Penalized maximum-likelihood image reconstruction for lesion detection

Detecting cancerous lesions is one major application in emission tomography. In this paper, we study penalized maximum-likelihood image reconstruction for this important clinical task. Compared to analytical reconstruction methods, statistical approaches can improve the image quality by accurately modelling the photon detection process and measurement noise in imaging systems. To explore the full potential of penalized maximum-likelihood image reconstruction for lesion detection, we derived simplified theoretical expressions that allow fast evaluation of the detectability of a random lesion. The theoretical results are used to design the regularization parameters to improve lesion detectability. We conducted computer-based Monte Carlo simulations to compare the proposed penalty function, conventional penalty function, and a penalty function for isotropic point spread function. The lesion detectability is measured by a channelized Hotelling observer. The results show that the proposed penalty function outperforms the other penalty functions for lesion detection. The relative improvement is dependent on the size of the lesion. However, we found that the penalty function optimized for a 5 mm lesion still outperforms the other two penalty functions for detecting a 14 mm lesion. Therefore, it is feasible to use the penalty function designed for small lesions in image reconstruction, because detection of large lesions is relatively easy.     

Evaluation of internal noise methods for Hotelling observer models

The inclusion of internal noise in model observers is a common method to allow for quantitative comparisons between human and model observer performance in visual detection tasks. In this article, we studied two different strategies for inserting internal noise into Hotelling model observers. In the first strategy, internal noise was added to the output of individual channels: (a) Independent nonuniform channel noise, (b) independent uniform channel noise. In the second strategy, internal noise was added to the decision variable arising from the combination of channel responses. The standard deviation of the zero mean internal noise was either constant or proportional to: (a) the decision variable's standard deviation due to the external noise, (b) the decision variable's variance caused by the external noise, (c) the decision variable magnitude on a trial to trial basis. We tested three model observers: square window Hotelling observer (HO), channelized Hotelling observer (CHO), and Laguerre-Gauss Hotelling observer (LGHO) using a four alternative forced choice (4AFC) signal known exactly but variable task with a simulated signal embedded in real x-ray coronary angiogram backgrounds. The results showed that the internal noise method that led to the best prediction of human performance differed across the studied model observers. The CHO model best predicted human observer performance with the channel internal noise. The HO and LGHO best predicted human observer performance with the decision variable internal noise. The present results might guide researchers with the choice of methods to include internal noise into Hotelling model observers when evaluating and optimizing medical image quality.     

Optimization of a tomosynthesis system for the detection of lung nodules

Mathematical observers that track human performance can be used to reduce the number of human observer studies needed to optimize imaging systems. The performance of human observers for the detection of a 3.6 mm lung nodule in anatomical backgrounds was measured as a function of varying tomosynthetic angle and compared with mathematical observers. The human observer results showed a dramatic increase in the percent of correct responses, from 80% in the projection images to 96% in the projection images with a tomosynthetic angle of just 3 degrees. This result suggests the potential usefulness of the scanned beam digital x-ray system for this application. Given the small number of images (40) used per tomosynthetic angle and the highly nonstationary statistical nature of the backgrounds, the nonprewhitening eye observer achieved a higher performance than the channelized Hotelling observer using a Laguerre-Gauss basis. The channelized Hotelling observer with internal noise and the eye filter matched to the projection data were shown to track human performance as the tomosynthetic angle changed. The validation of these mathematical observers extends their applicability to the optimization of tomosynthesis systems.     

Rapid calculation of detectability in Bayesian single photon emission computed tomography

We consider the calculation of lesion detectability using a mathematical model observer, the channelized Hotelling observer (CHO), in a signal-known-exactly/background-known-exactly detection task for single photon emission computed tomography (SPECT). We focus on SPECT images reconstructed with Bayesian maximum a posteriori methods. While model observers are designed to replace time-consuming studies using human observers, the calculation of CHO detectability is usually accomplished using a large number of sample images, which is still time consuming. We develop theoretical expressions for a measure of detectability, the signal-to-noise-ratio (SNR) of a CHO observer, that can be very rapidly evaluated. Key to our expressions are approximations to the reconstructed image covariance. In these approximations, we use methods developed in the PET literature, but modify them to reflect the different nature of attenuation and distance-dependent blur in SPECT. We validate our expressions with Monte Carlo methods. We show that reasonably accurate estimates of the SNR can be obtained at a computational expense equivalent to approximately two projection operations, and that evaluating SNR for subsequent lesion locations requires negligible additional computation.     

