By Sabine Van Huffel
Substantial progress in both data recording technologies and information processing in recent years has enabled the acquisition and analysis of large amounts of biomedical data. Extraction of the underlying information patterns relevant to patient health, called “sources,” is a core problem in biomedical information processing. Blind source separation (BSS) is the task of finding such source signals and the mixing mechanism, given only the raw signals, hence “blind.” Rapid advances in healthcare diagnostics and medical technologies are opening new challenges for information processing.
Biomedical signal processing has moved away from vector processing (classical single-channel time and frequency-domain analysis, despite the continued use in medical practice) to matrix processing. At that level, the BSS problem is solved by decomposition of the data matrix into a sum of interpretable rank-1 terms. This problem is underdetermined, however, unless strong assumptions on the sources are imposed, such as statistical independence (with use of independent component analysis (ICA) methods) or nonnegative components (with use of nonnegative matrix factorization (NMF) methods). Matrix-based BSS methods have become increasingly popular for artifact removal  and even for separating stimulus-related activity in multimodal data, such as electro-encephalography (EEG) and functional magnetic resonance imaging (fMRI) . These methods are too restrictive, however. It may be possible to develop more powerful information systems by facing the following challenges:
From matrix to tensor.
Figure 1. Decomposition of a third-order tensor. Top: CPD, atomic terms. Bottom: BTD, molecular terms. Adapted from .
For maintaining structural information, higher-order tensors are very attractive. They generalize vectors and matrices to multiway tables of numbers . The BSS problem is solved via tensor decomposition. Most well known is the canonical polyadic decomposition (CPD; also known as canonical decomposition or parallel factor decomposition) , which decomposes a tensor in rank-1 terms (see Figure 1, top). CPD is unique under mild conditions, which makes it a broadly applicable key tool for BSS. The assumptions needed for source extraction are very natural.
From rank-1 to low-rank. It is important to observe that a rank-1 structure is, in fact, very restrictive. Except for strength, no other source variations can be modeled. The decomposition may still be unique when it involves terms that are more general and/or realistic than rank-1 terms. We denote such generalized decompositions as block term decompositions (BTD)  (see Figure 1, bottom.) BTDs make it possible to model more variation (shape, delay, . . .) and provide more detail. In addition, BTDs allow broad source modelling .
De Lathauwer’s team implemented most tensor decompositions, including CPD and BTD, into a powerful open-source, easy-to-use, optimization-based Matlab toolbox; see www.tensorlab.net. Moreover, factorizations can be coupled or fused with each other. Factors can be shared, and any structure can be imposed. Tensorlab is the core of our solution of tensor-based BSS problems. A few case studies illustrate the process.
Extracting the Epileptic Component from EEG
Because seizures exhibit oscillatory behavior that is almost stable in localization and frequency during an entire episode (2-10 seconds), they fit a trilinear structure. In Figure 2, we construct a third-order tensor via wavelet transform of all EEG channels and show that CPD reliably identifies one epileptic source with \(R = 2\) and no need for artifact removal .
Figure 2. At left, CPD of wavelet transformed EEG showing a 10-second onset of a right temporal lobe complex partial seizure. Muscle (at five and nine seconds) and eye blink artefacts (sharp wave) are visible. At right, the spatial (top), frequency (middle), and temporal modes (bottom) are shown. The epileptic atom is on the left, the eye blink artefact on the right. Adapted from .
When seizures are nonstationary (e.g., varying in space or frequency during the episode), CPD is too restrictive. In that case, use of BTD improves the extracted seizure component, as shown in . We applied wavelet transform or Hankel expansion to organise the EEG data in a tensor. With the former approach, we were able to model nonstationary seizures evolving in either frequency or spatial distribution; the latter was useful for extracting the epileptic pattern obscured by severe artifacts. Nevertheless, successful use of this technique in practice depends on blind selection of appropriate model parameters.
