Noise estimation in single- and multiple-coil magnetic resonance data based on statistical models

Noise estimation is a challenging task in magnetic resonance imaging (MRI), with applications in quality assessment, filtering or diffusion tensor estimation. Main noise estimators based on the Rician model are revisited and classified in this article, and new useful methods are proposed. Additionally, all the surveyed estimators are extended to the noncentral chi model, which applies to multiple-coil MRI and some important parallel imaging algorithms for accelerated acquisitions. The proposed new noise estimation procedures, based on the distribution of local moments, show better performance in terms of smaller variance and unbiased estimation over a wide range of experiments, with the additional advantage of not needing to explicitly segment the background of the image.     

An MRI denoising method using image data redundancy and local SNR estimation

This paper presents an LMMSE-based method for the three-dimensional (3D) denoising of MR images assuming a Rician noise model. Conventionally, the LMMSE method estimates the noise-less signal values using the observed MR data samples within local neighborhoods. This is not an efficient procedure to deal with this issue while the 3D MR data intrinsically includes many similar samples that can be used to improve the estimation results. To overcome this problem, we model MR data as random fields and establish a principled way which is capable of choosing the samples not only from a local neighborhood but also from a large portion of the given data. To follow the similar samples within the MR data, an effective similarity measure based on the local statistical moments of images is presented. The parameters of the proposed filter are automatically chosen from the estimated local signal-to-noise ratio. To further enhance the denoising performance, a recursive version of the introduced approach is also addressed. The proposed filter is compared with related state-of-the-art filters using both synthetic and real MR datasets. The experimental results demonstrate the superior performance of our proposal in removing the noise and preserving the anatomical structures of MR images.     

Generalized total variation-based MRI Rician denoising model with spatially adaptive regularization parameters

Magnetic resonance imaging (MRI) is an outstanding medical imaging modality but the quality often suffers from noise pollution during image acquisition and transmission. The purpose of this study is to enhance image quality using feature-preserving denoising method. In current literature, most existing MRI denoising methods did not simultaneously take the global image prior and local image features into account. The denoising method proposed in this paper is implemented based on an assumption of spatially varying Rician noise map. A two-step wavelet-domain estimation method is developed to extract the noise map. Following a Bayesian modeling approach, a generalized total variation-based MRI denoising model is proposed based on global hyper-Laplacian prior and Rician noise assumption. The proposed model has the properties of backward diffusion in local normal directions and forward diffusion in local tangent directions. To further improve the denoising performance, a local variance estimator-based method is introduced to calculate the spatially adaptive regularization parameters related to local image features and spatially varying noise map. The main benefit of the proposed method is that it takes full advantage of the global MR image prior and local image features. Numerous experiments have been conducted on both synthetic and real MR data sets to compare our proposed model with some state-of-the-art denoising methods. The experimental results have demonstrated the superior performance of our proposed model in terms of quantitative and qualitative image quality evaluations.     

Influence of multichannel combination, parallel imaging and other reconstruction techniques on MRI noise characteristics

The statistical properties of background noise such as its standard deviation and mean value are frequently used to estimate the original noise level of the acquired data. This requires the knowledge of the statistical intensity distribution of the background signal, that is, the probability density of the occurrence of a certain signal intensity. The influence of many new MRI techniques and, in particular, of various parallel-imaging methods on the noise statistics has neither been rigorously investigated nor experimentally demonstrated yet. In this study, the statistical distribution of background noise was analyzed for MR acquisitions with a single-channel and a 32-channel coil, with sum-of-squares (SoS) and spatial-matched-filter (SMF) data combination, with and without parallel imaging using k-space and image-domain algorithms, with real-part and conventional magnitude reconstruction and with several reconstruction filters. Depending on the imaging technique, the background noise could be described by a Rayleigh distribution, a noncentral chi-distribution or the positive half of a Gaussian distribution. In particular, the noise characteristics of SoS-reconstructed multichannel acquisitions (with k-space-based parallel imaging or without parallel imaging) differ substantially from those with image-domain parallel imaging or SMF combination. These effects must be taken into account if mean values or standard deviations of background noise are employed for data analysis such as determination of local noise levels. Assuming a Rayleigh distribution as in conventional MR images or a noncentral chi-distribution for all multichannel acquisitions is invalid in general and may lead to erroneous estimates of the signal-to-noise ratio or the contrast-to-noise ratio.     