A channelized Hotelling observer study of lesion detection in SPECT MAP reconstruction using anatomical priors

In emission tomography, anatomical side information, in the form of organ and lesion boundaries, derived from intra-patient coregistered CT or MR scans can be incorporated into the reconstruction. Our interest is in exploring the efficacy of such side information for lesion detectability. To assess detectability we used the SNR of a channelized Hotelling observer and a signal-known exactly/background-known exactly detection task. In simulation studies, we incorporated anatomical side information into a SPECT MAP (maximum a posteriori) reconstruction by smoothing within but not across organ or lesion boundaries. A non-anatomical prior was applied by uniform smoothing across the entire image. We investigated whether the use of anatomical priors with organ boundaries alone or with perfect lesion boundaries alone would change lesion detectability relative to the case of a prior with no anatomical information. Furthermore, we investigated whether any such detectability changes for the organ-boundary case would be a function of the distance of the lesion to the organ boundary. We also investigated whether any detectability changes for the lesion-boundary case would be a function of the degree of proximity, i.e. a difference in the radius of the true functional lesion and the radius of the anatomical lesion boundary. Our results showed almost no detectability difference with versus without organ boundaries at any lesion-to-organ boundary distance. Our results also showed no difference in lesion detectability with and without lesion boundaries, and no variation of lesion detectability with degree of proximity.     

X-ray imaging optimization using virtual phantoms and computerized observer modelling

This study develops and demonstrates a realistic x-ray imaging simulator with computerized observers to maximize lesion detectability and minimize patient exposure. A software package, ViPRIS, incorporating two computational patient phantoms, has been developed for simulating x-ray radiographic images. A tomographic phantom, VIP-Man, constructed from Visible Human anatomical colour images is used to simulate the scattered portion using the ESGnrc Monte Carlo code. The primary portion of an x-ray image is simulated using the projection ray-tracing method through the Visible Human CT data set. To produce a realistic image, the software simulates quantum noise, blurring effects, lesions, detector absorption efficiency and other imaging artefacts. The primary and scattered portions of an x-ray chest image are combined to form a final image for computerized observer studies and image quality analysis. Absorbed doses in organs and tissues of the segmented VIP-Man phantom were also obtained from the Monte Carlo simulations. Approximately 25,000 simulated images and 2,500,000 data files were analysed using computerized observers. Hotelling and Laguerre-Gauss Hotelling observers are used to perform various lesion detection tasks. Several model observer tasks were used including SKE/BKE, MAFC and SKEV. The energy levels and fluence at the minimum dose required to detect a small lesion were determined with respect to lesion size, location and system parameters.     

Fully 4D motion-compensated reconstruction of cardiac SPECT images

In this paper, we investigate the benefits of a spatiotemporal approach for reconstruction of image sequences. In the proposed approach, we introduce a temporal prior in the form of motion compensation to account for the statistical correlations among the frames in a sequence, and reconstruct all the frames collectively as a single function of space and time. The reconstruction algorithm is derived based on the maximum a posteriori estimate, for which the one-step late expectation-maximization algorithm is used. We demonstrated the method in our experiments using simulated single photon emission computed tomography (SPECT) cardiac perfusion images. The four-dimensional (4D) gated mathematical cardiac-torso phantom was used for simulation of gated SPECT perfusion imaging with Tc-99m-sestamibi. In addition to bias-variance analysis and time activity curves, we also used a channelized Hotelling observer to evaluate the detectability of perfusion defects in the reconstructed images. Our experimental results demonstrated that the incorporation of temporal regularization into image reconstruction could significantly improve the accuracy of cardiac images without causing any significant cross-frame blurring that may arise from the cardiac motion. This could lead to not only improved detection of perfusion defects, but also improved reconstruction of the heart wall which is important for functional assessment of the myocardium.     

