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Inverse imaging problems that are ill-posed can be encountered across multiple domains of science and technology, ranging from medical diagnosis to astronomical studies. To reconstruct images from incomplete and distorted data, it is necessary to create algorithms that can take into account both, the physical mechanisms responsible for generating these measurements and the intrinsic characteristics of the images being analyzed. In this work, the sparse representation of images is reviewed, which is a realistic, compact and effective generative model for natural images inspired by the visual system of mammals. It enables us to address ill-posed linear inverse problems by training the model on a vast collection of images. Moreover, we extend the application of sparse coding to solve the non-linear and ill-posed problem in microwave tomography imaging, which could lead to a significant improvement of the state-of-the-arts algorithms., Comment: submitted to RevMexAA (conference series)

In this paper, we extend the Discrete Empirical Interpolation Method (DEIM) to the third-order tensor case based on the t-product and use it to select important/ significant lateral and horizontal slices/features. The proposed Tubal DEIM (TDEIM) is investigated both theoretically and numerically. The experimental results show that the TDEIM can provide more accurate approximations than the existing methods. An application of the proposed method to the supervised classification task is also presented.

In non-invasive brain-computer interface systems, pre-movement decoding plays an important role in the detection of movement before limbs actually move. Movement-related cortical potential is a kind of brain activity associated with pre-movement decoding. In current studies, patterns decoded from movement are mainly applied to the binary classification between movement state and resting state, such as elbow flexion and rest. The classifications between two movement states and among multiple movement states are still challenging. This study proposes a new method, the star-arrangement spectral filtering (SASF), to solve the multi-class pre-movement classification problem. We first design a referenced task-related component analysis (RTRCA) framework that consists of two modules. This first module is the classification between movement state and resting state; the second module is the classification of multiple movement states. SASF is developed by optimizing the features in RTRCA. In SASF, feature selection on filter banks is used on the first module of RTRCA, and feature selection on time windows is used on the second module of RTRCA. A linear discriminant analysis classifier is used to classify the optimized features. In the binary classification between two motions, the classification accuracy of SASF achieves 0.9670$\pm$0.0522, which is significantly higher than the result provided by the deep convolutional neural network (0.6247$\pm$0.0680) and the discriminative spatial pattern method (0.4400$\pm$0.0700). In the multi-class classification of 7 states, the classification accuracy of SASF is 0.9491$\pm$0.0372. The proposed SASF greatly improves the classification between two motions and enables the classification among multiple motions. The result shows that the movement can be decoded from EEG signals before the actual limb movement.

Pre-movement decoding plays an important role in movement detection and is able to detect movement onset with low-frequency electroencephalogram (EEG) signals before the limb moves. In related studies, pre-movement decoding with standard task-related component analysis (STRCA) has been demonstrated to be efficient for classification between movement state and resting state. However, the accuracies of STRCA differ among subbands in the frequency domain. Due to individual differences, the best subband differs among subjects and is difficult to be determined. This study aims to improve the performance of the STRCA method by a feature selection on multiple subbands and avoid the selection of best subbands. This study first compares three frequency range settings ($M_1$: subbands with equally spaced bandwidths; $M_2$: subbands whose high cut-off frequencies are twice the low cut-off frequencies; $M_3$: subbands that start at some specific fixed frequencies and end at the frequencies in an arithmetic sequence.). Then, we develop a mutual information based technique to select the features in these subbands. A binary support vector machine classifier is used to classify the selected essential features. The results show that $M_3$ is a better setting than the other two settings. With the filter banks in $M_3$, the classification accuracy of the proposed FBTRCA achieves 0.8700$\pm$0.1022, which means a significantly improved performance compared to STRCA (0.8287$\pm$0.1101) as well as to the cross validation and testing method (0.8431$\pm$0.1078).

Empirical mode decomposition (EMD) has developed into a prominent tool for adaptive, scale-based signal analysis in various fields like robotics, security and biomedical engineering. Since the dramatic increase in amount of data puts forward higher requirements for the capability of real-time signal analysis, it is difficult for existing EMD and its variants to trade off the growth of data dimension and the speed of signal analysis. In order to decompose multi-dimensional signals at a faster speed, we present a novel signal-serialization method (serial-EMD), which concatenates multi-variate or multi-dimensional signals into a one-dimensional signal and uses various one-dimensional EMD algorithms to decompose it. To verify the effects of the proposed method, synthetic multi-variate time series, artificial 2D images with various textures and real-world facial images are tested. Compared with existing multi-EMD algorithms, the decomposition time becomes significantly reduced. In addition, the results of facial recognition with Intrinsic Mode Functions (IMFs) extracted using our method can achieve a higher accuracy than those obtained by existing multi-EMD algorithms, which demonstrates the superior performance of our method in terms of the quality of IMFs. Furthermore, this method can provide a new perspective to optimize the existing EMD algorithms, that is, transforming the structure of the input signal rather than being constrained by developing envelope computation techniques or signal decomposition methods. In summary, the study suggests that the serial-EMD technique is a highly competitive and fast alternative for multi-dimensional signal analysis., Comment: 19 pages, 17 figures

