Christian Jutten

Multiclass Brain–Computer Interface Classification by Riemannian Geometry

This paper presents a new classification framework for brain-computer interface (BCI) based on motor imagery. This framework involves the concept of Riemannian geometry in the manifold of covariance matrices. The main idea is to use spatial covariance matrices as EEG signal descriptors and to rely on Riemannian geometry to directly classify these matrices using the topology of the manifold of symmetric and positive definite (SPD) matrices. This framework allows to extract the spatial information contained in EEG signals without using spatial filtering. Two methods are proposed and compared with a reference method [multiclass Common Spatial Pattern (CSP) and Linear Discriminant Analysis (LDA)] on the multiclass dataset IIa from the BCI Competition IV. The first method, named minimum distance to Riemannian mean (MDRM), is an implementation of the minimum distance to mean (MDM) classification algorithm using Riemannian distance and Riemannian mean. This simple method shows comparable results with the reference method. The second method, named tangent space LDA (TSLDA), maps the covariance matrices onto the Riemannian tangent space where matrices can be vectorized and treated as Euclidean objects. Then, a variable selection procedure is applied in order to decrease dimensionality and a classification by LDA is performed. This latter method outperforms the reference method increasing the mean classification accuracy from 65.1% to 70.2%.

Fetal ECG Extraction by Extended State Kalman Filtering Based on Single-Channel Recordings

In this paper, we present an extended nonlinear Bayesian filtering framework for extracting electrocardiograms (ECGs) from a single channel as encountered in the fetal ECG extraction from abdominal sensor. The recorded signals are modeled as the summation of several ECGs. Each of them is described by a nonlinear dynamic model, previously presented for the generation of a highly realistic synthetic ECG. Consequently, each ECG has a corresponding term in this model and can thus be efficiently discriminated even if the waves overlap in time. The parameter sensitivity analysis for different values of noise level, amplitude, and heart rate ratios between fetal and maternal ECGs shows its effectiveness for a large set of values of these parameters. This framework is also validated on the extractions of fetal ECG from actual abdominal recordings, as well as of actual twin magnetocardiograms.

Learning Overcomplete Dictionaries Based on Atom-by-Atom Updating

A dictionary learning algorithm learns a set of atoms from some training signals in such a way that each signal can be approximated as a linear combination of only a few atoms. Most dictionary learning algorithms use a two-stage iterative procedure. The first stage is to sparsely approximate the training signals over the current dictionary. The second stage is the update of the dictionary. In this paper we develop some atom-by-atom dictionary learning algorithms, which update the atoms sequentially. Specifically, we propose an efficient alternative to the well-known K-SVD algorithm, and show by various experiments that the proposed algorithm is much faster than K-SVD while its results are better. Moreover, we propose a novel algorithm that instead of alternating between the two dictionary learning stages, performs only the second stage. While in K-SVD each atom is updated along with the nonzero entries of its associated row vector in the coefficient matrix (which we name it its profile), in the new algorithm, each atom is updated along with the whole entries of its profile. As a result, contrary to K-SVD, the support of each profile can be changed while updating the dictionary. To further accelerate the convergence of this algorithm and to have a control on the cardinality of the representations, we then propose its two-stage counterpart by adding the sparse approximation stage. Experimental results on recovery of a known synthetic dictionary and dictionary learning for a class of auto-regressive signals demonstrate the promising performance of the proposed algorithms.

Fusion of hyperspectral and panchromatic images using multiresolution analysis and nonlinear PCA band reduction

This article presents a novel method for the enhancement of the spatial quality of hyperspectral (HS) images through the use of a high resolution panchromatic (PAN) image. Due to the high number of bands, the application of a pan-sharpening technique to HS images may result in an increase of the computational load and complexity. Thus a dimensionality reduction preprocess, compressing the original number of measurements into a lower dimensional space, becomes mandatory. To solve this problem, we propose a pan-sharpening technique combining both dimensionality reduction and fusion, making use of non-linear principal component analysis (NLPCA) and Indusion, respectively, to enhance the spatial resolution of a HS image. We have tested the proposed algorithm on HS images obtained from CHRIS-Proba sensor and PAN image obtained from World view 2 and demonstrated that a reduction using NLPCA does not result in any significant degradation in the pan-sharpening results.

Dictionary Learning for Sparse Representation: A Novel Approach

A dictionary learning problem is a matrix factorization in which the goal is to factorize a training data matrix, Y, as the product of a dictionary, D, and a sparse coefficient matrix, X, as follows, Y ≃ DX. Current dictionary learning algorithms minimize the representation error subject to a constraint on D (usually having unit column-norms) and sparseness of X. The resulting problem is not convex with respect to the pair (D,X). In this letter, we derive a first order series expansion formula for the factorization, DX. The resulting objective function is jointly convex with respect to D and X. We simply solve the resulting problem using alternating minimization and apply some of the previously suggested algorithms onto our new problem. Simulation results on recovery of a known dictionary and dictionary learning for natural image patches show that our new problem considerably improves performance with a little additional computational load.

