Towards Best Practice of Interpreting Deep Learning Models for EEG-based Brain Computer Interfaces

As deep learning has achieved state-of-the-art performance for many tasks of EEG-based BCI, many efforts have been made in recent years trying to understand what have been learned by the models. This is commonly done by generating a heatmap indicating to which extent each pixel of the input contributes to the final classification for a trained model. Despite the wide use, it is not yet understood to which extent the obtained interpretation results can be trusted and how accurate they can reflect the model decisions. In order to fill this research gap, we conduct a study to evaluate different deep interpretation techniques quantitatively on EEG datasets. The results reveal the importance of selecting a proper interpretation technique as the initial step. In addition, we also find that the quality of the interpretation results is inconsistent for individual samples despite when a method with an overall good performance is used. Many factors, including model structure and dataset types, could potentially affect the quality of the interpretation results. Based on the observations, we propose a set of procedures that allow the interpretation results to be presented in an understandable and trusted way. We illustrate the usefulness of our method for EEG-based BCI with instances selected from different scenarios

Model-Free, Regularized, Fast, and Robust Analytical Orientation Distribution Function Estimation

International audienceHigh Angular Resolution Imaging (HARDI) can better explore the complex micro-structure of white matter compared to Diffusion Tensor Imaging (DTI). Orientation Distribution Function (ODF) in HARDI is used to describe the probability of the fiber direction. There are two type definitions of the ODF, which were respectively proposed in Q-Ball Imaging (QBI) and Diffusion Spectrum Imaging (DSI). Some analytical reconstructions methods have been proposed to estimate these two type of ODFs from single shell HARDI data. However they all have some assumptions and intrinsic modeling errors. In this article, we propose, almost without any assumption, a uniform analytical method to estimate these two ODFs from DWI signals in q space, which is based on Spherical Polar Fourier Expression (SPFE) of signals. The solution is analytical and is a linear transformation from the q-space signal to the ODF represented by Spherical Harmonics (SH). It can naturally combines the DWI signals in dierent Q-shells. Moreover It can avoid the intrinsic Funk-Radon Transform (FRT) blurring error in QBI and it does not need any assumption of the signals, such as the multiple tensor model and mono/multi-exponential decay. We validate our method using synthetic data, phantom data and real data. Our method works well in all experiments, especially for the data with low SNR, low anisotropy and non-exponential decay

A Riemannian Framework for Orientation Distribution Function Computing

International audienceCompared with Diffusion Tensor Imaging (DTI), High Angular Resolution Imaging (HARDI) can better explore the complex microstructure of white matter. Orientation Distribution Function (ODF) is used to describe the probability of the fiber direction. Fisher information metric has been constructed for probability density family in Information Geometry theory and it has been successfully applied for tensor computing in DTI. In this paper, we present a state of the art Riemannian framework for ODF computing based on Information Geometry and sparse representation of orthonormal bases. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms. And the weighted Frechet mean exists uniquely on this manifold. We also propose a novel scalar measurement, named Geometric Anisotropy (GA), which is the Riemannian geodesic distance between the ODF and the isotropic ODF. The Renyi entropy H_{1/2} of the ODF can be computed from the GA. Moreover, we present an Affine-Euclidean framework and a Log-Euclidean framework so that we can work in an Euclidean space. As an application, Lagrange interpolation on ODF field is proposed based on weighted Frechet mean. We validate our methods on synthetic and real data experiments. Compared with existing Riemannian frameworks on ODF, our framework is model-free. The estimation of the parameters, i.e. Riemannian coordinates, is robust and linear. Moreover it should be noted that our theoretical results can be used for any probability density function (PDF) under an orthonormal basis representation

Riemannian Median and Its Applications for Orientation Distribution Function Computing

International audienceThe geometric median is a classic robust estimator of centrality for data in Euclidean spaces, and it has been generalized in analytical manifold in [1]. Recently, an intrinsic Riemannian framework for Orientation Distribution Function (ODF) was proposed for the calculation in ODF field [2]. In this work, we prove the unique existence of the Riemannian median in ODF space. Then we explore its two potential applications, median filtering and atlas estimation

Diffeomorphism Invariant Riemannian Framework for Ensemble Average Propagator Computing