Detection performance analysis for time-of-flight PET

In this paper, we investigate the performance of time-of-flight (TOF) positron emission tomography (PET) in improving lesion detectability. We present a theoretical approach to compare lesion detectability of TOF versus non-TOF systems and perform computer simulations to validate the theoretical prediction. A single-ring TOF PET tomograph is simulated using SimSET software, and images are reconstructed in 2D from list-mode data using a maximum a posteriori method. We use a channelized Hotelling observer to assess the detection performance. Both the receiver operating characteristic (ROC) and localization ROC curves are compared for the TOF and non-TOF PET systems. We first studied the SNR gains for TOF PET with different scatter and random fractions, system timing resolutions and object sizes. We found that the TOF information improves the lesion detectability and the improvement is greater with larger fractions of randoms, better timing resolution and bigger objects. The scatters by themselves have little impact on the SNR gain after correction. Since the true system timing resolution may not be known precisely in practice, we investigated the effect of mismatched timing kernels and showed that using a mismatched kernel during reconstruction always degrades the detection performance, no matter whether it is narrower or wider than the real value. Using the proposed theoretical framework, we also studied the effect of lumpy backgrounds on the detection performance. Our results indicated that with lumpy backgrounds, the TOF PET still outperforms the non-TOF PET, but the improvement is smaller compared with the uniform background case. More specifically, with the same correlation length, the SNR gain reduces with bigger number of lumpy patches and greater lumpy amplitudes. With the same variance, the SNR gain reaches the minimum when the width of the Gaussian lumps is close to the size of the tumor.     

Computer aided detection of masses in mammography using subregion Hotelling observers

We propose to investigate the use of the subregion Hotelling observer for the basis of a computer aided detection scheme for masses in mammography. A database of 1320 regions of interest (ROIs) was selected from the DDSM database collected by the University of South Florida using the Lumisys scanner cases. The breakdown of the cases was as follows: 656 normal ROIs, 307 benign ROIs, and 357 cancer ROIs. Each ROI was extracted at a size of 1024 x 1024 pixels and sub-sampled to 128 x 128 pixels. For the detection task, cancer and benign cases were considered positive and normal was considered negative. All positive cases had the lesion centered in the ROI. We chose to investigate the subregion Hotelling observer as a classifier to detect masses. The Hotelling observer incorporates information about the signal, the background, and the noise correlation for prediction of positive and negative and is the optimal detector when these are known. For our study, 225 subregion Hotelling observers were set up in a 15 x 15 grid across the center of the ROIs. Each separate observer was designed to "observe," or discriminate, an 8 x 8 pixel area of the image. A leave one out training and testing methodology was used to generate 225 "features," where each feature is the output of the individual observers. The 225 features derived from separate Hotelling observers were then narrowed down by using forward searching linear discriminants (LDs). The reduced set of features was then analyzed using an additional LD with receiver operating characteristic (ROC) analysis. The 225 Hotelling observer features were searched by the forward searching LD, which selected a subset of 37 features. This subset of 37 features was then analyzed using an additional LD, which gave a ROC area under the curve of 0.9412 +/- 0.006 and a partial area of 0.6728. Additionally, at 98% sensitivity the overall classifier had a specificity of 55.9% and a positive predictive value of 69.3%. Preliminary results suggest that using subregion Hotelling observers in combination with LDs can provide a strong backbone for a CAD scheme to help radiologists with detection. Such a system could be used in conjunction with CAD systems for false positive reduction.     

Diagnosis of Parkinsonian disorders using a channelized Hotelling observer model: proof of principle

Imaging dopamine transporters using PET and SPECT probes is a powerful technique for the early diagnosis of Parkinsonian disorders. In order to perform automated accurate diagnosis of these diseases, a channelized Hotelling observer (CHO) based model was developed and evaluated using the SPECT tracer [Tc-99m]TRODAT-1. Computer simulations were performed using a digitized striatal phantom to characterize early stages of the disease (20 lesion-present cases with varying lesion size and contrast). Projection data, modeling the effects of attenuation and geometric response function, were obtained for each case. Statistical noise levels corresponding to those observed clinically were added to the projection data to obtain 100 noise realizations for each case. All the projection data were reconstructed, and a subset of the transaxial slices containing the striatum was summed and used for further analysis. CHO models, using the Laguerre-Gaussian functions as channels, were designed for two cases: (1) By training the model using individual lesion-present samples and (2) by training the model using pooled lesion-present samples. A decision threshold obtained for each CHO model was used to classify the study population (n = 40). It was observed that individual lesion trained CHO models gave high diagnostic accuracy for lesions that were larger than those used to train the model and vice-versa. On the other hand, the pooled CHO model was found to give a high diagnostic accuracy for all the lesion cases (average diagnostic accuracy = 0.95 +/- 0.07; p < 0.0001 Fisher's exact test). Based on our results, we conclude that a CHO model has the potential to provide early and accurate diagnosis of Parkinsonian disorders, thereby improving patient management.     

