Towards non-invasive tissue hydration measurements with optical coherence tomography.

The attenuation coefficient ( µ OCT ) measured by optical coherence tomography (OCT) has been used to determine tissue hydration. Previous dual-wavelength OCT systems could not attain the needed precision, which we attribute to the absence of wavelength-dependent scattering of tissue in the underlying model. Assuming that scattering can be described using two parameters, we propose a triple/quadrupole-OCT system to achieve clinically relevant precision in water volume fraction. In this study, we conduct a quantitative analysis to determine the necessary precision of µ OCT measurements and compare it with numerical simulation. Our findings emphasize that achieving a clinically relevant assessment of a 2% water fraction requires determining the attenuation coefficient with a remarkable precision of 0.01 m m - 1 . This precision threshold is influenced by the chosen wavelength for attenuation measurement and can be enhanced through the inclusion of a fourth wavelength range.

Cramér-Rao Lower Bound for Magnetic Field Localization around Elementary Structures.

The determination of a mobile terminal's position with high accuracy and ubiquitous coverage is still challenging. Global satellite navigation systems (GNSSs) provide sufficient accuracy in areas with a clear view to the sky. For GNSS-denied environments like indoors, complementary positioning technologies are required. A promising approach is to use the Earth's magnetic field for positioning. In open areas, the Earth's magnetic field is almost homogeneous, which makes it possible to determine the orientation of a mobile device using a compass. In more complex environments like indoors, ferromagnetic materials cause distortions of the Earth's magnetic field. A compass usually fails in such areas. However, these magnetic distortions are location dependent and therefore can be used for positioning. In this paper, we investigate the influence of elementary structures, in particular a sphere and a cylinder, on the achievable accuracy of magnetic positioning methods. In a first step, we analytically calculate the magnetic field around a sphere and a cylinder in an outer homogeneous magnetic field. Assuming a noisy magnetic field sensor, we investigate the achievable positioning accuracy when observing these resulting fields. For our analysis, we calculate the Cramér-Rao lower bound, which is a fundamental lower bound on the variance of an unbiased estimator. The results of our investigations show the dependency of the positioning error variance on the magnetic sensor properties, in particular the sensor noise variance and the material properties, i.e., the relative permeability of the sphere with respect to the cylinder and the location of the sensor relative to the sphere with respect to the cylinder. The insights provided in this work make it possible to evaluate experimental results from a theoretical perspective.

Empirical optimization of energy bin weights for compressing measurements with realistic photon counting x-ray detectors.

BACKGROUND: Photon counting detectors (PCDs) provide higher spatial resolution, improved contrast-to-noise ratio (CNR), and energy discriminating capabilities. However, the greatly increased amount of projection data in photon counting computed tomography (PCCT) systems becomes challenging to transmit through the slip ring, process, and store. PURPOSE: This study proposes and evaluates an empirical optimization algorithm to obtain optimal energy weights for energy bin data compression. This algorithm is universally applicable to spectral imaging tasks including 2 and 3 material decomposition (MD) tasks and virtual monoenergetic images (VMIs). This method is simple to implement while preserving spectral information for the full range of object thicknesses and is applicable to different PCDs, for example, silicon detectors and CdTe detectors. METHODS: We used realistic detector energy response models to simulate the spectral response of different PCDs and an empirical calibration method to fit a semi-empirical forward model for each PCD. We numerically optimized the optimal energy weights by minimizing the average relative Cramér-Rao lower bound (CRLB) due to the energy-weighted bin compression, for MD and VMI tasks over a range of material area density ρ A , m ${\rho }_{A,m}$ (0-40 g/cm2 water, 0-2.16 g/cm2 calcium). We used Monte Carlo simulation of a step wedge phantom and an anthropomorphic head phantom to evaluate the performance of this energy bin compression method in the projection domain and image domain, respectively. RESULTS: The results show that for 2 MD, the energy bin compression method can reduce PCCT data size by 75% and 60%, with an average variance penalty of less than 17% and 3% for silicon and CdTe detectors, respectively. For 3 MD tasks with a K-edge material (iodine), this method can reduce the data size by 62.5% and 40% with an average variance penalty of less than 12% and 13% for silicon and CdTe detectors, respectively. CONCLUSIONS: We proposed an energy bin compression method that is broadly applicable to different PCCT systems and object sizes, with high data compression ratio and little loss of spectral information.

Uncertainty propagation in absolute metabolite quantification for in vivo MRS of the human brain.