Effective noise estimation and filtering from correlated multiple-coil MR data

Modern magnetic resonance (MR) imaging protocols based on multiple-coil acquisitions have carried on a new attention to noise and signal statistical modeling, as long as most of the existing techniques for data processing are model based. In particular, nonaccelerated multiple-coil and GeneRalized Autocalibrated Partially Parallel Acquisitions (GRAPPA) have brought noncentral-χ (nc-χ) statistics into stake as a suitable substitute for traditional Rician distributions. However, this model is only valid when the signals received by each coil are roughly uncorrelated. The recent literature on this topic suggests that this is often not the case, so nc-χ statistics are in principle not adequate. Fortunately, such model can be adapted through the definition of a set of effective parameters, namely, an effective noise power (greater than the actual power of thermal noise in the Radio Frequency receiver) and an effective number of coils (smaller than the actual number of RF receiving coils in the system). The implications of these artifacts in practical algorithms have not been discussed elsewhere. In the present paper, we aim to study their actual impact and suggest practical rules to cope with them. We define the main noise parameters in this context, introducing a new expression for the effective variance of noise which is of capital importance for the two image processing problems studied: first, we propose a new method to estimate the effective variance of noise from the composite magnitude signal of MR data when correlations are assumed. Second, we adapt several model-based image denoising techniques to the correlated case using the noise estimation techniques proposed. We show, through a number of experiments with synthetic, phantom, and in vivo data, that neglecting the correlated nature of noise in multiple-coil systems implies important errors even in the simplest cases. At the same time, the proper statistical characterization of noise through effective parameters drives to improved accuracy (both qualitatively and quantitatively) for both of the problems studied.     

3D MR image denoising using rough set and kernel PCA method

In this paper, we have presented a two stage method, using kernel principal component analysis (KPCA) and rough set theory (RST), for denoising volumetric MRI data. A rough set theory (RST) based clustering technique has been used for voxel based processing. The method groups similar voxels (3D cubes) using class and edge information derived from noisy input. Each clusters thus formed now represented via basis vector. These vectors now projected into kernel space and PCA is performed in the feature space. This work is motivated by idea that under Rician noise MRI data may be non-linear and kernel mapping will help to define linear separator between these clusters/basis vectors thus used for image denoising. We have further investigated various kernels for Rician noise for different noise levels. The best kernel is then selected on the performance basis over PSNR and structure similarity (SSIM) measures. The work has been compared with state-of-the-art methods under various measures for synthetic and real databases.     

Automatic correction of in-plane bulk motion artifacts in self-navigated radial MRI

Radial MRI is typically used for scans that are sensitive to unavoidable motion. While the translational motion artifact can be easily removed from the radial trajectory data by phase correction, correction of rotational motion still remains a challenge in radial MRI. We present a novel method to refocus the image corrupted by view-to-view motion in the view-interleaved radial MRI data. In this method, the error in rotational view angles was modeled as a polynomial function of the view order and the model parameters were estimated by minimizing the self-navigator image metrics such as image entropy, gradient entropy, normalized gradient squared and mean square difference. Translational motion correction was conducted by aligning the projection profiles. Simulation studies were conducted to demonstrate the robustness of both translational and rotational motion correction methods in different noise levels. The proposed method was successfully applied to correct for motion of healthy subjects. Substantial motion correction with relative error of less than 5% was achieved by using either first- or second-order model with the image metrics. This study demonstrates the potential of the method for motion-sensitive applications.     

An anisotropic images segmentation and bias correction method

Intensity inhomogeneities cause considerable difficulty in the quantitative analysis of magnetic resonance (MR) images. Thus, bias field correction is a necessary step before quantitative analysis of MR data can be undertaken. This paper presents an anisotropic approach to bias correction and segmentation for images with intensity inhomogeneities and noise. Intensity-based methods are usually applied to estimate the bias field; however, most of them only concern the intensity information. When the images have noise or slender topological objects, these methods cannot obtain accurate results or bias fields. We use structure information to construct an anisotropic Gibbs field and combine the anisotropic Gibbs field with the Bayesian framework to segment images while estimating the bias fields. Our method is able to capture bias of quite general profiles. Moreover, it is robust to noise and slender topological objects. The proposed method has been used for images of various modalities with promising results.     

