Line generalisation by repeated elimination of points

This paper presents a new approach to line generalisation which uses the concept of ‘effective area’ for progressive simplification of a line by point elimination. Two coastlines are used to compare the performance of this, with that of the widely used Douglas-Peucker, algorithm. The results from the area-based algorithm compare favourably with manual generalisation of the same lines. It is capable of achieving both imperceptible minimal simplifications and caricatural generalisations. By careful selection of cutoff values, it is possible to use the same algorithm for scale-dependent and scale-independent generalisations. More importantly, it offers scope for modelling cartographic lines as consisting of features within features so that their geometric manipulation may be modified by application- and/or user-defined rules and weights. The paper examines the merits and limitations of the algorithm and the opportunities it offers for further research and progress in the field of line generalisation. © 1993 Maney Publishing

An adaptative method for the smothing of curves edge detection application

We present a new approach to smooth discrete curves . The smoothing is realized by associating portions of regular curves which are defined on each points interval . The originality of the method consists in finding the portions of curves by minimizing the squared error over a restricted neighbourhood around each point . Adding continuity constraints at the junction points, we obtain a direct formulation of the solution . A unique parameter allows to easily control the smoothing amplitude which can be selected between two extreme cases : interpolation or approximation . It seems like a drawer behaviour trying to join points by a curve . He can choose to join each point by a curve or only take into account the global form of the set of points . The method is particularly adapted to fit contours defined on an image and is used as a final step of image segmentation process . The parameter controlling the smoothing amplitude is computed from the value of local gradient magnitude on each pixel .La méthode de lissage de courbes discrètes présentée est fondée sur la minimisation d'un critère d'erreur quadratique appliqué sur des portions jointives de la courbe à traiter. En imposant des contraintes géométriques au niveau des points de jonctions entre intervalles, on aboutit à une formulation directe de la solution. Un paramètre unique permet de façon simple de contrôler la force du lissage qui évolue ainsi entre 2 cas extrêmes: l'interpolation et l'approximation. La méthode simule le comportement d'un scripteur cherchant à unir des points par une courbe, il peut privilégier le passage du tracé par chaque point ou au contraire respecter la forme globale définie par l'ensemble des points. Cette méthode adaptative de lissage est utilisée comme étape finale d'un processus de segmentation d'images, le paramètre contrôlant la force du lissage étant défini à partir du gradient mesuré localement en chaque point

Towards extracting artistic sketches and maps from digital elevation models

The main trend of computer graphics is the creation of photorealistic images however, there is increasing interest in the simulation of artistic and illustrative techniques. This thesis investigates a profile based technique for automatically extracting artistic sketches from regular grid digital elevation models. The results resemble those drawn by skilled cartographers and artists.The use of cartographic line simplification algorithms, which are usually applied to complex two-dimensional lines such as coastlines, allow a set of most important points on the terrain surface to be identified, these form the basis for sketching.This thesis also contains a wide ranging review of terrain representation techniques and suggests a new taxonomy

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)