A certified conflict locator for the incremental maintenance of the Delaunay graph of semi-algebraic sets

Most of the curves and surfaces encountered in geometric modelling are defined as the set of solutions of a system of algebraic equations or inequalities (semi-algebraic sets). The Voronoi diagram of a set of sites is a decomposition of the space into proximal regions (one for each site). Voronoi diagrams have been used to answer proximity queries. The dual graph of the Voronoi diagram is called the Delaunay graph. Only approximations by conics can guarantee a proper continuity of the first order derivative at contact points, which is necessary for guaranteeing the exactness of the Delaunay graph. The central idea of this paper is that a (one time) symbolic preprocessing may accelerate the certified numerical evaluation of the Delaunay graph conflict locator. The symbolic preprocessing is the computation of the implicit equation of the generalised offset to conics. The certified computation of the Delaunay graph conflict locator relies on theorems on the uniqueness of a root in given intervals (Kantorovich, Moore-Krawczyk). For conics, the computations get much faster by considering only the implicit equations of the generalised offsets

CUDA based Level Set Method for 3D Reconstruction of Fishes from Large Acoustic Data

Acoustic images present views of underwater dynamics, even in high depths. With multi-beam echo sounders (SONARs), it is possible to capture series of 2D high resolution acoustic images. 3D reconstruction of the water column and subsequent estimation of fish abundance and fish species identification is highly desirable for planning sustainable fisheries. Main hurdles in analysing acoustic images are the presence of speckle noise and the vast amount of acoustic data. This paper presents a level set formulation for simultaneous fish reconstruction and noise suppression from raw acoustic images. Despite the presence of speckle noise blobs, actual fish intensity values can be distinguished by extremely high values, varying exponentially from the background. Edge detection generally gives excessive false edges that are not reliable. Our approach to reconstruction is based on level set evolution using Mumford-Shah segmentation functional that does not depend on edges in an image. We use the implicit function in conjunction with the image to robustly estimate a threshold for suppressing noise in the image by solving a second differential equation. We provide details of our estimation of suppressing threshold and show its convergence as the evolution proceeds. We also present a GPU based streaming computation of the method using NVIDIA’s CUDA framework to handle large volume data-sets. Our implementation is optimised for memory usage to handle large volumes

Polygon Feature Extraction from Satellite Imagery Based on Colour Image Segmentation and Medial Axis

Areal features are of great importance in applications like shore line mapping, boundary delineation and change detection. This research work is an attempt to automate the process of extracting feature boundaries from satellite imagery. This process is intended to eventually replace manual digitization by computer assisted boundary detection and conversion to a vector layer in a Geographic Information System. Another potential application is to be able to use the extracted linear features in image matching algorithms. In multi-spectral satellite imagery, various features can be distinguished based on their colour. There has been a good amount of work already done as far as boundary detection and skeletonization is concerned, but this research work is different from the previous ones in the way that it uses the Delaunay graph and the Voronoi tessellation to extract boundary and skeletons that are guaranteed to be topologically equivalent to the segmented objects. The features thus extracted as object border can be stored as vector maps in a Geographic Information System after labelling and editing. Here we present a complete methodology of the skeletonization process from satellite imagery using a colour image segmentation algorithm with examples of road networks and hydrographic networks.

A short ODE proof of the Fundamental Theorem of Algebra

We propose a short proof of the Fundamental Theorem of Algebra based on the ODE that describes the Newton flow and the fact that the value $|P(z)|$ is a Lyapunov function. It clarifies an idea that goes back to Cauchy