No abstract

We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

We give a provably correct algorithm to reconstruct a k-dimensional manifold embedded in d-dimensional Euclidean space. Input to our algorithm is a point sample coming from an unknown manifold. Our approach is based on two main ideas : the notion of tangential Delaunay complex defined in [6,19,20], and the technique of sliver removal by weighting the sample points [13]. Differently from previous methods, we do not construct any subdivision of the embedding d-dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k. To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced provided the transition functions are bi-Lipschitz with a constant close to 1, and the sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold.The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the ouput complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric.

Arsenic contamination of rice irrigated with contaminated groundwater contributes to the additional arsenic burden of the population where rice is the staple food. In an arsenic contaminated area, an experimental field-based study done on nine fields elucidated significant positive correlation between arsenic in irrigation water and soil, irrigation water and rice, and also soil and rice both for Boro (groundwater) and Aman (rainwater) rice. Speciation studies showed that for both Boro (cooked) and Aman (raw) rice from contaminated area, 90% of total recovered arsenic was inorganic. In arsenic contaminated, uncontaminated villages, and Kolkata city, daily quantities of arsenic ingested by adult population from cooked rice diet are equivalent to 6.5, 1.8, and 2.3 L respectively, of drinking water containing WHO guideline value. In contaminated area, daily intake only from cooked Boro rice for 34.6% of the samples exceeded the WHO recommended MTDI value (2 microg In-As day(-1) kg(-1) body wt), whereas daily intake from Aman rice was below MTDI value as was rice from uncontaminated areas and Kolkata city. Our study indicated that employing traditional rice cooking method as followed in Bengal delta and using water having arsenic <3 microg L(-1) for cooking, actual exposure to arsenic from rice would be much less.