Monitoring Neonatal Brain Recovery from EEG
We have worked to automate monitoring of the vulnerable brain in the Neonatal Intensive Care Unit following hypoxic insult or brain injury, after which seizures often emerge. We developed CPD-based algorithms similar to those described above and successfully extracted the onset of seizures . In addition, we sought to monitor brain recovery through automated quantification of the abnormality in one-hour EEG segments (called “EEG grading”). Our proposed holistic approach is shown in Figure 3. When we simultaneously decomposed all such tensors in a training set of 33 neonates via higher-order discriminant analysis (HODA), more relevant features were selected, thereby improving EEG grading accuracy (89%) and increasing robustness to the presence of artifacts .
Figure 3. The algorithm performs adaptive segmentation, according to the EEG amplitude (left: red, green, and blue indicate high, moderate, and low amplitude EEG respectively) and maps the segments’ features (amplitude, duration, spatial distribution) into a multidimensional histogram (middle) which is stored in a tensor (right). Image courtesy of Vladimir Matic.
Combining EEG and fMRI to Study Cognitive Function
Symmetric, data-driven fusion of EEG and fMRI has the potential to reveal and characterize the consecutive steps of cognitive brain processing with high spatiotemporal resolution. The methodological challenge is to define meaningful yet efficient multiway representations, decompositions, and constraints for the coupling of the two modalities.
Using a well-known visual detection task, we successfully applied Joint-ICA: structuring of EEG and fMRI data together in one matrix, followed by joint decomposition with ICA . This algorithm was extended to more complicated task paradigms (e.g., contour integration) and to a multichannel set-up. Extensions to a tensor-based framework are currently under investigation.
In summary, we have shown that tensor decompositions can be highly useful in biomedical data processing. CPD is now the main approach; the advantages of BTD are emerging. Tensors still have great unexplored potential in smart patient monitoring.
Acknowledgments. The author is supported by ERC Advanced Grant, #339804 BIOTENSORS. This article reflects only the author’s views, and the Union is not liable for any use that may be made of the contained information.
 A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Caiafa, and H.A. Phan, Tensor decompositions for signal processing applications: From two-way to multiway component analysis, IEEE Signal Processing Magazine, 32(2) (2015), 145-163.
 L. De Lathauwer, Blind separation of exponential polynomials and the decomposition of a tensor in rank-(Lr, Lr, 1) terms, SIAM J. Matrix Anal. Appl., 32 (2011), 1451-1474.
 L. De Lathauwer, Decompositions of a higher-order tensor in block terms—Part II: Definitions and uniqueness, SIAM J. Matrix Anal. Appl., 30 (2008), 1033-1066.
 M. De Vos, L. De Lathauwer, and S. Van Huffel, Decomposition methods in neuroscience, in Recent Advances in Biomedical Signal Processing, Bentham Science Publishers, Oak Park, Illinois, 2010, 1-27.
 B. Hunyadi, D. Camps, L. Sorber, W. Van Paesschen, M. De Vos, S. Van Huffel, and L. De Lathauwer, Block term decomposition for modelling epileptic seizures, EURASIP J. Adv. Signal Process, 139 (2014); doi:10.1186/1687-6180-2014-139.
 T. Kolda and B. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.
 V. Matic, P.J. Cherian, N. Koolen, G. Naulaers, R.M. Swarte, P. Govaert, S. Van Huffel, and M. De Vos, Holistic approach for automated background EEG assessment in asphyxiated full-term infants, J. Neural Engrg., 11(6) (2014), 1–14.
 B. Mijovic, K. Vanderperren, N. Novitskiy, B. Vanrumste, P. Stiers, B. Vand den Bergh, J. Wagemans, L. Lagae, S. Sunaert, S. Van Huffel, and M. De Vos, The why and how of JointICA: Results from a visual detection task, NeuroImage, 60(2) (2012), 1171-1185.
 M. Ullsperger and S. Debener, editors, Simultaneous EEG and fMRI: Recording, Analysis and Application, Oxford University Press, Oxford, 2010.