Thinning-free Polygonal Approximation of Thick Digital Curves Using Cellular Envelope

Since the inception of successful rasterization of curves and objects in the digital space, several algorithms have been proposed for approximating a given digital curve. All these algorithms, however, resort to thinning as preprocessing before approximating a digital curve with changing thickness. Described in this paper is a novel thinning-free algorithm for polygonal approximation of an arbitrarily thick digital curve, using the concept of "cellular envelope", which is newly introduced in this paper. The cellular envelope, defined as the smallest set of cells containing the given curve, and hence bounded by two tightest (inner and outer) isothetic polygons, is constructed using a combinatorial technique. This envelope, in turn, is analyzed to determine a polygonal approximation of the curve as a sequence of cells using certain attributes of digital straightness. Since a real-world curve=curve-shaped object with varying thickness, unexpected disconnectedness, noisy information, etc., is unsuitable for the existing algorithms on polygonal approximation, the curve is encapsulated by the cellular envelope to enable the polygonal approximation. Owing to the implicit Euclidean-free metrics and combinatorial properties prevailing in the cellular plane, implementation of the proposed algorithm involves primitive integer operations only, leading to fast execution of the algorithm. Experimental results that include output polygons for different values of the approximation parameter corresponding to several real-world digital curves, a couple of measures on the quality of approximation, comparative results related with two other well-referred algorithms, and CPU times, have been presented to demonstrate the elegance and efficacy of the proposed algorithm

Polygonal path simplification with angle constraints

We present efficient geometric algorithms for simplifying polygonal paths in R2 and R3 that have angle constraints, improving by nearly a linear factor over the graph-theoretic solutions based on known techniques. The algorithms we present match the time bounds for their unconstrained counterparts. As a key step in our solutions, we formulate and solve an off-line ball exclusion search problem, which may be of interest in its own right

Revisiting the Evolution and Application of Assignment Problem: A Brief Overview

The assignment problem (AP) is incredibly challenging that can model many real-life problems. This paper provides a limited review of the recent developments that have appeared in the literature, meaning of assignment problem as well as solving techniques and will provide a review on   a lot of research studies on different types of assignment problem taking place in present day real life situation in order to capture the variations in different types of assignment techniques. Keywords: Assignment problem, Quadratic Assignment, Vehicle Routing, Exact Algorithm, Bound, Heuristic etc

Applications of network optimization

Includes bibliographical references (p. 41-48).Ravindra K. Ahuja ... [et al.]

Modeling Surfaces from Volume Data Using Nonparallel Contours

Magnetic resonance imaging: MRI) and computed tomography: CT) scanners have long been used to produce three-dimensional samplings of anatomy elements for use in medical visualization and analysis. From such datasets, physicians often need to construct surfaces representing anatomical shapes in order to conduct treatment, such as irradiating a tumor. Traditionally, this is done through a time-consuming and error-prone process in which an experienced scientist or physician marks a series of parallel contours that outline the structures of interest. Recent advances in surface reconstruction algorithms have led to methods for reconstructing surfaces from nonparallel contours that could greatly reduce the manual component of this process. Despite these technological advances, the segmentation process has remained unchanged. This dissertation takes the first steps toward bridging the gap between the new surface reconstruction technologies and bringing those methods to use in clinical practice. We develop VolumeViewer, a novel interface for modeling surfaces from volume data by allowing the user to sketch contours on arbitrarily oriented cross-sections of the volume. We design the algorithms necessary to support nonparallel contouring, and we evaluate the system with medical professionals using actual patient data. In this way, we begin to understand how nonparallel contouring can aid the segmentation process and expose the challenges associated with a nonparallel contouring system in practice

The optimal assignment problem: an investigation into current solutions, new approaches and the doubly stochastic polytope

