2-D Shape Fitting for Locating Exploding Projectile from Explosion Patch

For test and evaluation of a proximity fuze, it is necessary to know the distance offset of the exploding ammunition round, fitted with the fuze, from a specific target. If the event is recorded using in-line high-speed photography the event of explosion can be resolved in time, and it becomes necessary to ascertain the position of the round, wrt the target, as it is exploding. An estimation of intensity centroid position fails as the flash is non-uniform in nature and is partially occluded by the exploding round. This paper is about an approach to find the location of the round using 2-D shapes fitting of the explosion patch.Defence Science Journal, 2010, 60(3), pp.238-243, DOI:http://dx.doi.org/10.14429/dsj.60.34

New Confocal Hyperbola-based Ellipse Fitting with Applications to Estimating Parameters of Mechanical Pipes from Point Clouds

This manuscript presents a new method for fitting ellipses to two-dimensional data using the confocal hyperbola approximation to the geometric distance of points to ellipses. The proposed method was evaluated and compared to established methods on simulated and real-world datasets. First, it was revealed that the confocal hyperbola distance considerably outperforms other distance approximations such as algebraic and Sampson. Next, the proposed ellipse fitting method was compared with five reliable and established methods proposed by Halir, Taubin, Kanatani, Ahn and Szpak. The performance of each method as a function of rotation, aspect ratio, noise, and arc-length were examined. It was observed that the proposed ellipse fitting method achieved almost identical results (and in some cases better) than the gold standard geometric method of Ahn and outperformed the remaining methods in all simulation experiments. Finally, the proposed method outperformed the considered ellipse fitting methods in estimating the geometric parameters of cylindrical mechanical pipes from point clouds. The results of the experiments show that the confocal hyperbola is an excellent approximation to the true geometric distance and produces reliable and accurate ellipse fitting in practical settings

Robust Detection of Non-overlapping Ellipses from Points with Applications to Circular Target Extraction in Images and Cylinder Detection in Point Clouds

This manuscript provides a collection of new methods for the automated detection of non-overlapping ellipses from edge points. The methods introduce new developments in: (i) robust Monte Carlo-based ellipse fitting to 2-dimensional (2D) points in the presence of outliers; (ii) detection of non-overlapping ellipse from 2D edge points; and (iii) extraction of cylinder from 3D point clouds. The proposed methods were thoroughly compared with established state-of-the-art methods, using simulated and real-world datasets, through the design of four sets of original experiments. It was found that the proposed robust ellipse detection was superior to four reliable robust methods, including the popular least median of squares, in both simulated and real-world datasets. The proposed process for detecting non-overlapping ellipses achieved F-measure of 99.3% on real images, compared to F-measures of 42.4%, 65.6%, and 59.2%, obtained using the methods of Fornaciari, Patraucean, and Panagiotakis, respectively. The proposed cylinder extraction method identified all detectable mechanical pipes in two real-world point clouds, obtained under laboratory, and industrial construction site conditions. The results of this investigation show promise for the application of the proposed methods for automatic extraction of circular targets from images and pipes from point clouds

A fast robust geometric fitting method for parabolic curves.

Fitting discrete data obtained by image acquisition devices to a curve is a common task in many fields of science and engineering. In particular, the parabola is some of the most employed shape features in electrical engineering and telecommunication applications. Standard curve fitting techniques to solve this problem involve the minimization of squared errors. However, most of these procedures are sensitive to noise. Here, we propose an algorithm based on the minimization of absolute errors accompanied by a normalization of the directrix vector that leads to an improved stability of the method. This way, our proposal is substantially resilient to noisy samples in the input dataset. Experimental results demonstrate the good performance of the algorithm in terms of speed and accuracy when compared to previous approaches, both for synthetic and real data.This work is partially supported by the Ministry of Economy and Competitiveness of Spain [grant number TIN2014-53465-R], project name Video surveillance by active search of anomalous events. It is also partially supported by the Autonomous Government of Andalusia (Spain) [grant number TIC-6213], project name Development of Self-Organizing Neural Networks for Information Technologies; and [grant number TIC-657], project name Self-organizing systems and robust estimators for video surveillance. All of them include funds from the European Regional Development Fund (ERDF). The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the SCBI (Supercomputing and Bioinformatics) center of the University of Málaga. They have also been supported by the Biomedic Research Institute of Málaga (IBIMA). They also gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU. Karl Thurnhofer-Hemsi is funded by a Ph.D. scholarship from the Spanish Ministry of Education, Culture and Sport under the FPU program [grant number FPU15/06512]

