We study the problem of how to cover a set of points by a small number of axis-parallel ellipses. This question is well motivated by a special pattern recognition task where one has to identify ellipse-shaped protein spots in 2-dimensional gel electrophoresis images. See [4] for a description of the application and for several algorithms for practical variations of the problem. Here we first investigate the covering problem from a theoretical point of view, and then consider a restricted variant induced by the application.

We study the problem of covering a polygonal region by a small number of axis{parallel ellipses. This question is well motivated by the special pattern recognition task of identifying ellipse shaped protein spots in 2{dimensional electrophoresis images. We implemented several algorithms solving this problem: a greedy brute force method and a linear programming formulation. Several other algorithms are presented and discussed from a more theoretical point of view. 1 Detecting Spots in 2{dimensional Gel Electrophoresis Images 1.1 Gel Electrophoresis: The Application Background Proteomics is a rapidly growing eld within computational molecular biology. In proteomics 2{ dimensional gel electrophoresis (2DE) is the best known and widely used technique to separate proteins. A 2DE gel is the product of two separations performed sequentially in acrylamide gel media: isoelectric focusing as the rst dimension and a separation by molecular size as the second dimension. A two-dimensional pattern of spots each representing a protein is the result of that process. Eventually, spots are made visible by staining or radiographic methods. By analyzing series of such 2DE images one hopes to identify the proteins that change their expression (size, intensity) and re ect/cause certain biochemical and biomedical conditions of an organism, see [17]. Ideally, in an gel image each spot has the shape of an axis{parallel ellipse, which is a widely accepted modeling assumption, see e.g. [3] or [12]. However, spots that are very close to each other can partially merge (modeled by overlapping ellipses) and form rather complicated regions as depicted in Figure 1. At Freie Universitat Berlin we have started a few years ago to develop the software system CAROL (see [6]) that answers local and global matching queries for gel images. The novelty of its matching tool is the automatization of the process of setting the landmarks, Alon says: Is landmark well de ned ? which is the process that is done by human in other systems. Alon says: Did I say it right ? Instead, our matching between a source and a target image uses the history of the incremental Delaunay triangulation ([13, 11]) of the target spots. The matching tool starts from the assumption that images are already given as spot lists with each spot represented by point coordinates of its center and a real value describing its intensity. However, since the accessible spot detection algorithms did not supply results precise enough for our approach we developed and included a new detection algorithm into the CAROL system, see [15] for details. Research has been partly supported by Deutsche Forschungsgemeinschaft, grant FL 165/4{1. y Department of Computer Science, the University of Arizona, email:alon@cs.arizona.EDU z Institut f ur Informatik, Freie Universitat Berlin, Takustr. 9, D-14195 Berlin, email:ho mann@inf.fu-berlin.de x Institut f ur Informatik, Freie Universitat Berlin, Takustr. 9, D-14195 Berlin, email:kriegel@inf.fu-berlin.de { Institut f ur Informatik, Freie Universitat Berlin, Takustr. 9, D-14195 Berlin, email:schultz@inf.fu-berlin.de

With the growing importance of proteomics in biomedical and pharmaceutical sciences a need has emerged for computing tools that are capable of digitally visualizing and analyzing protein spot patterns within two-dimensional electrophoresis (2-DE) gel. Matching programs need to meet requirements such as interlaboratory comparison and the comparison of samples from different origins. For such research purposes, we have developed the CAROL system that implements new algorithms for spot detection and matching, which enable researchers to take a different approach to protein spot identification and comparison. The present short communication discusses how the system deals with uncertain geometric spot information that arises from streaks and complex spot regions and how this can be amplified for the matching procedure.

We provide a new on-line strategy that enables a mobile robot with vision to explore an, unknown polygon by a tour less than 26.5 times as long as the shortest watchman tour. This improves considerably on the best upper bound of 133 known so far. Our strategy uses a new way of dynamically decomposing the polygon. The analysis is based on a novel geometric structure called the angle hull.

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.

A Tk-guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible from some point contained in G. We investigate the function r(n,h,k), which is the largest number of Tk-guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove new lower and upper bounds on parts of this function. In particular, we show the following upper bounds: 1. r(n,0,k)⩽ n k+4  , with equality for even k. 2. r(n,h,1)= n+ 4h 3 + 4 3 4+ 4 3  3. (n,h,2)⩾ n 6 These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k > 1. In the course of proving the upper bounds, new results on partitioning are obtained which also have efficient algorithmic versions.