基于CT模拟数据研究多层CHO的低对比检测性能

【摘 要】 目的:基于CT模拟数据,比较多层通道化Hotelling观察器（CHO）和单层CHO对低对比物体的检测性能。 方法:利用计算机模拟产生有、无信号两类三维图像,分别改变信号幅度及尺寸,研究单层、多层CHO对不同幅度、尺寸的信号检测性。通过比较观察者性能指标PC,定量评估多层CHO较单层CHO的低对比检测能力。 结果:当信号尺寸为2.5 mm,信号幅度分别为2.5、5.0、10.0 HU时,PC分别为0.79、0.96、1.00,层数为10时,多层CHO性能PC达饱和,对应单层CHO性能PC分别为0.67、0.78、0.93,性能分别提升了17.91%、23.08%、7.53%;当信号幅度固定为2.5 HU,信号尺寸分别为2.0、2.5、3.5、5.0、10.0 mm时,多层CHO在层数分别为8、10、8、6、3时,PC达最大,分别为0.69、0.79、0.95、0.99、1.00,对应单层CHO性能PC分别为0.63、0.67、0.74、0.86、0.96,性能分别提升9.5%、17.9%、28.4%、15.1%、4.2%。 结论:由于综合多层信息制定决策,多层CHO相比于单层CHO对低对比物体的检测能力更强。但利用多层CHO进行图像质量评估时,层数选择至关重要,应根据信号幅度、尺寸、背景噪音等因素选择合适的层数。     

CHO数字观察器在PET图像滤波方法评估中的应用

在临床应用中需要限制扫描时间和药物剂量,这往往会使正电子发射断层扫描(PET)的图像的分辨率变低,噪声变多。为提供可供临床诊断的图像,去噪是一个必须的手段,而在重建后增加一个滤波器是目前最常用的去噪方法。因此对不同滤波器滤波效果的比较是PET图像重建中的重要环节,其中最关键的是滤波参数的选取。目前采用的信噪比(SNR)以及恢复系数(RC)等评估方法可以用来非定量地选取参数,研究者们只能凭经验选取最优参数。而通道化霍特林观察器(CHO)作为一个比较通用的数字观察器,已被用于与PET图像质量相关的各种参数的选择,如重建算法参数、系统设计参数、临床协议参数等,然而其在评估不同滤波方法对图像重建质量的影响中的应用研究还比较少。通过比较CHO计算得到的ROC(receiver operating characteristic)曲线下面积(area under the ROC curve,AUC),选择两种常用的滤波器(即高斯滤波器和非局部均值(Non-Local Mean, NLM)滤波器)的最优参数,并评估它们在PET中的滤波效果。结果表明,对于13 mm球体,σ为1.1~1.4的高斯滤波器和f为0.5~0.9的NLM滤波器可以达到最大的检测能力值,而对于10 mm球体,σ为1.4~2.0的高斯滤波器和f为0.5~0.9的NLM滤波器可以达到最大的检测能力值。虽然两个滤波器所对应的AUC值都能高达0.9,但是NLM滤波器的AUC值高于高斯滤波器。通过IEC图像和病人图像也能发现,NLM滤波后的PET图像中的亮点比高斯滤波的更加清晰,噪声更少。该结论和传统滤波器评估方法得到的结论一致,这说明在PET的病灶检测任务中,CHO能够准确地比较这两种滤波器的性能。     

Hardware assessment using the multi-module, multi-resolution system (M3R): a signal-detection study

The multi-module, multi-resolution system (M3R) is used for hardware assessment in objective, task-based signal detection studies in projection data. A phantom capable of generating multiple realizations of a random textured background is introduced. Measured backgrounds from this phantom are used along with simulated lumpy and uniform backgrounds to investigate signal-to-noise ratio as a function of exposure time. Results are shown to agree with theoretical predictions, exhibiting a power-law like dependence previously seen for studies performed either in simulation or without an imaging system, and help validate the use of simulated lumpy backgrounds in observer studies. A second study looks at signal-detection performance, measured by AUC (area under the receiver operating characteristic curve), in lumpy backgrounds for 20 M3R aperture combinations as a function of lump size and signal size. Observer performance reveals an improvement in AUC for certain ranges of signal and lump combinations through the use of multiplexed, multiple-pinhole apertures, indicating a need for task-specific aperture optimization. The channelized Hotelling observer is used with Laguerre-Gauss channels for both observer studies. Methods for selection of number of channels and channel width are discussed.     

