Autoregressive moving average modeling for spectral parameter estimation from a multigradient echo chemical shift acquisition

Purpose: To integrate modern super‐resolution spectral quantification techniques with multi‐echo magnetic resonance imaging in order to provide multi‐parametric chemical shift imaging in real‐time. Method and Materials: A technique integrating the Steiglitz‐McBride algorithm and Cauchy's Calculus of Residues was developed to determine the proton resonance frequency(PRF), T2*, and T1‐wieghted amplitude for multiple species in a chemical shift MR signal encoded using multiple gradient echoes. Monte Carlo simulation was implemented evaluating the performance of the algorithm under varying acquisition parameters and results compared to the Cramer‐Rao lower bound (CRLB). The PRF values in a mayonnaise‐lemon juice phantom were measured during heating demonstrating the ability of performing PRF thermometry in the presence of lipid. Phantom scans were performed comparing the T2* values from this technique and from spoiled gradient echo(SPGR) acquisitions. To demonstrate the use of this technique for temperature imaging in vivo, an image‐guided laser treatment in a canine brain was performed. Temperature maps were created at spatial resolutions of 1.5×1.5×4mm3 every five seconds in addition to T1‐weighted and T2* images providing multi‐parametric monitoring during treatment. Results: The technique achieves the CRLB demonstrating robust estimation of the PRF and T2* in the presence of noise. The uncertainty of the technique was theoretically and experimentally shown to be inversely proportional to the image SNR. The mayonnaise‐lemon juice phantom showed a PRF shift with temperature consistent to published literature. T2* values provided by the technique did not statistically differ from SPGR‐based T2* mapping. Monitoring of the canine brain treatment was successful providing highly sensitive temperature measurements. T2*‐corrected T1‐W images detected a treatment‐induced hemorrhage demonstrating the importance of multi‐parametric monitoring. Conclusion: A technique is presented that provides chemical shift imaging at higher spatiotemporal resolutions than what was previously available. The ability of this technique to provide multiple parameters demonstrates great promise for image‐guided thermal therapies.

Backgroud: Echo Train Length (ETL) or turbo factor refers to the echo number of radiofrequency pulse refocusing 1800 after excitation RF pulse which used in Fast Spin Echo (FSE). ETL was is the single most important parameter on the image quality. Longer ETLs result in more T2-weighting because more late echoes with longer TE’s contribute to the overall signal. Longer ETL’s also associated with a decrease in the overall signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR) because the later echoes are weaker and the image having blurring. The ETL was influence with scanning time. PROPELLER sequences can be reducing the artefact motion. The aim of this research is to known how the effect of the ETLs variation on sequences T2WI axial PROPELLER to the best image quality and the ETLs value which have best quality.
 
 Methods: This type of study is a quantitative study conducted with an experimental approach in Radiology Installation of Santo Borromeo’s Bandung. Data is collected by the examination brain MRI in 6 (six) patients, after a brain MRI scanning with axial slice T2WI PROPELLER sequences, with variation of Echo Train Length (ETL). The variation of ETL we used are 12, 18, 24 and 30. And the estimation using Region of Interest (ROI) at the ventricle lateral, nucleus caudate, putamen, thalamus and internal capsule. The assessment results are summarized and then in a statistical test using SPSS version 23.0 with the method of Pearson correlate and linier regression.
 
 Results: The result showed that there is having effect of Echo Train Length (ETL) on sequences T2WI axial PROPELLER to the image quality. There is a strong correlation which the result of significant value 0,000 < p value 0.05. The relation effect variation of ETL with SNR very strong is nucleus caudate and CNR very strong between nucleus caudate and ventricle lateral. The mean rank value of SNR and CNR have been showed is ETL 12 with a mean rank up to 152,57 and 121,18
 
 Conclusion:There was the influence of the echo train length toward image quality of brain MRI on sequences T2WI axial PROPELLER.
 The best echo train length value of brain MRI on sequence T2WI axial PROPELLER is 12.

