We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

Art Gallery and Illumination Problems

Chapter 15 – Geometric Shortest Paths and Network Optimization

With applications in biology, the world-wide web, and several other areas, mining of graph-structured objects has received significant interest recently One of the major research directions in this field is concerned with predictive data mining in graph databases where each instance is represented by a graph Some of the proposed approaches for this task rely on the excellent classification performance of support vector machines To control the computational cost of these approaches, the underlying kernel functions are based on frequent patterns In contrast to these approaches, we propose a kernel function based on a natural set of cyclic and tree patterns independent of their frequency, and discuss its computational aspects To practically demonstrate the effectiveness of our approach, we use the popular NCI-HIV molecule dataset Our experimental results show that cyclic pattern kernels can be computed quickly and offer predictive performance superior to recent graph kernels based on frequent patterns

Cyclic pattern kernels for predictive graph mining

We present an elastic registration algorithm for the alignment of biological images. Our method combines and extends some of the best techniques available in the context of medical imaging. We express the deformation field as a B-spline model, which allows us to deal with a rich variety of deformations. We solve the registration problem by minimizing a pixelwise mean-square distance measure between the target image and the warped source. The problem is further constrained by way of a vector-spline regularization which provides some control over two independent quantities that are intrinsic to the deformation: its divergence, and its curl. Our algorithm is also able to handle soft landmark constraints, which is particularly useful when parts of the images contain very little information or when its repartition is uneven. We provide an optimal analytical solution in the case when only landmarks and smoothness considerations are taken into account. We have applied our approach to perform the elastic registration of images such as electrophoretic gels and fly embryos. The validation of the results by experts has been favorable in all cases.

/pdf/elastic-registration-of-biological-images-using-vector-56zca8fh12.pdf

Elastic registration of biological images using vector-spline regularization

We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line.
Our analysis is doubly founded on a novel geometric structure called angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D, ${\cal AH}(D)$, to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of ${\cal AH}(D)$ cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight.

The Polygon Exploration Problem

Protein spot identification in two-dimensional electrophoresis gels can be supported by the comparison of gel images accessible in different World Wide Web two-dimensional electrophoresis (2-DE) gel protein databases. The comparison may be performed either by visual cross-matching between gel images or by automatic recognition of similar protein spot patterns. A prerequisite for the automatic point pattern matching approach is the detection of protein spots yielding the x(s),y(s) coordinates and integrated spot intensities i(s). For this purpose an algorithm is developed based on a combination of hierarchical watershed transformation and feature extraction methods. This approach reduces the strong over-segmentation of spot regions normally produced by watershed transformation. Measures for the ellipticity and curvature are determined as features of spot regions. The resulting spot lists containing x(s),y(s),i(s)-triplets are calculated for a source as well as for a target gel image accessible in 2-DE gel protein databases. After spot detection a matching procedure is applied. Both the matching of a local pattern vs. a full 2-DE gel image and the global matching between full images are discussed. Preset slope and length tolerances of pattern edges serve as matching criteria. The local matching algorithm relies on a data structure derived from the incremental Delaunay triangulation of a point set and a two-step hashing technique. For the incremental construction of triangles the spot intensities are considered in decreasing order. The algorithm needs neither landmarks nor an a priori image alignment. A graphical user interface for spot detection and gel matching is written in the Java programming language for the Internet. The software package called CAROL (http://gelmatching.inf.fu-berlin.de) is realized in a client-server architecture.

New algorithmic approaches to protein spot detection and pattern matching in two-dimensional electrophoresis gel databases.

We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262n simple cycles and at least 2.0845n Hamilton cycles.

Based on counting arguments for perfect matchings we prove that 2.3404n is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46.

/pdf/on-the-number-of-cycles-in-planar-graphs-4hhuf2ggus.pdf

On the number of cycles in planar graphs

In proteomics, two-dimensional gel electrophoresis (2-DE) is a separation technique for proteins. The resulting protein spots can be identified either by using picking robots and subsequent mass spectrometry or by visual cross inspection of a new gel image with an already analyzed master gel. Difficulties especially arise from inherent noise and irregular geometric distortions in 2-DE images. Aiming at the automated analysis of large series of 2-DE images, or at the even more difficult interlaboratory gel comparisons, the bottleneck is to solve the two most basic algorithmic problems with high quality: Identifying protein spots and computing a matching between two images. For the development of the analysis software CAROl at Freie Universitat Berlin, we have reconsidered these two problems and obtained new solutions which rely on methods from computational geometry. Their novelties are: 1. Spot detection is also possible for complex regions formed by several "merged" (usually saturated) spots; 2. User-defined landmarks are not necessary for the matching. Furthermore, images for comparison are allowed to represent different parts of the entire protein pattern, which only partially "overlap." The implementation is done in a client server architecture to allow queries via the internet. We also discuss and point at related theoretical questions in computational geometry.

Geometric algorithms for the analysis of 2D-electrophoresis gels.

We provide a competitive strategy for a mobile robot with vision, that has to explore an unknown simple polygon starting from and returning to a given point on the boundary. Our strategy creates a tour that does not exceed in length 133 times the length of the optimal watchman route. This paper is the first to describe a complete strategy and to give a proof for such a constant competitive factor.

Klaus Kriegel

Papers

The Polygon Exploration Problem

New algorithmic approaches to protein spot detection and pattern matching in two-dimensional electrophoresis gel databases.

On the number of cycles in planar graphs

Geometric algorithms for the analysis of 2D-electrophoresis gels.

A competitive strategy for learning a polygon