On Covering Points with Conics and Strips in the Plane

Geometric covering problems have always been of focus in computer scientific research. The generic geometric covering problem asks to cover a set S of n objects with another set of objects whose cardinality is minimum, in a geometric setting. Many versions of geometric cover have been studied in detail, one of which is line cover: Given a set of points in the plane, find the minimum number of lines to cover them. In Euclidean space Rm, this problem is known as Hyperplane Cover, where lines are replaced by affine hyperplanes bounded by dimension d. Line cover is NP-hard, so is its hyperplane analogue. Our thesis focuses on few extensions of hyperplane cover and line cover. One of the techniques used to study NP-hard problems is Fixed Parameter Tractability (FPT), where, in addition to input size, a parameter k is provided for input instance. We ask to solve the problem with respect to k, such that the running time is a function in both n and k, strictly polynomial in n, while the exponential component is limited to k. In this thesis, we study FPT and parameterized complexity theory, the theory of classifying hard problems involving a parameter k. We focus on two new geometric covering problems: covering a set of points in the plane with conics (conic cover) and covering a set of points with strips or fat lines of given width in the plane (fat line cover). A conic is a non-degenerate curve of degree two in the plane. A fat line is defined as a strip of finite width w. In this dissertation, we focus on the parameterized versions of these two problems, where, we are asked to cover the set of points with k conics or k fat lines. We use the existing techniques of FPT algorithms, kernelization and approximation algorithms to study these problems. We do a comprehensive study of these problems, starting with NP-hardness results to studying their parameterized hardness in terms of parameter k. We show that conic cover is fixed parameter tractable, and give an algorithm of running time O∗ ((k/1.38)^4k), where, O∗ implies that the running time is some polynomial in input size. Utilizing special properties of a parabola, we are able to achieve a faster algorithm and show a running time of O∗ ((k/1.15)^3k). For fat line cover, first we establish its NP-hardness, then we explore algorithmic possibilities with respect to parameterized complexity theory. We show W [1]-hardness of fat line cover with respect to the number of fat lines, by showing a parameterized reduction from the problem of stabbing axis-parallel squares in the plane. A parameterized reduction is an algorithm which transforms an instance of one parameterized problem into an instance of another parameterized problem using a FPT-algorithm. In addition, we show that some restricted versions of fat line cover are also W [1]-hard. Further, in this thesis, we explore a restricted version of fat line cover, where the set of points are integer coordinates and allow only axis-parallel lines to cover them. We show that the problem is still NP-hard. We also show that this version is fixed parameter tractable having a kernel size of O (k^2) and give a FPT-algorithm with a running time of O∗ (3^k). Finally, we conclude our study on this problem by giving an approximation algorithm for this version having a constant approximation ratio 2

On the capture and representation of fonts

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The commercial need to capture, process and represent the shape and form of an outline has lead to the development of a number of spline routines. These use a mathematical curve format that approximates the contours of a given shape. The modelled outline lends itself to be used on, and for, a variety of purposes. These include graphic screens, laser printers and numerically controlled machines. The latter can be employed for cutting foil, metal. plastic and stone. One of the most widely used software design packages has been the lKARUS system. This, developed by URW of Hamburg (Gennany), employs a number of mathematical descriptions that facilitate the process of both modelling and representation of font characters. It uses a variety of curve formats, including Bezier cubics, general conics and parabolics. The work reported in this dissertation focuses on developing improved techniques, primarily. for the lKARUS system. This includes two algorithms which allow a Bezier cubic description, two for a general conic representation and, yet another, two for the parabolic case. In addition, a number of algorithms are presented which promote conversions between these mathematical forms; for example, Bezier cubics to a general conic form. Furthennore, algorithms are developed to assist the process of rasterising both cubic and quadratic arcs.This study was partly funded by the Science and Education Research Council (SERC)