On the Achievability of Cramér–Rao Bound in Noisy Compressed Sensing

Recently, it has been proved in Babadi [B. Babadi, N. Kalouptsidis, and V. Tarokh, “Asymptotic achievability of the Cramér-Rao bound for noisy compressive sampling,” IEEE Trans. Signal Process., vol. 57, no. 3, pp. 1233-1236, 2009] that in noisy compressed sensing, a joint typical estimator can asymptotically achieve the Cramér-Rao lower bound of the problem. To prove this result, Babadi used a lemma, which is provided in Akçakaya and Tarokh [M. Akçakaya and V. Trarokh, “Shannon theoretic limits on noisy compressive sampling,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 492-504, 2010] that comprises the main building block of the proof. This lemma is based on the assumption of Gaussianity of the measurement matrix and its randomness in the domain of noise. In this correspondence, we generalize the results obtained in Babadi by dropping the Gaussianity assumption on the measurement matrix. In fact, by considering the measurement matrix as a deterministic matrix in our analysis, we find a theorem similar to the main theorem of Babadi for a family of randomly generated (but deterministic in the noise domain) measurement matrices that satisfy a generalized condition known as “the concentration of measures inequality.” By this, we finally show that under our generalized assumptions, the Cramér-Rao bound of the estimation is achievable by using the typical estimator introduced in Babadi et al.

Recovery of Low-Rank Matrices Under Affine Constraints via a Smoothed Rank Function

In this paper, the problem of matrix rank minimization under affine constraints is addressed. The state-of-the-art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem. We propose an algorithm based on a smooth approximation of the rank function, which practically improves recovery limits on the rank of the solution. This approximation leads to a non-convex program; thus, to avoid getting trapped in local solutions, we use the following scheme. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy. On the theoretical side, benefiting from the spherical section property, we will show that the sequence of the solutions of the approximating programs converges to the minimum rank solution. On the experimental side, it will be shown that the proposed algorithm, termed SRF standing for smoothed rank function, can recover matrices, which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization. Furthermore, it will be demonstrated that, in completing partially observed matrices, the accuracy of SRF is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low-rank matrix.

Linear-quadratic blind source separating structure for removing show-through in scanned documents

Digital documents are usually degraded during the scanning process due to the contents of the backside of the scanned manuscript. This is often caused by the show-through effect, i.e. the backside image that interferes with the main front side picture due to the intrinsic transparency of the paper. This phenomenon is one of the degradations that one would like to remove especially in the field of Optical Character Recognition (OCR) or document digitalization which require denoised texts as inputs. In this paper, we first propose a novel and general nonlinear model for canceling the show-through phenomenon. A nonlinear blind source separation algorithm is used for this purpose based on a new recursive and extendible structure. However, the results are restricted due to a blurring effect that appears during the scanning process due to the light transfer function of the paper. Consequently, for improving the results, we introduce a refined separating architecture for simultaneously removing the show-through and blurring effects.

SRF: Matrix completion based on smoothed rank function

In this paper, we address the matrix completion problem and propose a novel algorithm based on a smoothed rank function (SRF) approximation. Among available algorithms like FPCA and OptSpace, there is no solution that can simultaneously cover wide range of easy and hard problems. This new algorithm provides accurate results in almost all scenarios with a reasonable run time. It especially has low execution time in hard problems where other methods need long time to converge. Furthermore, when the rank is known in advance and is high, our method is very faster than previous methods for the same accuracy. The main idea of the algorithm is based on a continuous and differentiable approximation of the rank function and then, using gradient projection approach to minimize it.

Directed Differential Connectivity Graph of Interictal Epileptiform Discharges

In this paper, we study temporal couplings between interictal events of spatially remote regions in order to localize the leading epileptic regions from intracerebral EEG (iEEG). We aim to assess whether quantitative epileptic graph analysis during interictal period may be helpful to predict the seizure onset zone of ictal iEEG. Using wavelet transform, cross-correlation coefficient, and multiple hypothesis test, we propose a differential connectivity graph (DCG) to represent the connections that change significantly between epileptic and nonepileptic states as defined by the interictal events. Post-processings based on mutual information and multiobjective optimization are proposed to localize the leading epileptic regions through DCG. The suggested approach is applied on iEEG recordings of five patients suffering from focal epilepsy. Quantitative comparisons of the proposed epileptic regions within ictal onset zones detected by visual inspection and using electrically stimulated seizures, reveal good performance of the present method.