Noise Effects in Various Quantitative Susceptibility Mapping Methods

Abstract

Various regularization methods have been proposed for single-orientation quantitative susceptibility mapping (QSM), which is an ill-posed magnetic field to susceptibility source inverse problem. Noise amplification, a major issue in inverse problems, manifests as streaking artifacts and quantification errors in QSM and has not been comparatively evaluated in these algorithms. In this paper, various QSM methods were systematically categorized for noise analysis. Six representative QSM methods were selected from four categories: two non-Bayesian methods with alteration or approximation of the dipole kernel to overcome the ill conditioning; four Bayesian methods using a general mathematical prior or a specific physical structure prior to select a unique solution, and using a data fidelity term with or without noise weighting. The effects of noise in these QSM methods were evaluated by reconstruction errors in simulation and image quality in 50 consecutive human subjects. Bayesian QSM methods with noise weighting consistently reduced root mean squared errors in numerical simulations and increased image quality scores in the human brain images, when compared to non-Bayesian methods and to corresponding Bayesian methods without noise weighting (p ≤ 0.001). In summary, noise effects in QSM can be reduced using Bayesian methods with proper noise weighting.

Inverse Problem in Quantitative Susceptibility Mapping

“…Finally, since there are various ways to solve (4.2) and (4.3), such as the conjugate gradient method [23], nonlinear conjugate gradient method [14], quasi-Newton method [27], split Bregman method [10], and augmented Lagrangian method [51], the choice of a method for solving the optimization problem may affect the reconstructed image as well. Further research on reconstruction methods will be needed to achieve more accurate and robust reconstructions for QSM [45].…”

Structure Prior Effects in Bayesian Approaches of Quantitative Susceptibility Mapping

“…The maximum a posterior solution [17, 23] is $\begin{array}{c}T1\mathbf{\chi }=\underset{\mathbf{\chi }}{\arg \min }\left[\mathbf{E}+\alpha \mathbf{R}\right],\end{array}$where E constitutes the data fidelity term. Since Gaussian noise in the complex MR signal domain should be accounted for in the data fidelity term and with proper noise weighting, noise effects in QSM can be reduced using Bayesian methods [17]. In the following section, we use E = ‖ w z ‖ 2 2 , with z = d ⊗ χ − δ b and w   as noise weighting.…”

“…The following examples of regularization terms have been explored [17]:The gradient ( G ) L2 norm (GL2)$\begin{array}{c}T1R={{‖G\chi ‖}_{\mathrm{2}}}^{\mathrm{2}}={{‖\frac{\partial \chi }{\partial x}‖}_{\mathrm{2}}}^{\mathrm{2}}+{{‖\frac{\partial \chi }{\partial y}‖}_{\mathrm{2}}}^{\mathrm{2}}+{{‖\frac{\partial \chi }{\partial z}‖}_{\mathrm{2}}}^{\mathrm{2}}\end{array}$ The gradient L1 norm (GL1)$\begin{array}{c}T1R={‖G\chi ‖}_1={‖\frac{\partial \chi }{\partial x}‖}_1+{‖\frac{\partial \chi }{\partial y}‖}_1+{‖\frac{\partial \chi }{\partial z}‖}_1\end{array}$ The total variation norm (TV)$\begin{array}{c}T1R=\mathrm{T}\mathrm{V}\left(\chi \right)=\underset{\mathrm{boldr}}{{\displaystyle false\sum }}\sqrt{{\left(\frac{\partial \chi }{\partial x}\left(\mathrm{boldr}\right)\right)}^{\mathrm{normal2}}+{\left(\frac{\partial \chi }{\partial y}\left(\mathrm{boldr}\right)\right)}^{\mathrm{normal2}}+{\left(\frac{\partial \chi }{\partial z}\left(\mathrm{boldr}\right)\right)}^{\mathrm{normal2}}}\end{array}$ A wavelet domain such as a Daubechies wavelet (Φ) L1 norm $\begin{array}{c}T1\mathrm{R<...><!--...-->}\end{array}$…”

“…For clinical data acquired using a single orientation, various regularization methods have been proposed for QSM [17]. The Bayesian formulation provides the main QSM approach to identify a susceptibility distribution of minimal streaking artifacts [20–22].…”

“…Previously, we examine the importance of noise whitening in constructing the data fidelity term [17]. Here, we examine the importance of including structure information in the prior term.…”