MSc(Eng),Faculty of Engineering and the Built Environment, University of the Witwatersrand, 2010This dissertation presents two important results: a novel algorithm that approximately solves the optimal assignment problem as well as a novel method of projecting matrices into the doubly stochastic polytope while preserving the optimal assignment. The optimal assignment problem is a classical combinatorial optimisation problem that has fuelled extensive research in the last century. The problem is concerned with a matching or assignment of elements in one set to those in another set in an optimal manner. It finds typical application in logistical optimisation such as the matching of operators and machines but there are numerous other applications. In this document a process of iterative weighted normalization applied to the benefit matrix associated with the Assignment problem is considered. This process is derived from the application of the Computational Ecology Model to the assignment problem and referred to as the OACE (Optimal Assignment by Computational Ecology) algorithm. This simple process of iterative weighted normalisation converges towards a matrix that is easily converted to a permutation matrix corresponding to the optimal assignment or an assignment close to optimality. The document also considers a method of projecting a matrix into the doubly stochastic polytope while preserving the optimal assignment. Various methods of projecting square matrices into the doubly stochastic polytope exist but none that preserve the assignment. This novel result could prove instrumental in solving assignment problems and promises applications in other optimisation algorithms similar to those that Sinkhorn’s algorithm finds

Applications of network optimization

Includes bibliographical references (p. 41-48).Ravindra K. Ahuja ... [et al.]

Algorithms for Geometric Covering and Piercing Problems

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)

On Flows, Paths, Roots, and Zeros

This thesis has two parts; in the first of which we give new results for various network flow problems. (1) We present a novel dual ascent algorithm for min-cost flow and show that an implementation of it is very efficient on certain instance classes. (2) We approach the problem of numerical stability of interior point network flow algorithms by giving a path following method that works with integer arithmetic solely and is thus guaranteed to be free of any nu-merical instabilities. (3) We present a gradient descent approach for the undirected transship-ment problem and its special case, the single source shortest path problem (SSSP). For distrib-uted computation models this yields the first SSSP-algorithm with near-optimal number of communication rounds. The second part deals with fundamental topics from algebraic computation. (1) We give an algorithm for computing the complex roots of a complex polynomial. While achieving a com-parable bit complexity as previous best results, our algorithm is simple and promising to be of practical impact. It uses a test for counting the roots of a polynomial in a region that is based on Pellet's theorem. (2) We extend this test to polynomial systems, i.e., we develop an algorithm that can certify the existence of a k-fold zero of a zero-dimensional polynomial system within a given region. For bivariate systems, we show experimentally that this approach yields signifi-cant improvements when used as inclusion predicate in an elimination method.Im ersten Teil dieser Dissertation präsentieren wir neue Resultate für verschiedene Netzwerkflussprobleme. (1)Wir geben eine neue Duale-Aufstiegsmethode für das Min-Cost-Flow- Problem an und zeigen, dass eine Implementierung dieser Methode sehr effizient auf gewissen Instanzklassen ist. (2)Wir behandeln numerische Stabilität von Innere-Punkte-Methoden fürNetwerkflüsse, indem wir eine solche Methode angeben die mit ganzzahliger Arithmetik arbeitet und daher garantiert frei von numerischen Instabilitäten ist. (3) Wir präsentieren ein Gradienten-Abstiegsverfahren für das ungerichtete Transshipment-Problem, und seinen Spezialfall, das Single-Source-Shortest-Problem (SSSP), die für SSSP in verteilten Rechenmodellen die erste mit nahe-optimaler Anzahl von Kommunikationsrunden ist. Der zweite Teil handelt von fundamentalen Problemen der Computeralgebra. (1) Wir geben einen Algorithmus zum Berechnen der komplexen Nullstellen eines komplexen Polynoms an, der eine vergleichbare Bitkomplexität zu vorherigen besten Resultaten hat, aber vergleichsweise einfach und daher vielversprechend für die Praxis ist. (2)Wir erweitern den darin verwendeten Pellet-Test zum Zählen der Nullstellen eines Polynoms auf Polynomsysteme, sodass wir die Existenz einer k-fachen Nullstelle eines Systems in einer gegebenen Region zertifizieren können. Für bivariate Systeme zeigen wir experimentell, dass eine Integration dieses Ansatzes in eine Eliminationsmethode zu einer signifikanten Verbesserung führt