PURPOSE: Absolute spectral quantification is the standard method for deriving estimates of the concentration from metabolite signals measured using in vivo proton MRS (1 H-MRS). This method is often reported with minimum variance estimators, specifically the Cramér-Rao lower bound (CRLB) of the metabolite signal amplitude's scaling factor from linear combination modeling. This value serves as a proxy for SD and is commonly reported in MRS experiments. Characterizing the uncertainty of absolute quantification, however, depends on more than simply the CRLB. The uncertainties of metabolite-specific (T1m , T2m ), reference-specific (T1ref , T2ref ), and sequence-specific (TR , TE ) parameters are generally ignored, potentially leading to an overestimation of precision. In this study, the propagation of uncertainty is used to derive a comprehensive estimate of the overall precision of concentrations from an internal reference. METHODS: The propagated uncertainty is calculated using analytical derivations and Monte Carlo simulations and subsequently analyzed across a set of commonly measured metabolites and macromolecules. The effect of measurement error from experimentally obtained quantification parameters is estimated using published uncertainties and CRLBs from in vivo 1 H-MRS literature. RESULTS: The additive effect of propagated measurement uncertainty from applied quantification correction factors can result in up to a fourfold increase in the concentration estimate's coefficient of variation compared to the CRLB alone. A case study analysis reveals similar multifold increases across both metabolites and macromolecules. CONCLUSION: The precision of absolute metabolite concentrations derived from 1 H-MRS experiments is systematically overestimated if the uncertainties of commonly applied corrections are neglected as sources of error.

Nonlinear event-based state estimation using particle filter under packet loss.

In this research paper, we investigate the problem of remote state estimation for nonlinear discrete systems. Specifically, we focus on scenarios where event-triggered sensor schedules are utilized and where packet drops occur between the sensor and the estimator. In the sensor scheduler, the SOD mechanism is proposed to decrease the amount of data transmitted from the sensor to a remote estimator and the phenomena of packet drops modeled with random variables obeying the Bernoulli distribution. As a consequence of packet drops, the assumption of Gaussianity no longer holds at the estimator side. By fully considering the non-linearity and non-Gaussianity of the dynamic system, this paper develops an event-trigger particle filter algorithm to relieve the communication burden and achieve an appropriate estimation accuracy. First, we derive an explicit expression for the likelihood function when an event trigger occurs and the possible occurrence of packet dropout is taken into consideration. Then, using a special form of sequential Monte-Carlo algorithm, the posterior distribution is approximated and the corresponding minimum mean-squared error is derived. By contrasting the error covariance matrix with the posterior Cramér-Rao lower bound, the estimator's performance is assessed. An illustrative numerical example shows the effectiveness of the proposed design.

Fast and robust quantification of uncertainty in non-linear diffusion MRI models.

Diffusion MRI (dMRI) allows for non-invasive investigation of brain tissue microstructure. By fitting a model to the dMRI signal, various quantitative measures can be derived from the data, such as fractional anisotropy, neurite density and axonal radii maps. We investigate the Fisher Information Matrix (FIM) and uncertainty propagation as a generally applicable method for quantifying the parameter uncertainties in linear and non-linear diffusion MRI models. In direct comparison with Markov Chain Monte Carlo (MCMC) sampling, the FIM produces similar uncertainty estimates at much lower computational cost. Using acquired and simulated data, we then list several characteristics that influence the parameter variances, including data complexity and signal-to-noise ratio. For practical purposes we investigate a possible use of uncertainty estimates in decreasing intra-group variance in group statistics by uncertainty-weighted group estimates. This has potential use cases for detection and suppression of imaging artifacts.

An Improved Orthogonal Matching Pursuit Algorithm for CS-Based Channel Estimation.

Wireless broadband transmission channels usually have time-domain-sparse properties, and the reconstruction of these channels using a greedy search-based orthogonal matching pursuit (OMP) algorithm can effectively improve channel estimation performance while decreasing the length of the reference signal. In this research, the improved OMP and SOMP algorithms for compressed-sensing (CS)-based channel estimation are proposed for single-carrier frequency domain equalization (SC-FDE) systems, which, in comparison with conventional algorithms, calculate the path gain after obtaining the path delay and updating the observation matrices. The reliability of the communication system is further enhanced because the channel path gain is calculated using longer observation vectors, which lowers the Cramér-Rao lower bound (CRLB) and results in better channel estimation performance. The developed method can also be applied to time-domain-synchronous OFDM (TDS-OFDM) systems, and it is applicable to the improvement of other matching pursuit algorithms.

