Covering Simple Polygonal Regions by Ellipses

We study the problem of how to cover simple polygonal rectilinear regions by a small set 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 electrophoresis images. We present and discuss various algorithmic approaches towards this problem ranging from a brute force method, to a linear programming formulation and an efficient theoretical solution. 1 Detecting Spots in 2–dimensional Gel Electrophoresis Images 1.1 Gel Electrophoresis: The Application Background With the growing importance of proteomics in biomedical and pharmaceutical sciences , see [7], there is an increasing need for efficient, reliable, and robust algorithmic solutions for various tasks that are part of an automatic analysis tool for 2–dimensional electrophoresis (2DE) gel images. With a resolution separation of several thousand proteins in real samples this method is almost two orders of magnitude better than competing Part of a joint research project with Deutsches Herzzentrum Berlin, supported by Deutsche Forschungsgemeinschaft, grant FL 165/4–1. yCS Dept. Stanford University, email:[email protected] zInstitut für Informatik, Freie Universität Berlin, Takustr. 9, D14195 Berlin, email:[email protected] techniques. A 2DE gel is the product of two separations performed sequentially in acrylamide gel media: isoelectric focusing as the first 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. Ideally, each spot has the shape of an axis–parallel ellipse. However, as outlined below spots that are very close to each other can partially overlap and form rather complex regions. From the application point of view the overall goal is to analyse images from a whole gel series in order to identify those proteins that changes their expression (size, intensity) what could be a hint that reflects/causes certain biochemical and biomedical conditions of an organism. There are several commercial software packages available (like Melanie, PD Quest, Phoretix), which, together with their corresponding hardware, offer complete solutions to the gel analysis task including statistical analysis. However, these also have drawbacks since they do not support the comparison of images drawn from various sources (like databases in the Internet) which may have different size for example and, on the other side, they are too expensive for somebody who wants too evaluate just a few gels once in a while. With this background we have started a few years ago to develop a softare system CAROL (see [1]) that is able to perform local and global matching queries for gel images (say, given in gif format) via the Internet. Its novel algorithmic idea was to avoid setting landmarks by hand. Instead, the matching between a source and a target image uses the history of the incremental Delaunay triangulation, [3], [4], of the target spots. To this end we assumed 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, as provided e. g. by the PDQuest system. Then, matching criteria are both geometric resemblance of locally intensive spot patterns as well as spot neighborhood comparisons. Meanwhile it has become lucid that wrongly detected spots like twin spots, spots within so called streaks or other complex regions are the main obstacle towards a better performance of the matching tool compared to influence of geometric distortions . That is why we decided to develop and include a new spot detection algorithm into the CAROL system to overcome these difficulties. In Figure 1 a part of a gel image is shown and aside the ellipses representing spots as computed by our spot detection algorithm. The main steps of the detection can be summarized as follows, compare with Figure 2 (numbered left to right). The algorithm, see [6] for details, starts smoothing the pixel image by a Gaussian filter (1) and computes a gradient image (2). Applying the watershed transformation to the gradient image we get a segmentation of the original image (3). Ideally, every obtained segment should represent a spot. In reality many more segments than there are spots are produced. By comparing segments with their neighborhood (using neighborhood graphs) it is possible to select those segments which really are part of a spot (4). All these segments should be merged to spots. Typical simple situations are shown in Figure 3. Some of the segments are isolated and form a single spot (top left). It can happen that two or three neighboring segments have to be merged. This case (top right) is solved by covering each subset by axis–parallel ellipses and comparing which covering fits best, i. e. , minimizes the symetric difference. However, there are situations like the bottom one in Figure 3, where the combinatorial complexity does not allow such an exhaustive search. Nevertheless, each such region has to be interpreted as union of ellipses, since it is typically oversaturated (so gray level values do not help here) and, most important, such intensive regions play an important role in the matching procedure. Typically, in complicated 2DE gel images there are up to 10 such complex regions. In fact, not only for very complex regions but also for twin spots and streaks there is an inherent uncertainty in the 2DE gel images due to the electrophoresis process itself which is highly susceptible to faults and distortions. Consequently, in the recent CAROL version we cope with that problem by maintaining a list of proposals how such an ambiguous region could be covered instead of computing only the best covering. In the matching algorithm part we then have the possibility to accept the matching of two complex region if there is a pair of proposed coverings that match. Figure 1 shows only 15 percent of a full 2DE gel image in GIF format. The total size is 811 900 pixel and our spot detection algorithm used about 9 sec to compute a total of 553 spots. 1.2 Modelling the Covering Problem We assume that a simply connected pixel pattern R is given. In the application it is usually a subpattern of a 100 100–square. Let R denote the polygonal curve describing its boundary. Identifying a pixel with its center point p we define the pixel sets R