Klaus Kriegel

The Polygon Exploration Problem

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,

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

{\cal AH}(D)

The art gallery theorem for polygons with holes

, 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

Geometric algorithms for the analysis of 2D-electrophoresis gels

{\cal AH}(D)

On the number of cycles in planar graphs

cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight.

An applied point pattern matching problem: comparing 2D patterns of protein spots

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.

A Graph-Coloring Result and Its Consequences For Polygon-Guarding Problems

Art gallery problems which have been extensively studied over the last decade ask how to station a small (minimum) set of guards in a polygon such that every point of the polygon is watched by at least one guard. The graph-theoretic formulation and solution to the gallery problem for polygons in standard form is given. A complexity analysis is carried out, and open problems are discussed.<<ETX>>

An alternative approach to deal with geometric uncertainties in computer analysis of two-dimensional electrophoresis gels.

In proteomics 2-dimensional gel electrophoresis (2-DE) is a separation technique for proteins. The resulting protein spots can be identified by either 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 Universität 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.

Embedding rectilinear graphs in linear time

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.

Generalized guarding and partitioning for rectilinear polygons

Abstract It is shown how to use various ideas from computational geometry to derive a new algorithmic solution to the matching problem of 2D patterns of protein spots obtained by the 2D gel electrophoresis technique. The algorithm especially relies on a data structure derived from the incremental Delaunay triangulation of a point set and several heuristics to cope with distortions and noise inherent to the electrophoresis process. The main feature of the presented solution is that interactive landmark setting is optional and not necessary.