Sensor Placement in an Irregular 3D Surface for Improving Localization Accuracy Using a Multi-Objective Memetic Algorithm.

Accurate localization is a critical task in underwater navigation. Typical localization methods use a set of acoustic sensors and beacons to estimate relative position, whose geometric configuration has a significant impact on the localization accuracy. Although there is much effort in the literature to define optimal 2D or 3D sensor placement, the optimal sensor placement in irregular and constrained 3D surfaces, such as autonomous underwater vehicles (AUVs) or other structures, is not exploited for improving localization. Additionally, most applications using AUVs employ commercial acoustic modems or compact arrays, therefore the optimization of the placement of spatially independent sensors is not a considered issue. This article tackles acoustic sensor placement optimization in irregular and constrained 3D surfaces, for inverted ultra-short baseline (USBL) approaches, to improve localization accuracy. The implemented multi-objective memetic algorithm combines an evaluation of the geometric sensor's configuration, using the Cramer-Rao Lower Bound (CRLB), with the incidence angle of the received signal. A case study is presented over a simulated homing and docking scenario to demonstrate the proposed optimization algorithm.

Accuracy and precision of depth-resolved estimation of attenuation coefficients in optical coherence tomography.

Significance: Parametric imaging of the attenuation coefficient µOCT using optical coherence tomography (OCT) is a promising approach for evaluating abnormalities in tissue. To date, a standardized measure of accuracy and precision of µOCT by the depth-resolved estimation (DRE) method, as an alternative to least squares fitting, is missing. Aim: We present a robust theoretical framework to determine accuracy and precision of the DRE of µOCT. Approach: We derive and validate analytical expressions for the accuracy and precision of µOCT determination by the DRE using simulated OCT signals in absence and presence of noise. We compare the theoretically achievable precisions of the DRE method and the least-squares fitting approach. Results: Our analytical expressions agree with the numerical simulations for high signal-to-noise ratios and qualitatively describe the dependence on noise otherwise. A commonly used simplification of the DRE method results in a systematic overestimation of the attenuation coefficient in the order of µOCT2×Δ, where Δ is the pixel stepsize. When µOCT·|AFR|≲1.8, µOCT is reconstructed with higher precision by the depth-resolved method compared to fitting over the length of an axial fitting range |AFR|. Conclusions: We derived and validated expressions for the accuracy and precision of DRE of µOCT. A commonly used simplification of this method is not recommended as being used for OCT-attenuation reconstruction. We give a rule of thumb providing guidance in the choice of estimation method.

Information-theoretic analysis of Hierarchical Temporal Memory-Spatial Pooler algorithm with a new upper bound for the standard information bottleneck method.

Hierarchical Temporal Memory (HTM) is an unsupervised algorithm in machine learning. It models several fundamental neocortical computational principles. Spatial Pooler (SP) is one of the main components of the HTM, which continuously encodes streams of binary input from various layers and regions into sparse distributed representations. In this paper, the goal is to evaluate the sparsification in the SP algorithm from the perspective of information theory by the information bottleneck (IB), Cramer-Rao lower bound, and Fisher information matrix. This paper makes two main contributions. First, we introduce a new upper bound for the standard information bottleneck relation, which we refer to as modified-IB in this paper. This measure is used to evaluate the performance of the SP algorithm in different sparsity levels and various amounts of noise. The MNIST, Fashion-MNIST and NYC-Taxi datasets were fed to the SP algorithm separately. The SP algorithm with learning was found to be resistant to noise. Adding up to 40% noise to the input resulted in no discernible change in the output. Using the probabilistic mapping method and Hidden Markov Model, the sparse SP output representation was reconstructed in the input space. In the modified-IB relation, it is numerically calculated that a lower noise level and a higher sparsity level in the SP algorithm lead to a more effective reconstruction and SP with 2% sparsity produces the best results. Our second contribution is to prove mathematically that more sparsity leads to better performance of the SP algorithm. The data distribution was considered the Cauchy distribution, and the Cramer-Rao lower bound was analyzed to estimate SP's output at different sparsity levels.

Technical note: Accuracy and precision of T2 and T2* with a gradient-echo spin-echo (GESE) sequence for cardiac imaging.