A k-means-based classification algorithm is investigated to assess suitability for rapidly separating and classifying fat/water spectral peaks from a fast chemical shift imaging technique for magnetic resonance temperature imaging. Algorithm testing is performed in simulated mathematical phantoms and agar gel phantoms containing mixed fat/water regions. Proton resonance frequencies (PRFs), apparent spin-spin relaxation (T2*) times, and T1-weighted (T1-W) amplitude values were calculated for each voxel using a single-peak autoregressive moving average (ARMA) signal model. These parameters were then used as criteria for k-means sorting, with the results used to determine PRF ranges of each chemical species cluster for further classification. To detect the presence of secondary chemical species, spectral parameters were recalculated when needed using a two-peak ARMA signal model during the subsequent classification steps. Mathematical phantom simulations involved the modulation of signal-to-noise ratios (SNR), maximum PRF shift (MPS) values, analysis window sizes, and frequency expansion factor sizes in order to characterize the algorithm performance across a variety of conditions. In agar, images were collected on a 1.5T clinical MR scanner using acquisition parameters close to simulation, and algorithm performance was assessed by comparing classification results to manually segmented maps of the fat/water regions. Performance was characterized quantitatively using the Dice Similarity Coefficient (DSC), sensitivity, and specificity. The simulated mathematical phantom experiments demonstrated good fat/water separation depending on conditions, specifically high SNR, moderate MPS value, small analysis window size, and low but nonzero frequency expansion factor size. Physical phantom results demonstrated good identification for both water (0.997 ± 0.001, 0.999 ± 0.001, and 0.986 ± 0.001 for DSC, sensitivity, and specificity, respectively) and fat (0.763 ± 0.006, 0.980 ± 0.004, and 0.941 ± 0.002 for DSC, sensitivity, and specificity, respectively). Temperature uncertainties, based on PRF uncertainties from a 5 × 5-voxel ROI, were 0.342 and 0.351°C for pure and mixed fat/water regions, respectively. Algorithm speed was tested using 25 × 25-voxel and whole image ROIs containing both fat and water, resulting in average processing times per acquisition of 2.00 ± 0.07 s and 146 ± 1 s, respectively, using uncompiled MATLAB scripts running on a shared CPU server with eight Intel Xeon(TM) E5640 quad-core processors (2.66 GHz, 12 MB cache) and 12 GB RAM. Results from both the mathematical and physical phantom suggest the k-means-based classification algorithm could be useful for rapid, dynamic imaging in an ROI for thermal interventions. Successful separation of fat/water information would aid in reducing errors from the nontemperature sensitive fat PRF, as well as potentially facilitate using fat as an internal reference for PRF shift thermometry when appropriate. Additionally, the T1-W or R2* signals may be used for monitoring temperature in surrounding adipose tissue.

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper

It has recently been found that by jointly processing multiple (sum, azimuth-, and elevation-difference) matched filter samples, it is possible to extract and localize several (more than two) targets spaced more closely than the classical interpretation of radar resolution. This paper derives the Cramer-Rao lower bound (CRLB) for sampled monopulse radar data. It is worthwhile to know the limits of such procedures; and in addition to its role in delivering the measurement accuracies required by a target tracker, the CRLB reveals an estimator's efficiency. We interrogate the CRLB expressions for cases of interest. Of particular interest are the CRLB's implications on the number of targets localizable: assuming a sampling-period equal to a rectangular pulse's length, five targets can be isolated between two matched filter samples given that the target's signal-to-noise ratios (SNRs) are known. This reduced to three targets when the SNRs are not known, but the number of targets increases back to five (and beyond) when a dithered boresight strategy is used. Further insight to the impact of pulse shape and of the benefits of oversampling are given.

Considering the influence of acceleration and the Gaussian envelope for a laser Doppler velocimeter (LDV), parameter estimation of a Doppler signal with a Gaussian envelope was investigated based on introducing acceleration. According to the theory of mathematics statistics, the Cramer-Rao lower bounds (CRLBs) of Doppler circular frequency and its first order rate were analyzed, formulas of CRLBs were given, and the power spectrum estimation with adjustment was discussed. The results of theory and the simulation show that the CRLBs are related to the data length, the signal-to-noise ratio (SNR), and the width of the Gaussian envelope, and they can be decreased by increasing the data length or improving the SNR; the larger the acceleration is and the narrower the Gaussian envelope is, the larger the CRLBs of Doppler circular frequency and its first order rate are; the gap between the variances of the measuring results and the CRLBs narrows when the SNR of the signal is improved, and is almost eliminated when the SNR is higher than 6 dB. It is concluded that the model presented is much more suitable for a LDV than that acquired by Rife and Boorstyn [IEEE Trans. Inform. Theory 20, 591 (1974)].