Noise figure characterization of preamplifiers at NMR frequencies

A method for characterizing the noise figure of preamplifiers at NMR frequencies is presented. The noise figure of preamplifiers as used for NMR and MRI detection varies with source impedance and with the operating frequency. Therefore, to characterize a preamplifier's noise behavior, it is necessary to perform noise measurements at the targeted frequency while varying the source impedance with high accuracy. At high radiofrequencies, such impedance variation is typically achieved with transmission-line tuners, which however are not available for the relatively low range of typical NMR frequencies. To solve this issue, this work describes an alternative approach that relies on lumped-element circuits for impedance manipulation. It is shown that, using a fixed-impedance noise source and suitable ENR correction, this approach permits noise figure characterization for NMR and MRI purposes. The method is demonstrated for two preamplifiers, a generic BF998 MOSFET module and an MRI-dedicated, integrated preamplifier, which were both studied at 128MHz, i.e., at the Larmor frequency of protons at 3 Tesla. Variations in noise figure of 0.01dB or less over repeated measurements reflect high precision even for small noise figures in the order of 0.4dB. For validation, large sets of measured noise figure values are shown to be consistent with the general noise-parameter model of linear two-ports. Finally, the measured noise characteristics of the superior preamplifier are illustrated by SNR measurements in MRI data.     

Motion predicted online dynamic MRI reconstruction from partially sampled k-space data

In this work we address the problem of reconstructing dynamic MRI sequences in an online fashion, i.e. reconstructing the current frame given that the previous frames have been already reconstructed. The reconstruction consists of a prediction and a correction step. The prediction step is based on an Auto-Regressive AR(1) model. Assuming that the prediction is good, the difference between the predicted frame and the actual frame (to be reconstructed) will be sparse. In the correction step, the difference between the predicted frame and the actual frame is estimated from partially sampled K-space data via a sparsity promoting least squares minimization problem. We have compared the proposed method with state-of-the-art methods in online dynamic MRI reconstruction. The experiments have been carried out on 2D and 3D Dynamic Contrast Enhanced (DCE) MRI datasets. Results show that our method yields the least reconstruction error.     

Signal quality improvement of holographic data storage using adaptive two-dimensional filter

Holographic data storage is being widely studied for the purpose of developing next-generation large optical memories. A prospective use of this type of memory is in building image archives in large-scale data centers. In particular, demand for energy conservation at data centers, and therefore for holographic data storage, is growing. In holographic data storage, interference between bits occurs owing to wave aberration in the optical system, shrinkage of the medium, and crosstalk noise from neighboring holograms during multiplex recording; as a result of the interference, the reproduced image deteriorates and the bit error rate (BER) increases. In this study, to reduce the BER in both off-axis-type recording and coaxial-type recording, a two-dimensional finite impulse response (FIR) filter is applied to a reproduced image that has been recorded by angle multiplex recording and shift multiplex recording. First, for the optimization of the FIR filter coefficients, the linear minimum mean square error (LMMSE) method is applied; this method optimizes the coefficients by reducing the BER. Furthermore, for evaluating the optimization performance of the LMMSE method, the optimization performance is compared with that of the real-coded genetic algorithm (RCGA), which has the capability to search a wide range of coefficients. The optimization by the LMMSE method has been found to be excellent for off-axis-type recording but not for coaxial-type recording. It is speculated that this is because of the brightness irregularity in the reproduced image, resulting from crosstalk. On the other hand, a marked reduction in the BER is observed using the RCGA, despite the brightness irregularity. In this study, the effectiveness of the LMMSE method for signals recorded by coaxial-type recording, in which large brightness irregularity is expected, is examined using automatic gain control (AGC). It is found that the application of AGC reduces the BER even in the case of coaxial-type recording.     

Respiratory noise correction using phase information

Respiratory noise is a confounding factor in functional magnetic resonance imaging (MRI) data analysis. A novel method called Respiratory noise Correction using Phase information is proposed to retrospectively correct for the respiratory noise in functional MRI (fMRI) time series. It is demonstrated that the respiratory movement and the phase of functional MRI images are highly correlated in time. The signal fluctuation due to respiratory movements can be effectively estimated from the phase variation and removed from the functional MRI time series using a Wiener filtering technique. In our experiments, this new method is compared with RETROICOR, which requires recording respiration signal simultaneously in an fMRI experiment. The two techniques show comparable performance with respect to the respiratory noise correction for fMRI time series. However, this technique is more advantageous because there is no need for monitoring the subjects’ respiration or changing functional MRI protocols. This technique is also potentially useful for correcting respiratory noise from abnormal breathing or when the respiration is not periodic.     

