The chromatic art gallery problem asks for the minimum number of “colors” t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility—two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is in the worst case Θ(log⁡log⁡n). By contrast, the best known upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log⁡n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is, again in the worst case, Θ(log⁡n). Finally, our techniques also help us establish the first non-trivial lower bound of Ω(log⁡log⁡n/log⁡log⁡log⁡n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.

We address recently proposed chromatic versions of the classic Art Gallery Problem. Assume a simple polygon P is guarded by a finite set of point guards and each guard is assigned one of t colors. Such a chromatic guarding is said to be conflict-free if each point p∈ P sees at least one guard with a unique color among all guards visible from p. The goal is to establish bounds on the function χ_cf(n) of the number of colors sufficient to guarantee the existence of a conflict-free chromatic guarding for any n-vertex polygon. Bärtschi and Suri showed χ_cf(n)∈ O(log n) (Algorithmica, 2014) for simple orthogonal polygons and the same bound applies to general simple polygons (Bärtschi et al., SoCG 2014). In this paper, we assume the r-visibility model instead of standard line visibility. Points p and q in an orthogonal polygon are r-visible to each other if the rectangle spanned by the points is contained in P. For this model we show χ_cf(n)∈ O(loglog n) and χ_cf(n)∈Ω(loglog n /logloglog n). Most interestingly, we can show that the lower bound proof extends to guards with line visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau. This is the first non-trivial lower bound for this problem setting.Furthermore, for the strong chromatic version of the problem, where all guards r-visible from a point must have distinct colors, we prove a Θ(log n)-bound. Our results can be interpreted as coloring results for special geometric hypergraphs.

Abstract. We study the problem of how to cover a polygonal region by a sma ll number of axis–parallel ellipses. This question is well mot ivated by a special pattern recognition task where one has to identify ellipse s haped protein spots in 2–dimensional electrophoresis images. We present and di scuss two algorithmic approaches solving this problem: a greedy brute force me thod and a linear programming formulation. Furthermore we discuss related t h oretical questions.

We study online strategies for autonomous mobile robots with vision to explore unknown polygons with at most h holes. Our main contribution is an (h+c_0)!-competitive strategy for such polygons under the assumption that each hole is marked with a special color, where c_0 is a universal constant. The strategy is based on a new hybrid approach. Furthermore, we give a new lower bound construction for small h.

Let D be a connected region inside a simple polygon, P. We define the angle hull, 𝒜ℋ (D), of 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 𝒜ℋ (D) cannot exceed the perimeter of the relative convex hull of D by more than a factor of 2. A special case occurs when P equals the full plane. Here we prove a bound of π/2. Both bounds are tight, and corresponding results are obtained for any other angle.

We prove the following graph coloring result: Let G be a 2-connected bipartite planar graph. Then one can triangulate G in such a way that the resulting graph is 3-colorable. This result implies several new upper bounds for guarding problems including the first non—trivial upper bound for the rectilinear Prison Yard Problem: Moreover, we show a new lower bound of [5n/16] vertex guards for the rectilinear Prison Yard Problem and prove it to be asymptotically tight for the class of orthoconvex polygons.

We address the problem of how to cover a set of required points by a small number of axis-parallel ellipses that avoid a second set of forbidden points. We study geometric properties of such covers and present an efficient randomized approximation algorithm for the cover construction. This question is motivated by a special pattern recognition task where one has to identify ellipse-shaped protein spots in two-dimensional electrophoresis images.

Bei neurochirurgischen Eingriffen an der Wirbelsäule wird die Fluoroskopie als bildgebendes Verfahren eingesetzt, um die räumliche Lage von Instrumenten und Implantaten zu erkennen und gegebenenfalls zu korrigieren. Die häufige Wiederholung solcher Aufnahmen hat eine Reihe von Nachteilen für Patient und Operateur. Zur Vermeidung dieser Probleme wird eine Technik zur virtuellen Navigation vorgestellt, die es in Kombination mit einem Trackingsystem erlaubt, die Lage von Instrumenten in vorher aufgenommene Fluoroskopiebilder zu projizieren und diese dem Operateur auf einem Bildschirm anzuzeigen.

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.

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:alon@Graphics.Stanford.EDU zInstitut fur Informatik, Freie Universitat Berlin, Takustr. 9, D14195 Berlin, email:name@inf.fu-berlin.de 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

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. (C) 1999 Elsevier Science B.V. All rights reserved.

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.