This paper discusses the estimation of time-frequency representations of non-stationary signals, by means of ARMA (autoregressive moving-average) models with time-dependent coefficients. In order to allow the estimation of the model on a single realisation of the random signal, it is assumed that the coefficients are weighted sums of known functions, for instance polynomials, cosines. Two ARMA models will be discussed in this paper. The first one is a white noise driven ARMA model, which is rather usual. The second one has been designed for deterministic or speech-like signals and consists of a linear system driven by intermittent inputs with additive output (white) noise. The time-frequency representation of the signal is then obtained from the state-space realisation of the time-dependent model. After a description of the algorithms involved in both estimators, the paper focuses on a comparison of the two ARMA models on modulated sinusoids corrupted by white noise. This shows that these models behave equally well for high SNR (signal-to-noise ratio), but the deterministic model gives better results when the SNR decreases.

Introduction: Phosphorus magnetic resonance spectroscopy ( 31 P-MRS) allows insight into cardiac energetics in vivo , but is a technique with an intrinsically low signal to noise ratio (SNR). As ultra-high magnetic field MRI scanners become more widely available this limitation will lessen, allowing detection of smaller changes in metabolite concentrations or measurement of changes in smaller patient groups, which will further the understanding of cardiac energetics. Hypothesis: We hypothesised that the 2.4x increase in SNR, predicted by theory at 7T compared to 3T, will allow more precise metabolite concentration measurements in dilated cardiomyopathy (DCM) patients. Methods: 26 idiopathic DCM patients (69% male, mean age 55±2 yr, mean LVEF 35±2%) underwent 31 P-MRS at 3T and 7T field strengths. Patients were NYHA Class I (46%), II (50%) or III (4%). Results: The mean SNR increased from 6.5 ± 0.5 to 16.2 ± 1.5 for PCr (p <0.0001, paired t-test) and from 3.4 ± 0.2 to 11.4 ± 0.8 for γATP (p <0.0001) as the field strength increased from 3T to 7T. The PCr/ATP ratio was consistent between the 2 field strengths; at 3T PCr/ATP was 1.5 ± 0.1 and at 7T 1.5 ± 0.1; there was as expected no statistically significant difference (p=0.67) (fig a). The Cramér-Rao lower bounds (CRLB) quantify the minimum precision of fitted parameters. A desirable decrease in the PCr/ATP CRLB was seen from 3T to 7T (47 ± 3% vs 25 ± 2% p<0.0001)(fig b), i.e.-metabolites are indeed quantified more precisely at 7T compared to 3T. Conclusion: This study represents the first cardiac 31 P-MRS patient data acquired at a 7T field strength. Compared to a 2.8x SNR increase from 3T to 7T previously reported in normal volunteers, we observed a 2.5x increase in this patient population. There is no significant bias at 7T as the PCr/ATP ratios are similar. In conclusion 7T 31 P-MRS is feasible in DCM patients and more accurate than at 3T, which in future should allow more accurate quantification and understanding of myocardial energetics.

