Detection algorithm of paroxysmal atrial fibrillation with sparse coding based on Riemannian manifold / 生物医学工程学杂志

In order to solve the problem that the early onset of paroxysmal atrial fibrillation is very short and difficult to detect, a detection algorithm based on sparse coding of Riemannian manifolds is proposed. The proposed method takes into account that the nonlinear manifold geometry is closer to the real feature space structure, and the computational covariance matrix is used to characterize the heart rate variability (RR interval variation), so that the data is in the Riemannian manifold space. Sparse coding is applied to the manifold, and each covariance matrix is represented as a sparse linear combination of Riemann dictionary atoms. The sparse reconstruction loss is defined by the affine invariant Riemannian metric, and the Riemann dictionary is learned by iterative method. Compared with the existing methods, this method used shorter heart rate variability signal, the calculation was simple and had no dependence on the parameters, and the better prediction accuracy was obtained. The final classification on MIT-BIH AF database resulted in a sensitivity of 99.34%, a specificity of 95.41% and an accuracy of 97.45%. At the same time, a specificity of 95.18% was realized in MIT-BIH NSR database. The high precision paroxysmal atrial fibrillation detection algorithm proposed in this paper has a potential application prospect in the long-term monitoring of wearable devices.

Focal rigidity of flat tori

Given a closed Riemannian manifold (M, g), i.e. compact and boundaryless, there is a partition of its tangent bundle TM = ∪iΣi called the focal decomposition of TM. The sets Σi are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that flat n-tori, n > 2, are focally rigid in the sense that if two flat tori are focally equivalent then the tori are isometric up to rescaling. The case n = 2 was considered before by F. Kwakkel.

Dada uma variedade Riemanniana (M, g) fechada, isto é, compacta e sem bordo, existe uma partição de seu fibrado tangente TM = ∪iΣi chamada decomposição focal de TM. Os conjuntos Σi estão intimamente associados ao modo como focalizam as geodésicas de (M,g), isto é, à situação em que existem exatamente i arcos de geodésica de mesmo comprimento unindo pontos p e q em M. Nesta nota, estudamos a estrutura topológica da decomposição focal de uma variedade Riemanniana fechada e sua relação com a estrutura métrica de M. Nosso principal resultado é que n-toros planos, n > 2, são focalmente rigidos, isto é, se dois toros planos são focalmente equivalentes, então os dois toros são isométricos módulo mudança de escala. O caso n = 2 foi considerado anteriormente por F. Kwakkel.