 component is influenced by the recovery time series but with limited effects. The inversion precision is greatly affected by the quality of the data. The inversed spectrum deviates from the model seriously when the SNR of the artificial noise is lower than 50, and the T2 component is more sensitive to the noise than the T1 component. What's more, the maximal iteration time can also affect the inversion result, especially when the maximal iteration time is smaller than 500. Proper acquisition and inversion parameters for the characterization of the kerogen are obtained considering the precision and the computational cost.

Noise reduction of nuclear magnetic resonance (NMR) transversal data using improved wavelet transform and exponentially weighted moving average (EWMA)

NMR logging and core NMR signals acts as an effective way of pore structure evaluation and fluid discrimination, but it is greatly contaminated by noise for samples with low magnetic resonance intensity. Transversal relaxation time (T(2)) spectrum obtained by inversion of decay signals intrigued by Carr-Purcell-Meiboom-Gill (CPMG) sequence may deviate from the truth if the signal-to-noise ratio (SNR) is imperfect. A method of combing the improved wavelet thresholding with the EWMA is proposed for noise reduction of decay data. The wavelet basis function and decomposition level are optimized in consideration of information entropy and white noise estimation firstly. Then a hybrid threshold function is developed to avoid drawbacks of hard and soft threshold functions. To achieve the best thresholding values of different levels, a nonlinear objective function based on SNR and mean square error (MSE) is constructed, transforming the problem to a task of finding optimal solutions. Particle swarm optimization (PSO) is used to ensure the stability and global convergence. EWMA is carried out to eliminate unwanted peaks and sawtooths of the wavelet denoised signal. With validations of numerical simulations and experiments, it is demonstrated that the proposed approach can reduce the noise of T(2) decay data perfectly.

New denoising scheme for magnetic resonance spectroscopy signals

A new scheme for denoising magnetic resonance spectroscopy (MRS) signals is presented. This scheme is based on projecting noisy MRS signals in different domains, consecutively, and performing noise filtering operations in these domains. The domains are chosen such that the noise portion, which is inseparable from the desired signal in one domain, is separable in the other. A set of stable, linear, time-frequency (SLTF) transforms with different resolutions was selected for these projections as an example. Scheme evaluation was performed using extensive MRS signals with various noise levels. Compared with one domain denoising, it was observed that the proposed scheme gives superior results that compensate for the excess computational requirements. The proposed scheme supersedes also the wavelet packet denoising schemes.

NMR signal enhancement via a new time-frequency transform

In this paper, a reliable method to reduce the noise from nuclear magnetic resonance (NMR) signals using a recently developed linear critically sampled time-frequency transform is proposed. In addition to its low computational requirements, this transform has many theoretical advantages that make it a good candidate for NMR signal enhancement. NMR signals in the transform domain are concentrated in a few coefficients while the noise is well distributed. Performing a thresholding technique in the transform domain, therefore, significantly enhances the signal. A comparison with other signal enhancement techniques shows that this technique has a superior performance, thus confirming the theoretical expectations.

Wavelets and related time-frequency techniques in magnetic resonance spectroscopy

We survey the various applications in MRS of the wavelet transform and related time-frequency methods. For the sake of completeness, we first quickly review the mathematical tools needed.

Time-frequency analyses of TEOAE recordings from normals and SNHL patients

This study evaluated the characteristics of transient evoked otoacoustic emission (TEOAE) time-frequency (TF) representations from normal and hearing-impaired ears. Linear and non-linear TEOAE recordings from normally-hearing subjects (40) and non-linear recordings from patients with sensorineural hearing loss (SNHL) (40) were analysed using the short-time-Fourier-transform spectrogram, the Gabor spectrogram, and the adaptive spectrogram. The TF representations of the TEOAE recordings indicated a considerable dispersion of energy across frequencies and TEOAE time segments >4.0 ms. The linear and non-linear recordings from the normal subjects showed common frequency peaks. The TF representations from the patients with SNHL indicated that the significantly reduced energy in the mid-to-high TEOAE frequencies did not correlate closely with the threshold elevation. As in the recordings from the normal subjects, a high percentage of the TEOAE cumulative energy was found within a short TEOAE segment (4-14 ms).