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1

Sleep EEG analysis utilizing inter-channel covariance matrices

Gopan K. Gopika, Prabhu Sathvik S., Sinha Neelam

Biocybernetics and Biomedical Engineering

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2020

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Vol. 40, no. 1

527--545

EN

Background: Sleep is vital for normal body functions as sleep disorders can adversely affect a person. Electroencephalographic (EEG) signals indicate brain functions and have characteristic signatures for various sleep stages. These enable the use of EEG as an effective tool for in-depth studies about sleep. Sleep stages are broadly divided as rapid eye movement (REM) and non-rapid eye movement (NREM). NREM is further divided into 3 stages. The objective of the work is to distinguish the given EEG epoch as wake, NREM1, NREM2, NREM3 and REM. DREAMS Subject Database containing 5 EEG channels is used here. This work focuses on utilizing EEG by exploiting variations in inter-dependencies of different brain regions during sleep. New method: Covariance matrices of the wavelet-decomposed channels are used to obtain the variations in inter-dependencies. The feature sets are: (1) simple matrix properties(MF) like trace, determinant and norm, (2) eigen-values (E1), (3) eigen-vector corresponding to the largest eigen-value (E2) and (4) tangent vectors obtained using Riemann geometry (RG-TS). The features are input to ensemble classifier with bagging. Subject-specific, All-subjects-combined and Leave-one-subject-out methods of analysis are carried out. Results: In all methods of analysis, RG-TS features give maximum accuracy (80.05%, 83.05% and 61.79%), closely followed by E1 (79.49%, 77.14% and 58.34%). Comparison with existing method: The proposed method obtains higher and/or comparable accuracy. This work also ensures no biasing of classifier due to unequal class distribution. Conclusion: The performances of RG-TS and E1 features reveal that the changes in interdependencies of pre-frontal and occipital lobe along with the central lobe can be used to distinguish the different sleep stages.

2

Kernel-based Fisher discriminant analysis on the Riemannian manifold for nuclear atypia scoring of breast cancer

Das Asha, Nair Madhu S., Peter David

Biocybernetics and Biomedical Engineering

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2019

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Vol. 39, no. 3

728--741

EN

Breast carcinoma is the most prevalent type of malignancy among women worldwide. Breast cancer grading often termed as Nuclear Atypia Scoring (NAS) forms a significant factor in determining individualized treatment plans and in the prognosis of the disease. For addressing the problem of breast cancer grading, we attempt to model the variations in features between histopathological images of different cancer grades and thereby explore the discriminative information concealed in these variations. In this regard, we aggregate multiple correlated features from the images using the geodesic geometric mean of the region covariances, to obtain the gmRC descriptors. As these gmRC descriptors are symmetric positive definite (SPD) matrices lying on the non-Euclidean Riemannian manifold, the discriminant analysis techniques developed for the Euclidean framework may not be appropriate. Hence, we propose a kernel-based Fisher discriminant analysis on the Riemannian manifold (KFDAR), that exploits the kernel trick for embedding the non-linear Riemannian manifold M into a higher dimensional linear Hilbert space H, which are then reduced to a low-dimensional and more discriminative subspace, where the samples become linearly separable. The kernel approach formulated for the Hilbert space embedding and for the kernel discriminant analysis is based on three Riemannian distance metrics: the log-Euclidean metric and the two symmetrized Bregman divergences – Stein and Jeffrey divergences. The experimental results show that this mapping to a highly discriminative space has succeeded in well-separating the histopathological images belonging to different cancer grades and hence it qualitatively and quantitatively outperforms the existing algorithms for cancer grading.

3

Semi-slant submersions from almost product Riemannian manifolds

Gündüzalp Y.

Demonstratio Mathematica

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2016

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Vol. 49, nr 3

345--356

EN

In this paper, we introduce semi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give some examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion. We also find necessary and sufficient conditions for a semi-slant submersion to be totally geodesic.

4

Radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds

Shukla S. S., Yadav A.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

994--1011

EN

In this paper, we study radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds giving some non-trivial examples of these submanifolds. Integrability conditions of distributions D and RadTM on radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds, have been obtained. We also study totally contact umbilical radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds.

5

On almost pseudo conformally symmetric manifolds

De U., Gazi A.

Demonstratio Mathematica

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2009

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Vol. 42, nr 4

869-886

EN

The object of the present paper is to study a type of non-conformally flat semi-Riemannian manifolds called almost pseudo conform ally symmetric manifold. The existence of an almost pseudo conformally symmetric manifold is also shown by a non-trivial example.

6

Contact horizontally conformal submersions

Sahin B.

Demonstratio Mathematica

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2009

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Vol. 42, nr 4

857-868

EN

Using the notion of horizontally conformal submersion, we generalize the contact metric submersions and obtain classification theorems for this submersion when the total manifold has some special almost contact structures.

7

Almost locally conformal Kaehler product manifolds

Sahin B.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

211-218

EN

It is known that the product of two locally conformal Kaehler manifolds is not a locally conformal Kaehler manifold, ([1], P:46 ). In this paper we introduce an almost locally conformal Kaehler product manifold and show that the product of two locally conformal Kaehler manifolds is an almost locally conformal Kaehler manifold. Moreover, we investigate properties of curvature tensors of an almost locally conformal Kaehler Product manifold.