BACKGROUND: The use of a gradient echo spin echo (GESE) method to obtain rapid T2 and T2* estimation in the heart has been proposed. The effect of acquisition parameter settings on T2 and T2* bias and precision have not been investigated in depth. PURPOSE: To understand factors impacting the quantification of T2 and T2* values with a gradient echo spin echo (GESE) method using echo planar imaging (EPI) readouts in a reduced field of view acquisition. METHODS: The GESE method is implemented with a reduced field-of-view using an outer volume suppression (OVS) technique to minimize the time for multi-echo EPI readouts. The number of EPI readouts (images) for the GESE is optimized using Cramer-Rao Lower Bound (CRLB) and Monte Carlo simulations with a nonlinear least-square (NLLS) estimator. The SNR requirements were studied using the latter simulation method for a selected range of T2 and T2* values and T2/T2* ratios. Two healthy control subjects were imaged with the proposed GESE sequence and evaluated with the NLLS estimation method. In addition, the proposed OVS method was compared with a saturation bands OVS method in one subject. Clinical T2 and T2* mappings were used as the reference. RESULTS: The optimal number of EPI readouts is five and the performance is slightly better when the refocusing pulse is placed between the 2nd and 3rd readouts. The SNR requirement for achieving a target bias < 1 ms and standard deviation (SD) < 5 ms is more demanding when T2/T2* ratio increases. The minimum SNR requirement in the GESE acquisition should vary from 6 to 20 depending on specific myocardial T2 and T2* values at 3T. The T2 and T2* estimates using the proposed OVS method and the saturation bands OVS method are both similar to the reference. CONCLUSION: The GESE sequence with five EPI readouts is a feasible and efficient technique that can estimate T2 and T2* values in the septal myocardium within a heartbeat when the SNR requirement can be satisfied.

Optimized three-dimensional ultrashort echo time: Magnetic resonance fingerprinting for myelin tissue fraction mapping.

Quantitative assessment of brain myelination has gained attention for both research and diagnosis of neurological diseases. However, conventional pulse sequences cannot directly acquire the myelin-proton signals due to its extremely short T2 and T2* values. To obtain the myelin-proton signals, dedicated short T2 acquisition techniques, such as ultrashort echo time (UTE) imaging, have been introduced. However, it remains challenging to isolate the myelin-proton signals from tissues with longer T2. In this article, we extended our previous two-dimensional ultrashort echo time magnetic resonance fingerprinting (UTE-MRF) with dual-echo acquisition to three dimensional (3D). Given a relatively low proton density (PD) of myelin-proton, we utilized Cramér-Rao Lower Bound to encode myelin-proton with the maximal SNR efficiency for optimizing the MR fingerprinting design, in order to improve the sensitivity of the sequence to myelin-proton. In addition, with a second echo of approximately 3 ms, myelin-water component can be also captured. A myelin-tissue (myelin-proton and myelin-water) fraction mapping can be thus calculated. The optimized 3D UTE-MRF with dual-echo acquisition is tested in simulations, physical phantom and in vivo studies of both healthy subjects and multiple sclerosis patients. The results suggest that the rapidly decayed myelin-proton and myelin-water signal can be depicted with UTE signals of our method at clinically relevant resolution (1.8 mm isotropic) in 15 min. With its good sensitivity to myelin loss in multiple sclerosis patients demonstrated, our method for the whole brain myelin-tissue fraction mapping in clinical friendly scan time has the potential for routine clinical imaging.

Cramér-Rao lower bound of dark-field imaging using a grating interferometer / 中国医学物理学杂志

In grating-based phase contrast imaging,the phase stepping technique is commonly utilized for data acquisition and signal retrieval from acquired intensity data.However,the algorithm efficiency with respect to the dark-field retrieval has yet to be sufficiently evaluated.Herein the algorithm efficiency of dark-field retrieval based on Cramér-Rao lower bound is evaluated.The theoretical analysis and numerical results demonstrates that fully efficient algorithm is currently available only for 3-step phase stepping technique,and other techniques with more phase steps are all sub-optimal.Quantitatively,the dependence of the algorithm efficiency on the phase step number and the visibility is investigated.It is shown that the phase stepping technique can nearly approach its theoretical optimal efficiency in the case of a low visibility.With a phase step greater than 5,the algorithm efficiency is only 77.4%in the case of a high visibility.The study can provide some reference for signal-to-noise ratio improvement and potential dose optimization in X-ray and neutron grating-based dark-field imaging.

Analysis and Accuracy Improvement of UWB-TDoA-Based Indoor Positioning System.