Maximum-likelihood (ML) estimation is an established paradigm for the assessment of imaging system performance in nonlinear quantitation tasks. At high signal-to-noise ratio (SNR), ML estimates are asymptotically Gaussian-distributed, unbiased and efficient, thereby attaining the Cramer–Rao bound (CRB). Therefore, at high SNR the CRB is useful as a predictor of the variance of ML estimates and, consequently, as a basis for measures of estimation performance. At low SNR, however, the achievable parameter variances are often substantially larger than the CRB and the estimates are no longer Gaussian-distributed. These departures imply that inference about the estimates that is based on the CRB and the assumption of a normal distribution will not be valid. We have found previously that for some tasks these effects arise at noise levels considered clinically acceptable. We have derived the mathematical relationship between a new measure, χ2pdf-ML, and the expected probability density of the ML estimates, and have justified the use of χ2pdf-ML-isocontours in parameter space to describe the ML estimates. We validated this approach by simulation experiments using spherical objects imaged with a Gaussian point spread function. The parameters, activity concentration and size, were estimated simultaneously by ML, and variances and covariances calculated over 1000 replications per condition from 3D image volumes and from 2D tomographic projections of the same object. At low SNR, where the CRB is no longer achievable, χ2pdf-ML-isocontours provide a robust prediction of the distribution of the ML estimates. At high SNR, the χ2pdf-ML-isocontours asymptotically approach the analogous χ2pdf-F-contours derived from the Fisher information matrix. The χ2pdf-ML model appears to be suitable for characterization of the influence of the noise level and characteristics, the task, and the object on the shape of the probability density of the ML estimates at low SNR. Furthermore, it provides unique insights into the causes of the variability of estimation performance.

A method is provided for determining necessary conditions on sample size or signal to noise ratio (SNR) to obtain accurate parameter estimates from remote sensing measurements in fluctuating environments. These conditions are derived by expanding the bias and covariance of maximum likelihood estimates (MLEs) in inverse orders of sample size or SNR, where the first-order covariance term is the Cramer-Rao lower bound (CRLB). Necessary sample sizes or SNRs are determined by requiring that (i) the first-order bias and the second-order covariance are much smaller than the true parameter value and the CRLB, respectively, and (ii) the CRLB falls within desired error thresholds. An analytical expression is provided for the second-order covariance of MLEs obtained from general complex Gaussian data vectors, which can be used in many practical problems since (i) data distributions can often be assumed to be Gaussian by virtue of the central limit theorem, and (ii) it allows for both the mean and variance of the measurement to be functions of the estimation parameters. Here, conditions are derived to obtain accurate source localization estimates in a fluctuating ocean waveguide containing random internal waves, and the consequences of the loss of coherence on their accuracy are quantified.

In this paper, an expression for the Cramer–Rao bound (CRB) on the frequency and damping of electromechanical modes is derived. This provides an invaluable tool when documenting the performance capabilities of mode meter algorithms, as the CRB is used as a benchmark to which mode meter variance may be compared. It is shown that the derived CRB expression is supported by several well-known mode estimation phenomena, and it is numerically validated using the asymptotically efficient Prediction Error Method (PEM) on Monte Carlo simulations of an autoregressive moving average (ARMA) model. Finally, a practical use case is given, in which the CRB on the frequency and damping of an inter-area mode from the miniWECC system is used to compare the performance of several well-known mode meters in a study of how sample size affects accuracy.

Fast T2-weighted MR imaging: impact of variation in pulse sequence parameters on image quality and artifacts.

Publication in the conference proceedings of EUSIPCO, Tampere, Finland, 2000

The main contribution of our work consists in developing for the first time a method of estimating the direction of arrival (DOA) parameters assuming noncircular and temporally and spatially correlated signals. This new approach, based on a significant enhancement of the two-sided instrumental variable signal subspace fitting (IV-SSF) method, outperforms its classical version in terms of lower bias and error variance. Moreover, it will be shown that our new method is statistically more efficient than the MODE method especially in the case of partly and fully coherent signals where only the extended and the classical two-sided IV-SSF methods are applicable. We also derive an explicit expression for the stochastic Cramér-Rao bound (CRB) of the DOA estimates from temporally and spatially correlated signals generated from noncircular sources. The new CRB is compared to those of circular temporally correlated and noncircular independent and identically distributed signals to show that the CRB obtained assuming both noncircular sources and temporally correlated signals is lower than the CRBs derived considering only one of these two assumptions. This illustrates the potential gain that both noncircularity and temporal correlation provide when considered together. It will also be proven that the difference between the three CRBs increases with the number of snapshots. However, as the signal-to-noise ratio (SNR) increases, the CRBs merge together and decrease linearly. Moreover, at low SNR values it will be shown that temporal correlation is more informative about the unknown DOA parameters than noncircularity. Finally, the CRB derived assuming noncircular and temporally correlated signals depends on the noncircularity rate, the circularity phase separation, and the DOA separation.