s-SMOOTH: Sparsity and Smoothness Enhanced EEG Brain Tomography

Abstract

EEG source imaging enables us to reconstruct current density in the brain from the electrical measurements with excellent temporal resolution (~ ms). The corresponding EEG inverse problem is an ill-posed one that has infinitely many solutions. This is due to the fact that the number of EEG sensors is usually much smaller than that of the potential dipole locations, as well as noise contamination in the recorded signals. To obtain a unique solution, regularizations can be incorporated to impose additional constraints on the solution. An appropriate choice of regularization is critically important for the reconstruction accuracy of a brain image. In this paper, we propose a novel Sparsity and SMOOthness enhanced brain TomograpHy (s-SMOOTH) method to improve the reconstruction accuracy by integrating two recently proposed regularization techniques: Total Generalized Variation (TGV) regularization and ℓ1−2 regularization. TGV is able to preserve the source edge and recover the spatial distribution of the source intensity with high accuracy. Compared to the relevant total variation (TV) regularization, TGV enhances the smoothness of the image and reduces staircasing artifacts. The traditional TGV defined on a 2D image has been widely used in the image processing field. In order to handle 3D EEG source images, we propose a voxel-based Total Generalized Variation (vTGV) regularization that extends the definition of second-order TGV from 2D planar images to 3D irregular surfaces such as cortex surface. In addition, the ℓ1−2 regularization is utilized to promote sparsity on the current density itself. We demonstrate that ℓ1−2 regularization is able to enhance sparsity and accelerate computations than ℓ1 regularization. The proposed model is solved by an efficient and robust algorithm based on the difference of convex functions algorithm (DCA) and the alternating direction method of multipliers (ADMM). Numerical experiments using synthetic data demonstrate the advantages of the proposed method over other state-of-the-art methods in terms of total reconstruction accuracy, localization accuracy and focalization degree. The application to the source localization of event-related potential data further demonstrates the performance of the proposed method in real-world scenarios.

Simultaneous Sensing Error Recovery and Tomographic Inversion Using an Optimization-Based Approach

“…We also include reconstructions obtained by adding standard regularizers. In particular, we consider L2-and T V -regularizers [25,15] where L ∈ R N 2 ×N 2 is the Laplacian operator, ∇ denotes the spatial gradient, and λ > 0 is the regularizer parameter that balances the misfit term and regularization term. One approach to choose λ would be the L-curve method [16].…”

Task-Related EEG Source Localization via Graph Regularized Low-Rank Representation Model

“…To further consider the spatial smoothness, a total variation term can be imposed as another penalty term, such as first order total variation (TV) regularization in Ref. [30], [42], fractional order TV in [29], [31], and similar algorithm can be derived under the framework of ADMM, however further investigation with constraints of TV is our future work.…”

“…By replacing 2 -norm by 1 -norm, minimum current estimate (MCE) [28] is proposed to overcome overestimation of active area sizes incurred by 2norm. Recent development of compressive sensing algorithm proved the p (p ≤ 1) regularization on the original source signal usually provides a set of discrete sources distributed across the cortex due to high coherence of lead field matrix, in order to encourage reconstruction of extended source patches, it has been found that by enforcing sparsity in a transformed domain, e.g., total variation (TV) regularization [29], [30], [22], [31], focal source extents can be better estimated.…”

Poisson image denoising based on fractional-order total variation

“…However, it causes staircasing artifacts and the loss of image contrasts, latter of which is particularly obvious for piecewise smooth images [32]. To address these issues, some high-order regularizations for the Gaussian denoising problem have been introduced, including non-local total variation [21], TV combined with a fourth-order diffusive term [8], Euler's elastic models [15,49,50,61], a mean curvature model [69], a total generalized variation (TGV) model [4,55], a second-order TGV [26], and TGV together with sparsity and/or shearlets [19,27,40]. Specifically for the Poisson denoising, some high-order methods include PDE based models [62,67], a hybrid regularization combining TV with a fourth-order variation [20], and TGV-based Poisson denoising model [29].…”