International audienceBackground: In Diffusion Tensor Imaging (DTI), Riemannian framework based on Information Geometry theory has been proposed for processing tensors on estimation, interpolation, smoothing, regularization, segmentation, statistical test and so on. Recently Riemannian framework has been generalized to Orientation Distribution Function (ODF) and it is applicable to any Probability Density Function (PDF) under orthonormal basis representation. Spherical Polar Fourier Imaging (SPFI) was proposed for ODF and Ensemble Average Propagator (EAP) estimation from arbitrary sampled signals without any assumption. Purpose: Tensors only can represent Gaussian EAP and ODF is the radial integration of EAP, while EAP has full information for diffusion process. To our knowledge, so far there is no work on how to process EAP data. In this paper, we present a Riemannian framework as a mathematical tool for such task. Method: We propose a state-of-the-art Riemannian framework for EAPs by representing the square root of EAP, called wavefunction based on quantum mechanics, with the Fourier dual Spherical Polar Fourier (dSPF) basis. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and weighted Riemannian mean and median uniquely exist. We analyze theoretically the similarities and differences between Riemannian frameworks for EAPs and for ODFs and tensors. The Riemannian metric for EAPs is diffeomorphism invariant, which is the natural extension of the affine-invariant metric for tensors. We propose Log-Euclidean framework to fast process EAPs, and Geodesic Anisotropy (GA) to measure the anisotropy of EAPs. With this framework, many important data processing operations, such as interpolation, smoothing, atlas estimation, Principal Geodesic Analysis (PGA), can be performed on EAP data. Results and Conclusions: The proposed Riemannian framework was validated in synthetic data for interpolation, smoothing, PGA and in real data for GA and atlas estimation. Riemannian median is much robust for atlas estimation

Fast, Model-Free , Analytical ODF Reconstruction from the Q-Space Signal

International audienceThe orientation distribution function (ODF) is very important in diffusion MRI. There are two types of ODFs. One is proposed using radial projection in Q-ball imaging [7]. Another one is the marginal pdf proposed in diffusion spectrum imaging (DSI) [8]. Since the marginal pdf is much sharper and mathematically correct, it could be more useful. Recently some reconstruction methods were proposed for this kind of ODF [1, 6]. They are both based on mono-exponential model, globally or locally, which has some intrinsic modeling error [4]. Although the authors in [1] extended the mono-exponential model to multi-exponential model, this multi-exponential model needs to be estimated non-linearly for every voxel and only in some special sampling scheme it has a analytical solution. Here we give a model-free analytical reconstruction method based on the Spherical Polar Fourier expression of the signal. It can estimate the ODF fast and analytically from the signal

Non-Negative Spherical Deconvolution (NNSD) for Fiber Orientation Distribution Function Estimation

International audienceIn diffusion Magnetic Resonance Imaging (dMRI), Spherical Deconvolution (SD) is a commonly used approach for estimating the fiber Orientation Distribution Function (fODF). As a Probability Density Function (PDF) that characterizes the distribution of fiber orientations, the fODF is expected to be non-negative and to integrate to unity on the continuous unit sphere S2 . However, many existing approaches, despite using continuous representation such as Spherical Harmonics (SH), impose non-negativity only on discretized points of S2. Therefore, non-negativity is not guaranteed on the whole S2 Existing approaches are also known to .exhibit false positive fODF peaks, especially in regions with low anisotropy, causing an over-estimation of the number of fascicles that traverse each voxel. This paper proposes a novel approach, called Non-Negative SD (NNSD), to overcome the above limitations. NNSD offers the following advantages. First, NNSD is the first SH based method that guarantees non-negativity of the fODF throughout the unit sphere. Second, unlike approaches such as Maximum Entropy SD (MESD), Cartesian Tensor Fiber Orientation Distribution (CT-FOD), and discrete representation based SD (DR-SD) techniques, the SH representation allows closed form of spherical integration, efficient computation in a low dimensional space resided by the SH coefficients, and accurate peak detection on the continuous domain defined by the unit sphere. Third, NNSD is significantly less susceptible to producing false positive peaks in regions with low anisotropy. Evaluations of NNSD in comparison with Constrained SD (CSD), MESD, and DR-SD (implemented using L1-regularized least-squares with non-negative constraint), indicate that NNSD yields improved performance for both synthetic and real data. The performance gain is especially prominent for high resolution (1.25 mm)^3 data

