Given a finite set of points on the Euclidean sphere, the worst case quadrature error in Sobolev spaces has recently been shown to provide upper bounds on the covering radius of the point set. Moreover, quasi-Monte Carlo integration points on the sphere achieve the asymptotically optimal covering radius. Here, we extend these results to points on compact smooth Riemannian manifolds and provide numerical experiments illustrating our findings for the Grassmannian manifold.
Numerical integration and function approximation on compact Riemannian manifolds based on eigenfunctions of the Laplace-Beltrami operator have been widely studied in the recent literature. The standard example in numerical experiments is the Euclidean sphere. Here, we derive numerically feasible expressions for the approximation schemes on the Grassmannian manifold, and we present the associated numerical experiments on the Grassmannian. Indeed, our experiments illustrate and match the corresponding theoretical results in the literature.
Diabetic macular edema (DME) and retina vein occlusion (RVO) are macular diseases in which central photoreceptors are affected due to pathological accumulation of fluid. Optical coherence tomography allows to visually assess and evaluate photoreceptor integrity, whose alteration has been observed as an important biomarker of both diseases. However, the manual quantification of this layered structure is challenging, tedious and time-consuming. In this paper we introduce a deep learning approach for automatically segmenting and characterising photoreceptor alteration. The photoreceptor layer is segmented using an ensemble of four different convolutional neural networks. En-face representations of the layer thickness are produced to characterize the photoreceptors. The pixel-wise standard deviation of the score maps produced by the individual models is also taken to indicate areas of photoreceptor abnormality or ambiguous results. Experimental results showed that our ensemble is able to produce results in pair with a human expert, outperforming each of its constitutive models. No statistically significant differences were observed between mean thickness estimates obtained from automated and manually generated annotations. Therefore, our model is able to reliable quantify photoreceptors, which can be used to improve prognosis and managment of macular diseases.
PurposeThe purpose of the present study is to develop fast automated quantification of retinal fluid in optical coherence tomography (OCT) image sets.MethodsWe developed an image analysis pipeline tailored towards OCT images that consists of five steps for binary retinal fluid segmentation. The method is based on feature extraction, pre-segmention, dimension reduction procedures, and supervised learning tools.ResultsFluid identification using our pipeline was tested on two separate patient groups: one associated to neovascular age-related macular degeneration, the other showing diabetic macular edema. For training and evaluation purposes, retinal fluid was annotated manually in each cross-section by human expert graders of the Vienna Reading Center. Compared with the manual annotations, our pipeline yields good quantification, visually and in numbers.ConclusionsBy demonstrating good automated retinal fluid quantification, our pipeline appears useful to expert graders within their current grading processes. Owing to dimension reduction, the actual learning part is fast and requires only few training samples. Hence, it is well-suited for integration into actual manufacturer's devices, further improving segmentation by its use in daily clinical life.
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