Cheng Guan Koay

A signal transformational framework for breaking the noise floor and its applications in MRI

A long-standing problem in magnetic resonance imaging (MRI) is the noise-induced bias in the magnitude signals. This problem is particularly pressing in diffusion MRI at high diffusion-weighting. In this paper, we present a three-stage scheme to solve this problem by transforming noisy nonCentral Chi signals to noisy Gaussian signals. A special case of nonCentral Chi distribution is the Rician distribution. In general, the Gaussian-distributed signals are of interest rather than the Gaussian-derived (e.g., Rayleigh, Rician, and nonCentral Chi) signals because the Gaussian-distributed signals are generally more amenable to statistical treatment through the principle of least squares. Monte Carlo simulations were used to validate the statistical properties of the proposed framework. This scheme opens up the possibility of investigating the low signal regime (or high diffusion-weighting regime in the case of diffusion MRI) that contains potentially important information about biophysical processes and structures of the brain.

Investigation of anomalous estimates of tensor-derived quantities in diffusion tensor imaging.

The diffusion tensor is typically assumed to be positive definite. However, noise in the measurements may cause the eigenvalues of the tensor estimate to be negative, thereby violating this assumption. Negative eigenvalues in diffusion tensor imaging (DTI) data occur predominately in regions of high anisotropy and may cause the fractional anisotropy (FA) to exceed unity. Two constrained least squares methods for eliminating negative eigenvalues are explored. These methods, the constrained linear least squares method (CLLS) and the constrained nonlinear least squares method (CNLS), are compared with other commonly used algebraic constrained methods. The CLLS tensor estimator can be shown to be equivalent to the linear least squares (LLS) tensor estimator when the LLS tensor estimate is positive definite. Similarly, the CNLS tensor estimator can be shown to be equivalent to the nonlinear least squares (NLS) tensor estimator when the NLS tensor estimate is positive definite. The constrained least squares methods for eliminating negative eigenvalues are evaluated with both simulations and in vivo human brain DTI data. Simulation results show that the CNLS method is, in terms of mean squared error for estimating trace and FA, the most effective method for correcting negative eigenvalues. Magn Reson Med, 2006. Published 2006 Wiley‐Liss, Inc.

Effects of physiological noise in population analysis of diffusion tensor MRI data.

The goal of this study is to characterize the potential effect of artifacts originating from physiological noise on statistical analysis of diffusion tensor MRI (DTI) data in a population. DTI derived quantities including mean diffusivity (Trace(D)), fractional anisotropy (FA), and principal eigenvector (ε(1)) are computed in the brain of 40 healthy subjects from tensors estimated using two different methods: conventional nonlinear least-squares, and robust fitting (RESTORE). RESTORE identifies artifactual data points as outliers and excludes them on a voxel-by-voxel basis. We found that outlier data points are localized in specific spatial clusters in the population, indicating a consistency in brain regions affected across subjects. In brain parenchyma RESTORE slightly reduces inter-subject variance of FA and Trace(D). The dominant effect of artifacts, however, is bias. Voxel-wise analysis indicates that inclusion of outlier data points results in clusters of under- and over-estimation of FA, while Trace(D) is always over-estimated. Removing outliers affects ε(1) mostly in low anisotropy regions. It was found that brain regions known to be affected by cardiac pulsation - cerebellum and genu of the corpus callosum, as well as regions not previously reported, splenium of the corpus callosum-show significant effects in the population analysis. It is generally assumed that statistical properties of DTI data are homogenous across the brain. This assumption does not appear to be valid based on these results. The use of RESTORE can lead to a more accurate evaluation of a population, and help reduce spurious findings that may occur due to artifacts in DTI data.

Variance of Estimated DTI-Derived Parameters via First- Order Perturbation Methods

In typical applications of diffusion tensor imaging (DTI), DT‐derived quantities are used to make a diagnostic, therapeutic, or scientific determination. In such cases it is essential to characterize the variability of these tensor‐derived quantities. Parametric and empirical methods have been proposed to estimate the variance of the estimated DT, and quantities derived from it. However, the former method cannot be generalized since a parametric distribution cannot be found for all DT‐derived quantities. Although powerful empirical methods, such as the bootstrap, are available, they require oversampling of the diffusion‐weighted imaging (DWI) data. Statistical perturbation methods represent a hybrid between parametric and empirical approaches, and can overcome the primary limitations of both methods. In this study we used a first‐order perturbation method to obtain analytic expressions for the variance of DT‐derived quantities, such as the trace, fractional anisotropy (FA), eigenvalues, and eigenvectors, for a given experimental design. We performed Monte Carlo (MC) simulations of DTI experiments to test and validate these formulae, and to determine their range of applicability for different experimental design parameters, including the signal‐to‐noise ratio (SNR), diffusion gradient sampling scheme, and number of DWI acquisitions. This information should be useful for designing DTI studies and assessing the quality of inferences drawn from them. Magn Reson Med 57:141–149, 2007. Published 2006 Wiley‐Liss, Inc.