Regularized Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning

International audienceOne important problem in diffusion MRI (dMRI) is to recover the diffusion weighted signal from only a limited number of samples in q-space. An ideal framework for solving this problem is Compressed Sensing (CS), which takes advantage of the signal's sparseness or compressibility, allowing the entire signal to be reconstructed from relatively few measurements. CS theory requires a suitable dictionary that sparsely represents the signal. To date in dMRI there are two kinds of Dictionary Learning (DL) methods: 1) discrete representation based DL (DR-DL), and 2) continuous representation based DL (CR-DL). Due to the discretization in q-space, DR-DL suffers from the numerical errors in interpolation and regridding. By considering a continuous representation using Spherical Polar Fourier (SPF) basis, this paper proposes a novel CR-DL based Spherical Polar Fourier Imaging, called DL-SPFI, to recover the diffusion signal as well as the Ensemble Average Propagator (EAP) in continuous 3D space with closed form. DL-SPFI learns an optimal dictionary from the space of Gaussian diffusion signals. Then the learned dictionary is adaptively applied for different voxels in a weighted LASSO framework to robustly recover the di ffusion signal and the EAP. Compared with the start-of-the-art CR-DL method by Merlet et al. and DRDL by Bilgic et al., DL-SPFI has several advantages. First, the learned dictionary, which is proved to be optimal in the space of Gaussian diffusion signal, can be applied adaptively for different voxels. To our knowledge, this is the first work to learn a voxel-adaptive dictionary. The importance of this will be shown theoretically and empirically in the context of EAP estimation. Second, based on the theoretical analysis of SPF basis, we devise an efficient learning process in a small subspace of SPF coefficients, not directly in q-space as done by Merlet et al.. Third, DL-SPFI also devises different regularization for different atoms in the learned dictionary for robust estimation, by considering the structural prior in the space of signal exemplars. We evaluate DL-SPFI in comparison to L1-norm regularized SPFI (L1-SPFI) with fixed SPF basis, and the DR-DL by Bilgic et al. The experiments on synthetic data and real data demonstrate that the learned dictionary is sparser than SPF basis and yields lower reconstruction error than Bilgic's method, even though only simple synthetic Gaussian signals were used for training in DL-SPFI in contrast to real data used by Bilgic et al

Theoretical Analysis and Practical Insights on EAP Estimation via a Unified HARDI Framework

International audienceSince Diffusion Tensor Imaging (DTI) cannot describe complex non-Gaussian diffusion process, many techniques, called as single shell High Angular Resolution Diffusion Imaging (sHARDI) methods, reconstruct the Ensemble Average Propagator (EAP) or its feature Orientation Distribution Function (ODF) from diffusion weighted signals only in single shell. Q-Ball Imaging (QBI) and Diffusion Orientation Transform (DOT) are two famous sHARDI methods. However, these sHARDI methods have some intrinsic modeling errors or need some unreal assumptions. Moreover they are hard to deal with signals from different q-shells. Most recently several novel multiple shell HARDI (mHARDI) methods, including Diffusion Propagator Imaging (DPI), Spherical Polar Fourier Imaging (SPFI) and Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE), were proposed to analytically estimate EAP or ODF from multiple shell (or arbitrarily sampled) signals. These three methods all represent diffusion signal with some basis functions in spherical coordinate and use plane wave formula to analytically solve the Fourier transform. To our knowledge, there is no theoretical analysis and practical comparison among these sHARDI and mHARDI methods. In this paper, we propose a unified computational framework, named Analytical Fourier Transform in Spherical Coordinate (AFT-SC), to perform such theoretical analysis and practical comparison among all these five state-of-the-art diffusion MRI methods. We compare these five methods in both theoretical and experimental aspects. With respect to the theoretical aspect, some criteria are proposed for evaluation and some differences together with some similarities among the methods are highlighted. Regarding the experimental aspect, all the methods are compared in synthetic, phantom and real data. The shortcomings and advantages of each method are highlighted from which SPFI appears to be among the best because it uses an orthonormal basis that completely separates the spherical and radial information