Positioning systems are used in a wide range of applications which require determining the position of an object in space, such as locating and tracking assets, people and goods; assisting navigation systems; and mapping. Indoor Positioning Systems (IPSs) are used where satellite and other outdoor positioning technologies lack precision or fail. Ultra-WideBand (UWB) technology is especially suitable for an IPS, as it operates under high data transfer rates over short distances and at low power densities, although signals tend to be disrupted by various objects. This paper presents a comprehensive study of the precision, failure, and accuracy of 2D IPSs based on UWB technology and a pseudo-range multilateration algorithm using Time Difference of Arrival (TDoA) signals. As a case study, the positioning of a 4×4m2 area, four anchors (transceivers), and one tag (receiver) are considered using bitcraze's Loco Positioning System. A Cramér-Rao Lower Bound analysis identifies the convex hull of the anchors as the region with highest precision, taking into account the anisotropic radiation pattern of the anchors' antennas as opposed to ideal signal distributions, while bifurcation envelopes containing the anchors are defined to bound the regions in which the IPS is predicted to fail. This allows the formulation of a so-called flyable area, defined as the intersection between the convex hull and the region outside the bifurcation envelopes. Finally, the static bias is measured after applying a built-in Extended Kalman Filter (EKF) and mapped using a Radial Basis Function Network (RBFN). A debiasing filter is then developed to improve the accuracy. Findings and developments are experimentally validated, with the IPS observed to fail near the anchors, precision around ±3cm, and accuracy improved by about 15cm for static and 5cm for dynamic measurements, on average.

Single Transmitter Direction Finding Using a Single Moving Omnidirectional Antenna.

Traditional direction-finding systems are based on processing the outputs of multiple spatially separated antennas. The impinging signal Angle-of-Arrival (AOA) is estimated using the relative phase and amplitude of the multiple outputs that are sampled simultaneously. Here, we explore the potential of a single moving antenna to provide useful direction finding of a single transmitter. If the transmitted signal frequency is steady enough during the collection of data, a single antenna can be moved while tracking the phase changes to provide an Angle-of-Arrival measurement. The advantages of a single-antenna sensor include the sensor size, the lack of a need for multiple-receiver synchronization in time and frequency, the lack of mutual antenna coupling, and the cost of the system. However, a single-antenna sensor requires an accurate knowledge of its position during the data collection and it is challenged by transmitter phase instability, signal modulation, and transmitter movement during the measurement integration time. We analyze the performance of the proposed sensor, support the analysis with simulations and finally, present measurements performed by hardware configured to check the validity of the proposed single-antenna sensor.

Mobile Location in Wireless Sensor Networks Based on Multi Spot Measurements Model.

The localization of sensors in wireless sensor networks has recently gained considerable attention. The existing location methods are based on a one-spot measurement model. It is difficult to further improve the positioning accuracy of existing location methods based on single-spot measurements. This paper proposes two location methods based on multi-spot measurements to reduce location errors. Because the multi-spot measurements model has more measurement equations than the single-spot measurements model, the proposed methods provide better performance than the traditional location methods using one-spot measurement in terms of the root mean square error (RMSE) and Cramer-Rao lower bound (CRLB). Both closed-form and iterative algorithms are proposed in this paper. The former performs suboptimally with less computational burden, whereas the latter has the highest positioning accuracy in attaining the CRLB. Moreover, a novel CRLB for the proposed multi-spot measurements model is also derived in this paper. A theoretical proof shows that the traditional CRLB in the case of single-spot measurements performs worse than the proposed CRLB in the case of multi-spot measurements. The simulation results show that the proposed methods have a lower RMSE than the traditional location methods.

Precision of attenuation coefficient measurements by optical coherence tomography.

SIGNIFICANCE: Optical coherence tomography (OCT) is an interferometric imaging modality, which provides tomographic information on the microscopic scale. Furthermore, OCT signal analysis facilitates quantification of tissue optical properties (e.g., the attenuation coefficient), which provides information regarding the structure and organization of tissue. However, a rigorous and standardized measure of the precision of the OCT-derived optical properties, to date, is missing. AIM: We present a robust theoretical framework, which provides the Cramér -Rao lower bound σµOCT for the precision of OCT-derived optical attenuation coefficients. APPROACH: Using a maximum likelihood approach and Fisher information, we derive an analytical solution for σµOCT when the position and depth of focus are known. We validate this solution, using simulated OCT signals, for which attenuation coefficients are extracted using a least-squares fitting procedure. RESULTS: Our analytical solution is in perfect agreement with simulated data without shot noise. When shot noise is present, we show that the analytical solution still holds for signal-to-noise ratios (SNRs) in the fitting window being above 20 dB. For other cases (SNR<20 dB, focus position not precisely known), we show that the numerical calculation of the precision agrees with the σµOCT derived from simulated signals. CONCLUSIONS: Our analytical solution provides a fast, rigorous, and easy-to-use measure for OCT-derived attenuation coefficients for signals above 20 dB. The effect of uncertainties in the focal point position on the precision in the attenuation coefficient, the second assumption underlying our analytical solution, is also investigated by numerical calculation of the lower bounds. This method can be straightforwardly extended to uncertainty in other system parameters.