In this paper, the problem of training a classifier on a dataset with incomplete features is addressed. We assume that different subsets of features (random or structured) are available at each data instance. This situation typically occurs in the applications when not all the features are collected for every data sample. A new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary. Sufficient conditions are identified, such that, if it is possible to train a classifier on incomplete observations so that their reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples. Extensive simulation results on synthetic and well-known datasets are presented that validate our theoretical findings and demonstrate the effectiveness of the proposed method compared to traditional data imputation approaches and one state-of-the-art algorithm., Comment: 11 pages, 7 figures, paper accepted for presentation at L2ID Workshop at CVPR 2021 (19-25 June, 2021)

One of the current issues in Brain-Computer Interface is how to deal with noisy Electroencephalography measurements organized as multidimensional datasets. On the other hand, recently, significant advances have been made in multidimensional signal completion algorithms that exploit tensor decomposition models to capture the intricate relationship among entries in a multidimensional signal. We propose to use tensor completion applied to EEG data for improving the classification performance in a motor imagery BCI system with corrupted measurements. Noisy measurements are considered as unknowns that are inferred from a tensor decomposition model. We evaluate the performance of four recently proposed tensor completion algorithms plus a simple interpolation strategy, first with random missing entries and then with missing samples constrained to have a specific structure (random missing channels), which is a more realistic assumption in BCI Applications. We measured the ability of these algorithms to reconstruct the tensor from observed data. Then, we tested the classification accuracy of imagined movement in a BCI experiment with missing samples. We show that for random missing entries, all tensor completion algorithms can recover missing samples increasing the classification performance compared to a simple interpolation approach. For the random missing channels case, we show that tensor completion algorithms help to reconstruct missing channels, significantly improving the accuracy in the classification of motor imagery, however, not at the same level as clean data. Tensor completion algorithms are useful in real BCI applications. The proposed strategy could allow using motor imagery BCI systems even when EEG data is highly affected by missing channels and/or samples, avoiding the need of new acquisitions in the calibration stage., Comment: 21 pages, 3 tables, 4 figures

The number of neuroimaging data sets publicly available is growing at fast rate. The increase in availability and resolution of neuroimaging data requires modern approaches to signal processing for data analysis and results validation. We introduce the application of sparse multiway decomposition methods (Caiafa and Cichocki, 2012) to linearized neuroimaging models. We show that decomposed models are more compact but as accurate as full models and can be successfully used for fast data analysis. We focus as example on a recent model for the evaluation of white matter connectomes (Pestilli et al, 2014). We show that the multiway decomposed model achieves accuracy comparable to the full model, while requiring only a small fraction of the memory and compute time. The approach has implications for a majority of neuroimaging methods using linear approximations to measured signals., Comment: 19 pages, 1 table, 9 figures

In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an $N$th-order $(I_1\times I_2\times \cdots \times I_N)$ data tensor $\underline{\mathbf{X}}$ from a reduced set of multi-way compressive measurements by exploiting its low multilinear-rank structure. Moreover, we show that, an interesting property of multi-way measurements allows us to build the reconstruction based on compressive linear measurements taken only in two selected modes, independently of the tensor order $N$. In addition, it is proved that, in the matrix case and in a particular case with $3$rd-order tensors where the same 2D sensor operator is applied to all mode-3 slices, the proposed reconstruction $\underline{\mathbf{X}}_\tau$ is stable in the sense that the approximation error is comparable to the one provided by the best low-multilinear-rank approximation, where $\tau$ is a threshold parameter that controls the approximation error. Through the analysis of the upper bound of the approximation error we show that, in the 2D case, an optimal value for the threshold parameter $\tau=\tau_0 > 0$ exists, which is confirmed by our simulation results. On the other hand, our experiments on 3D datasets show that very good reconstructions are obtained using $\tau=0$, which means that this parameter does not need to be tuned. Our extensive simulation results demonstrate the stability and robustness of the method when it is applied to real-world 2D and 3D signals. A comparison with state-of-the-arts sparsity based CS methods specialized for multidimensional signals is also included. A very attractive characteristic of the proposed method is that it provides a direct computation, i.e. it is non-iterative in contrast to all existing sparsity based CS algorithms, thus providing super fast computations, even for large datasets., Comment: Submitted to IEEE Transactions on Signal Processing

A new generalized multilinear regression model, termed the Higher-Order Partial Least Squares (HOPLS), is introduced with the aim to predict a tensor (multiway array) $\tensor{Y}$ from a tensor $\tensor{X}$ through projecting the data onto the latent space and performing regression on the corresponding latent variables. HOPLS differs substantially from other regression models in that it explains the data by a sum of orthogonal Tucker tensors, while the number of orthogonal loadings serves as a parameter to control model complexity and prevent overfitting. The low dimensional latent space is optimized sequentially via a deflation operation, yielding the best joint subspace approximation for both $\tensor{X}$ and $\tensor{Y}$. Instead of decomposing $\tensor{X}$ and $\tensor{Y}$ individually, higher order singular value decomposition on a newly defined generalized cross-covariance tensor is employed to optimize the orthogonal loadings. A systematic comparison on both synthetic data and real-world decoding of 3D movement trajectories from electrocorticogram (ECoG) signals demonstrate the advantages of HOPLS over the existing methods in terms of better predictive ability, suitability to handle small sample sizes, and robustness to noise.