Effect of dose reduction on the detection of mammographic lesions: a mathematical observer model analysis

The effect of reduction in dose levels normally used in mammographic screening procedures on the detection of breast lesions were analyzed. Four types of breast lesions were simulated and inserted into clinically-acquired digital mammograms. Dose reduction by 50% and 75% of the original clinically-relevant exposure levels were simulated by adding corresponding simulated noise into the original mammograms. The mammograms were converted into luminance values corresponding to those displayed on a clinical soft-copy display station and subsequently analyzed by Laguerre-Gauss and Gabor channelized Hotelling observer models for differences in detectability performance with reduction in radiation dose. Performance was measured under a signal known exactly but variable detection task paradigm in terms of receiver operating characteristics (ROC) curves and area under the ROC curves. The results suggested that luminance mapping of digital mammograms affects performance of model observers. Reduction in dose levels by 50% lowered the detectability of masses with borderline statistical significance. Dose reduction did not have a statistically significant effect on detection of microcalcifications. The model results indicate that there is room for optimization of dose level in mammographic screening procedures.     

Incorporation of a Laguerre–Gauss Channelized Hotelling Observer for False-Positive Reduction in a Mammographic Mass CAD System

Previously, we developed a simple Laguerre-Gauss (LG) channelized Hotelling observer (CHO) for incorporation into our mass computer-aided detection (CAD) system. This LG-CHO was trained using initial detection suspicious region data and was empirically optimized for free parameters. For the study presented in this paper, we wish to create a more optimal mass detection observer based on a novel combination of LG channels. A large set of LG channels with differing free parameters was created. Each of these channels was applied to the suspicious regions, and an output test statistic was determined. A stepwise feature selection algorithm was used to determine which LG channels would combine best to detect masses. These channels were combined using a HO to create a single template for the mass CAD system. Results from free-response receiver operating characteristic curves demonstrated that the incorporation of the novel LG-CHO into the CAD system slightly improved performance in high-sensitivity regions.     

A mathematical model platform for optimizing a multiprojection breast imaging system

Multiprojection imaging is a technique in which a plurality of digital radiographic images of the same patient are acquired within a short interval of time from slightly different angles. Information from each image is combined to determine the final diagnosis. Projection data are either reconstructed into slices as in the case of tomosynthesis or analyzed directly as in the case of multiprojection correlation imaging technique, thereby avoiding reconstruction artifacts. In this study, the authors investigated the optimum geometry of acquisitions of a multiprojection breast correlation imaging system in terms of the number of projections and their total angular span that yield maximum performance in a task that models clinical decision. Twenty-five angular projections of each breast from 82 human subjects in our breast tomosynthesis database were each supplemented with a simulated 3 mm mass. An approach based on Laguerre-Gauss channelized Hotelling observer was developed to assess the detectability of the mass in terms of receiver operating characteristic (ROC) curves. Two methodologies were developed to integrate results from individual projections into one combined ROC curve as the overall figure of merit. To optimize the acquisition geometry, different components of acquisitions were changed to investigate which one of the many possible configurations maximized the area under the combined ROC curve. Optimization was investigated under two acquisition dose conditions corresponding to a fixed total dose delivered to the patient and a variable dose condition, based on the number of projections used. In either case, the detectability was dependent on the number of projections used, the total angular span of those projections, and the acquisition dose level. In the first case, the detectability approximately followed a bell curve as a function of the number of projections with the maximum between 8 and 16 projections spanning angular arcs of about 23 degrees-45 degrees, respectively. In the second case, the detectability increased with the number of projections approaching an asymptote at 11-17 projections for an angular span of about 45 degrees. These results indicate the inherent information content of the multi-projection image data reflecting the relative role of quantum and anatomical noise in multiprojection breast imaging. The optimization scheme presented here may be applied to any multiprojection imaging modalities and may be extended by including reconstruction in the case of digital breast tomosynthesis and breast computed tomography.     