Linear least‐squares method for unbiased estimation of T1 from SPGR signals

The longitudinal relaxation time, T1, can be estimated from two or more spoiled gradient recalled echo images (SPGR) acquired with different flip angles and/or repetition times (TRs). The function relating signal intensity to flip angle and TR is nonlinear; however, a linear form proposed 30 years ago is currently widely used. Here we show that this linear method provides T1 estimates that have similar precision but lower accuracy than those obtained with a nonlinear method. We also show that T1 estimated by the linear method is biased due to improper accounting for noise in the fitting. This bias can be significant for clinical SPGR images; for example, T1 estimated in brain tissue (800 ms < T1 < 1600 ms) can be overestimated by 10% to 20%. We propose a weighting scheme that correctly accounts for the noise contribution in the fitting procedure. Monte Carlo simulations of SPGR experiments are used to evaluate the accuracy of the estimated T1 from the widely‐used linear, the proposed weighted‐uncertainty linear, and the nonlinear methods. We show that the linear method with weighted uncertainties reduces the bias of the linear method, providing T1 estimates comparable in precision and accuracy to those of the nonlinear method while reducing computation time significantly. Magn Reson Med 60:496–501, 2008.

Three-dimensional analytical magnetic resonance imaging phantom in the Fourier domain.

This work presents a basic framework for constructing a 3D analytical MRI phantom in the Fourier domain. In the image domain the phantom is modeled after the work of Kak and Roberts on a 3D version of the famous Shepp‐Logan head phantom. This phantom consists of several ellipsoids of different sizes, orientations, locations, and signal intensities (or gray levels). It will be shown that the k‐space signal derived from the phantom can be analytically expressed. As a consequence, it enables one to bypass the need for interpolation in the Fourier domain when testing image‐reconstruction algorithms. More importantly, the proposed framework can serve as a benchmark for contrasting and comparing different image‐reconstruction techniques in 3D MRI with a non‐Cartesian k‐space trajectory. The proposed framework can also be adapted for 3D MRI simulation studies in which the MRI parameters of interest may be introduced to the signal intensity from the ellipsoid. Magn Reson Med, 2007. Published 2007 Wiley‐Liss, Inc.

Error Propagation Framework for Diffusion Tensor Imaging via Diffusion Tensor Representations

An analytical framework of error propagation for diffusion tensor imaging (DTI) is presented. Using this framework, any uncertainty of interest related to the diffusion tensor elements or to the tensor-derived quantities such as eigenvalues, eigenvectors, trace, fractional anisotropy (FA), and relative anisotropy (RA) can be analytically expressed and derived from the noisy diffusion-weighted signals. The proposed framework elucidates the underlying geometric relationship between the variability of a tensor-derived quantity and the variability of the diffusion weighted signals through the nonlinear least squares objective function of DTI. Monte Carlo simulations are carried out to validate and investigate the basic statistical properties of the proposed framework.

Probabilistic Identification and Estimation of Noise (PIESNO): A self-consistent approach and its applications in MRI

Data analysis in MRI usually entails a series of processing procedures. One of these procedures is noise assessment, which in the context of this work, includes both the identification of noise-only pixels and the estimation of noise variance (standard deviation). Although noise assessment is critical to many MRI processing techniques, the identification of noise-only pixels has received less attention than has the estimation of noise variance. The main objectives of this paper are, therefore, to demonstrate (a) that the identification of noise-only pixels has an important role to play in the analysis of MRI data, (b) that the identification of noise-only pixels and the estimation of noise variance can be combined into a coherent framework, and (c) that this framework can be made self-consistent. To this end, we propose a novel iterative approach to simultaneously identify noise-only pixels and estimate the noise standard deviation from these identified pixels in a commonly used data structure in MRI. Experimental and simulated data were used to investigate the feasibility, the accuracy and the stability of the proposed technique.

The Elliptical Cone of Uncertainty and Its Normalized Measures in Diffusion Tensor Imaging

Diffusion tensor magnetic resonance imaging (DT-MRI) is capable of providing quantitative insights into tissue microstructure in the brain. An important piece of information offered by DT-MRI is the directional preference of diffusing water molecules within a voxel. Building upon this local directional information, DT-MRI tractography attempts to construct global connectivity of white matter tracts. The interplay between local directional information and global structural information is crucial in understanding changes in tissue microstructure as well as in white matter tracts. To this end, the right circular cone of uncertainty was proposed by Basser as a local measure of tract dispersion. Recent experimental observations by Jeong et al. and Lazar et al. that the cones of uncertainty in the brain are mostly elliptical motivate the present study to investigate analytical approaches to quantify their findings. Two analytical approaches for constructing the elliptical cone of uncertainty, based on the first-order matrix perturbation and the error propagation method via diffusion tensor representations, are presented and their theoretical equivalence is established. We propose two normalized measures, circumferential and areal, to quantify the uncertainty of the major eigenvector of the diffusion tensor. We also describe a new technique of visualizing the cone of uncertainty in 3-D.

Optimization of a free water elimination two-compartment model for diffusion tensor imaging

Diffusion tensor imaging is used to measure the diffusion of water in tissue. The diffusion properties carry information about the relative organization and structure of the underlying tissue. In the case of a single voxel containing both tissue and a fast diffusing component such as free water, a single diffusion tensor is no longer appropriate. A two-tensor free water elimination model has previously been proposed to correct for the case of volume mixing. Here, this model was implemented in a straightforward but novel manner without the use of spatial constraints. The optimal acquisition parameters were investigated through Monte Carlo simulations and human brain imaging studies. At a signal-to-noise ratio of 40 with 64 diffusion-weighted encoding images, the most accurate estimates of fast diffusion signal were obtained with two diffusion-weighted shells (b-value in s/mm(2)×number of directions) of 500×32 and 1500×32. The potential bias in fractional anisotropy induced by this two-compartment model was more than an order of magnitude less than the error of using the single diffusion tensor model in the presence of partial volume effects with free water. This strategy may be useful for characterizing the diffusion of tissues adjacent to cerebral spinal fluid (CSF), tissues affected by edema, and removing artifacts from blurring and ghosting of the CSF signal.