Chaos-Enhanced Adaptive Hybrid Butterfly Particle Swarm Optimization Algorithm for Passive Target Localization.

This paper considers the problem of finding the position of a passive target using noisy time difference of arrival (TDOA) measurements, obtained from multiple transmitters and a single receiver. The maximum likelihood (ML) estimator's objective function is extremely nonlinear and non-convex, making it impossible to use traditional optimization techniques. In this regard, this paper proposes the chaos-enhanced adaptive hybrid butterfly particle swarm optimization algorithm, named CAHBPSO, as the hybridization of butterfly optimization (BOA) and particle swarm optimization (PSO) algorithms, to estimate passive target position. In the proposed algorithm, an adaptive strategy is employed to update the sensory fragrance of BOA algorithm, and chaos theory is incorporated into the inertia weight of PSO algorithm. Furthermore, an adaptive switch probability is employed to combine global and local search phases of BOA with the PSO algorithm. Additionally, the semidefinite programming is employed to convert the considered problem into a convex one. The statistical comparison on CEC2014 benchmark problems shows that the proposed algorithm provides a better performance compared to well-known algorithms. The CAHBPSO method surpasses the BOA, PSO and semidefinite programming (SDP) algorithms for a broad spectrum of noise, according to simulation findings, and achieves the Cramer-Rao lower bound (CRLB).

Bayesian Cramér-Rao Lower Bounds for Prediction and Smoothing of Nonlinear TASD Systems.

The performance evaluation of state estimators for nonlinear regular systems, in which the current measurement only depends on the current state directly, has been widely studied using the Bayesian Cramér-Rao lower bound (BCRLB). However, in practice, the measurements of many nonlinear systems are two-adjacent-states dependent (TASD) directly, i.e., the current measurement depends on the current state as well as the most recent previous state directly. In this paper, we first develop the recursive BCRLBs for the prediction and smoothing of nonlinear systems with TASD measurements. A comparison between the recursive BCRLBs for TASD systems and nonlinear regular systems is provided. Then, the recursive BCRLBs for the prediction and smoothing of two special types of TASD systems, in which the original measurement noises are autocorrelated or cross-correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the proposed recursive BCRLBs for the prediction and smoothing of TASD systems.

Improving the reproducibility of proton magnetic resonance spectroscopy brain thermometry: Theoretical and empirical approaches.

In proton magnetic resonance spectroscopy (1 H MRS)-based thermometry of brain, averaging temperatures measured from more than one reference peak offers several advantages, including improving the reproducibility (i.e., precision) of the measurement. This paper proposes theoretically and empirically optimal weighting factors to improve the weighted average of temperatures measured from three references. We first proposed concepts of equivalent noise and equivalent signal-to-noise ratio in terms of frequency measurement and a concept of relative frequency that allows the combination of different peaks in a spectrum for improving the precision of frequency measurement. Based on these, we then derived a theoretically optimal weighting factor and proposed an empirical weighting factor, both involving equivalent noise levels, for a weighted average of temperatures measured from three references (i.e., the singlets of NAA, Cr, and Ch in the 1 H MR spectrum). We assessed these two weighting factors by comparing their errors in measurement of temperatures with the errors of temperatures measured from individual references; we also compared these two new weighting factors with two previously proposed weighting factors. These errors were defined as the standard deviations in repeated measurements or in Monte Carlo studies. Both the proposed theoretical and empirical weighting factors outperformed the two previously proposed weighting factors as well as the three individual references in all phantom and in vivo experiments. In phantom experiments with 4- or 10-Hz line broadening, the theoretical weighting factor outperformed the empirical one, but the latter was superior in all other repeated and Monte Carlo tests performed on phantom and in vivo data. The proposed weighting factors are superior to the two previously proposed weighting factors and can improve the reproducibility of temperature measurement using 1 H MRS-based thermometry.