Comparison study of temporal regularization methods for fully 5D reconstruction of cardiac gated dynamic SPECT

Temporal regularization plays a critical role in cardiac gated dynamic SPECT reconstruction, of which the goal is to obtain an image sequence from a single acquisition which simultaneously shows both cardiac motion and tracer distribution change over the course of imaging (termed 5D). In our recent work, we explored two different approaches for temporal regularization of the dynamic activities in gated dynamic reconstruction without the use of fast camera rotation: one is the dynamic EM (dEM) approach which is imposed on the temporal trend of the time activity of each voxel, and the other is a B-spline modeling approach in which the time activity is regulated by a set of B-spline basis functions. In this work, we extend the B-spline approach to fully 5D reconstruction and conduct a thorough quantitative comparison with the dEM approach. In the evaluation of the reconstruction results, we apply a number of quantitative measures on two major aspects of the reconstructed dynamic images: (1) the accuracy of the reconstructed activity distribution in the myocardium and (2) the ability of the reconstructed dynamic activities to differentiate perfusion defects from normal myocardial wall uptake. These measures include the mean square error (MSE), bias-variance analysis, accuracy of time-activity curves (TAC), contrast-to-noise ratio of a defect, composite kinetic map of the left ventricle wall and perfusion defect detectability with channelized Hotelling observer. In experiments, we simulated cardiac gated imaging with the NURBS-based cardiac-torso phantom and Tc99m-Teboroxime as the imaging agent, where acquisition with the equivalent of only three full camera rotations was used during the imaging period. The results show that both dEM and B-spline 5D could achieve similar overall accuracy in the myocardium in terms of MSE. However, compared to dEM 5D, the B-spline approach could achieve a more accurate reconstruction of the voxel TACs; in particular, B-spline 5D could achieve a much smaller bias level in the early uptake stage of the imaging period. Furthermore, it could allow better separation of the perfusion defect from the normal at both the early and the late stages of the imaging period.     

La théorie statistique de la décision: application à l''étude du rapport signal sur bruit en tomographie d''émission monophotoniqueStatistical decision theory: application to the study of signal to noise ratio in single photon emission tomography

The statistical decision theory (SDT) is applied in all direct imaging modalities. The detection stage can be quantified exactly from the ideal observer concept. The ideal observer depend on the noise power spectrum (NPS). An expression of NPS in single photon emission tomography (SPECT) of the uncertainty over the number of counts is derived in order to estimate the noise and the signal to noise ratio (SNR). The reconstruction algorithm is the method of back-projection of filtered projection. It is shown that derived expressions of noise and SNR compare well with expressions derived from classical approaches. So, we can affirm that the SDT can be considered in tomography (indirect imaging).     

Spatiotemporal noise covariance estimation from limited empirical magnetoencephalographic data

The performance of parametric magnetoencephalography (MEG) and electroencephalography (EEG) source localization approaches can be degraded by the use of poor background noise covariance estimates. In general, estimation of the noise covariance for spatiotemporal analysis is difficult mainly due to the limited noise information available. Furthermore, its estimation requires a large amount of storage and a one-time but very large (and sometimes intractable) calculation or its inverse. To overcome these difficulties, noise covariance models consisting of one pair or a sum of multi-pairs of Kronecker products of spatial covariance and temporal covariance have been proposed. However, these approaches cannot be applied when the noise information is very limited, i.e., the amount of noise information is less than the degrees of freedom of the noise covariance models. A common example of this is when only averaged noise data are available for a limited prestimulus region (typically at most a few hundred milliseconds duration). For such cases, a diagonal spatiotemporal noise covariance model consisting of sensor variances with no spatial or temporal correlation has been the common choice for spatiotemporal analysis. In this work, we propose a different noise covariance model which consists of diagonal spatial noise covariance and Toeplitz temporal noise covariance. It can easily be estimated from limited noise information, and no time-consuming optimization and data-processing are required. Thus, it can be used as an alternative choice when one-pair or multi-pair noise covariance models cannot be estimated due to lack of noise information. To verify its capability we used Bayesian inference dipole analysis and a number of simulated and empirical datasets. We compared this covariance model with other existing covariance models such as conventional diagonal covariance, one-pair and multi-pair noise covariance models, when noise information is sufficient to estimate them. We found that our proposed noise covariance model yields better localization performance than a diagonal noise covariance, while it performs slightly worse than one-pair or multi-pair noise covariance models - although these require much more noise information. Finally, we present some localization results on median nerve stimulus empirical MEG data for our proposed noise covariance model.