Wavelet domain non-linear filtering for MRI denoising

Feature-preserved denoising is of great interest in medical image processing. This article presents a wavelet-based bilateral filtering scheme for noise reduction in magnetic resonance images. Undecimated wavelet transform is employed to provide effective representation of the noisy coefficients. Bilateral filtering of the approximate coefficients improves the denoising efficiency and effectively preserves the edge features. Denoising is done in the square magnitude domain, where the noise tends to be signal independent and is additive. The proposed method has been adapted specifically to Rician noise. The visual and the diagnostic quality of the denoised image is well preserved. The quantitative and the qualitative measures used as the quality metrics demonstrate the ability of the proposed method for noise suppression.     

Laplacian based non-local means denoising of MR images with Rician noise

Magnetic Resonance (MR) image is often corrupted with a complex white Gaussian noise (Rician noise) which is signal dependent. Considering the special characteristics of Rician noise, we carry out nonlocal means denoising on squared magnitude images and compensate the introduced bias. In this paper, we propose an algorithm which not only preserves the edges and fine structures but also performs efficient denoising. For this purpose we have used a Laplacian of Gaussian (LoG) filter in conjunction with a nonlocal means filter (NLM). Further, to enhance the edges and to accelerate the filtering process, only a few similar patches have been preselected on the basis of closeness in edge and inverted mean values. Experiments have been conducted on both simulated and clinical data sets. The qualitative and quantitative measures demonstrate the efficacy of the proposed method.     

星载差分吸收光谱仪噪声分析及处理方法

噪声对星载大气痕量气体差分吸收光谱仪信噪比具有重大影响,其是衡量光谱仪成像质量和痕量气体反演能力的标准。为量化并去除光谱仪系统噪声从而提高信噪比,分析了光谱仪噪声来源并给出了相应噪声模型,在此基础上建立了光谱仪信噪比模型。研究了入射光强度、积分时间和系统增益对系统信噪比的影响。通过光谱仪辐射定标试验数据对不同工作模式和不同参数与信噪比的关系进行了对比验证。并提出主要系统噪声的处理方法：利用线性拟合取截距法确定系统偏置噪声;在地面利用暗电流温度相关性获得温度修正因子实现载荷在轨暗电流校正;在探测器响应线性范围内利用两点校正法对PRNU噪声进行修正。结果表明：全幅成像模式下,可见1通道电子学偏置噪声响应DN值2 625,可见2通道电子学偏置噪声响应DN值2 763;暗电流噪声在CCD成像面温度高于0 ℃时占主要部分,温度下降至-20 ℃时其余噪声起主导作用,验证了CCD最佳制冷温度;光谱仪信噪比随着入射光响应和积分时间的增加而增大,系统增益不会影响信噪比;PRNU噪声通过校正得到明显改善,由3.32%下降到0.47%,提高了光谱仪成像质量。该噪声分析和处理方法对后期光谱数据的痕量气体反演提供帮助。     

Thermodynamics of ion-containing polymer blends and block copolymers

We develop a theory for the thermodynamics of ion-containing polymer blends and diblock copolymers, taking polyethylene oxide (PEO), polystyrene and lithium salts as an example. We account for the tight binding of Li^{+} ions to the PEO, the preferential solvation energy of anions in the PEO domain, the translational entropy of anions, and the ion-pair equilibrium between EO-complexed Li^{+} and anion. Our theory is able to predict many features observed in experiments, particularly the systematic dependence in the effective χ parameter on the size of the anions. Furthermore, comparison with the observed linear dependence in the effective χ on salt concentration yields an upper limit for the binding constant of the ion pair.     

胡杨叶片水分含量的近红外光谱检测

胡杨叶片水分含量是评价胡杨健康状况的重要指标。光谱检测法是一种常用的手段,但在近红外光谱的测量过程中,在一定程度上必然受到仪器噪声、摆放形态差异和环境的干扰,为避免噪声、散射对近红外光谱的影响,减少数据维数,采用多元散射校正(MSC)算法对原始光谱数据进行预处理,去除散射和基线漂移的影响,增加了光谱数据的信噪比,使有效光谱信息较为明显,谱带特征得到加强,有利于特征波长的选择。为降低模型的复杂度,防止过拟合现象,减小共线性影响,利用连续投影算法(SPA)进行特征变量选择,并通过多元线性回归模型,分析各个波长模拟的残差平方,评价各个波长的贡献,剔除贡献较小的波长,最终获得用于建模的特征波长,改善建模条件。最后使用偏最小二乘回归算法建立胡杨叶片水分含量检测模型。实验表明,直接使用原始光谱,利用SPA算法筛选变量个数为6个,模型预测精度为90.144%,相关系数r=0.674 24,RMSE=0.021 434,MSC处理后,利用SPA算法选定最终变量数为5个,预测精度为97.734%,相关系数r=0.781 63,RMSE=0.016 776。MSC和SPA算法有效的消除了散射噪声、减小了共线性干扰,模型的预测精度和相关性增加,误差减小,可用于胡杨叶片水分的快速无损检测,而且对其他作物叶片水分检测也具有一定的借鉴意义。     

Self-consistent tomography of temporally correlated errors

The error model of a quantum computer is essential for optimizing quantum algorithms to minimize the impact of errors using quantum error correction or error mitigation. Noise with temporal correlations, e.g. low-frequency noise and context-dependent noise, is common in quantum computation devices and sometimes even significant. However, conventional tomography methods have not been developed for obtaining an error model describing temporal correlations. In this paper,we propose self-consistent tomography protocols to obtain a model of temporally correlated errors, and we demonstrate that our protocols are efficient for low-frequency noise and context-dependent noise.     

A wavelet-based method for improving signal-to-noise ratio and contrast in MR images

Magnetic resonance (MR) images acquired with fast measurement often display poor signal-to-noise ratio (SNR) and contrast. With the advent of high temporal resolution imaging, there is a growing need to remove these noise artifacts. The noise in magnitude MR images is signal-dependent (Rician), whereas most de-noising algorithms assume additive Gaussian (white) noise. However, the Rician distribution only looks Gaussian at high SNR. Some recent work by Nowak employs a wavelet-based method for de-noising the square magnitude images, and explicitly takes into account the Rician nature of the noise distribution. In this article, we apply a wavelet de-noising algorithm directly to the complex image obtained as the Fourier transform of the raw k-space two-channel (real and imaginary) data. By retaining the complex image, we are able to de-noise not only magnitude images but also phase images. A multiscale (complex) wavelet-domain Wiener-type filter is derived. The algorithm preserves edges better when the Haar wavelet rather than smoother wavelets, such as those of Daubechies, are used. The algorithm was tested on a simulated image to which various levels of noise were added, on several EPI image sequences, each of different SNR, and on a pair of low SNR MR micro-images acquired using gradient echo and spin echo sequences. For the simulated data, the original image could be well recovered even for high values of noise (SNR approximately 0 dB), suggesting that the present algorithm may provide better recovery of the contrast than Nowak's method. The mean-square error, bias, and variance are computed for the simulated images. Over a range of amounts of added noise, the present method is shown to give smaller bias than when using a soft threshold, and smaller variance than a hard threshold; in general, it provides a better bias-variance balance than either hard or soft threshold methods. For the EPI (MR) images, contrast improvements of up to 8% (for SNR = 33 dB) were found. In general, the improvement in contrast was greater the lower the original SNR, for example, up to 50% contrast improvement for SNR of about 20 dB in micro-imaging. Applications of the algorithm to the segmentation of medical images, to micro-imaging and angiography (where the correct preservation of phase is important for flow encoding to be possible), as well as to de-noising time series of functional MR images, are discussed.     

A robust algorithm for high-resolution dynamic MRI based on the partially separable functions model

A recently developed partially separable functions (PSF) model can be used to generate high-resolution dynamic magnetic resonance imaging (MRI). However, this method could not robustly reconstruct high-quality MR images because the estimation of the PSF parameters is often interfered by the noise of the sampled MR data. To improve the robustness of MRI reconstruction using the PSF model, we proposed a new algorithm to estimate the PSF parameters by jointly using robust principal component analysis and modified truncated singular value decomposition regularization methods, instead of using the least square fitting method in the original PSF model. The experiment results of in vivo cardiac MRI demonstrated that the proposed algorithm can robustly reconstruct dynamic MR images with higher signal-to-noise ratio and clearer anatomical structures in comparison with the previous PSF model.