Article Matching 2D patterns of protein spots Share on Authors: Frank Hoffmann Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 Berlin Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 BerlinView Profile , Klaus Kriegel Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 Berlin Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 BerlinView Profile , Carola Wenk Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 Berlin Department of Computer Science, Institute fü Informatik, Freie Universität Berlin, Takustr. 9, D-14195 BerlinView Profile Authors Info & Claims SCG '98: Proceedings of the fourteenth annual symposium on Computational geometryJune 1998 Pages 231–239https://doi.org/10.1145/276884.276911Online:07 June 1998Publication History 10citation455DownloadsMetricsTotal Citations10Total Downloads455Last 12 Months1Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Let D be a connected region inside a simple polygon, P. We define the angle hull of D, 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 AH(D) cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight. Our result can be generalized to angles different from 90°, and to settings where region D is surrounded by obstacles other than a simple polygon.

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 xo on the boundary. Our strategy creates a tour that does not exceed in length 133 times the length of the shortest watchman route from x0. It has been claimed before by other authors that a competitive strategy with factor 2016 exists for this problem; but no proof has appeared except for the easy rectilinear case.

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 the angle hull. This structure is presented in Part II of this paper.

We study the problem of how to cover a polygonal region 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 electrophoresis images. We present and discuss two algorithmic approaches solving this problem: a greedy brute force method and a linear programming formulation. Furthermore we discuss related theoretical questions. 1 Detecting Spots in 2–dimensional Gel Electrophoresis Images 1.1 Gel Electrophoresis: The Application Background In proteomics 2–dimensional gel electrophoresis (2DE) is a 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 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. By analyzing series of such 2DE images one hopes to identify those proteins that change their expression (size, intensity) and reflect/cause certain biochemical and biomedical conditions of an organism, see [15]. 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 [6]. However, spots that are very close to each other can partially merge and form rather complicated regions as indicated in Figure 1. At Freie Universit ät Berlin we have started a few years ago to develop a software system CAROL (see [1]) that is able to perform local and global matching queries for gel images (given in GIF format). 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, [7], [8], 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 commercial PDQuest system. Meanwhile it has become lucid that wrongly detected spots are the main obstacle towards a better performance of the matching tool compared to influence of pure geometric distortions. That is why we decided to develop and include a new spot detection Research has been partly supported by Deutsche Forschungsgemeinschaft, grant FL 165/4–1. email:alon@Graphics.Stanford.EDU email: hoffmann,kriegel,schultz @inf.fu-berlin.de Fig. 1. Twin spots, streaks and complex region algorithm into the CAROL system to overcome these difficulties. This algorithm, see [14] for details, relies on a watershed transformation applied to the gradient image. The most difficult part is how to interpret twin spots, streaks (left side in Fig. 1) and so called complex regions (right side in Fig. 1) as unions of ellipses. In fact, in [14] the latter case was left open and in the implementation the user had to edit these complex regions by hand. To present an algorithmic solution to this question is the subject of this paper and to our knowledge it is the first algorithmic approach that deals with complex regions. In fact, just by looking at the images it is clear that there is an inherent uncertainty in the 2DE gel images due to the electrophoresis process itself which is highly susceptible to faults and geometric distortions, so there is no hope to come up with a perfect spot detection algorithm. Consequently, in the upcoming CAROL version we try to 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 which could be erroneous. In the matching algorithm part we then have the possibility to accept the matching of two ambiguous regions if there is a pair of proposed coverings that match. 1.2 Modeling the Covering Problem The following formalization is a compromise stemming from discussions with practitioners who solve these covering instances by hand. We assume that a connected pixel pattern is given, which is fat in the sense that there are no short cuts consisting of two pixels only. In the application is usually a 1–connected subpattern of a –square. Let denote the rectilinear polygonal curve describing its boundary. We shrink and expand the boundary in both directions as follows. Identifying a pixel with its center point we define the two pixel sets ! #" $ %'& and, analogously, )(* + -, . ' / 0 1 2 3 4 5$ %'& . Now we can formulate the approximative covering problem we are interested in from the application point of view. Fig. 2. Part of a gel image and spots computed For numbers 1 and a natural number find a smallest possible set of axis–parallel ellipses fulfilling the following conditions. 1. (Shape) For each + the ratio of its halfaxes is in the interval $ . 2. (Fitting) Each ellipse + respects ( , i.e., it does not intersect ( . 3. (Intersection) The boundaries of any pair of ellipses intersects in at most 2 points, and area area area & . 4. (Covering) ! "$#&%' covers at least a )( portion of pixels in . We especially emphasize that the somehow strange intersection condition 3 is justified by the application because two spots (ellipses) can only partially merge and do not form a cross. 2 Brute Force Solution vs. Linear Programming In this section we present two algorithmic solutions to our covering problem that have been implemented and tested on 2D gel images. In the implementation we assumed that the regions are 1–connected and rectilinear, however it is straightforward how to extend the algorithms to arbitrary polygonal regions. 2.1 Computing a Brute Force Solution The basic idea behind the naive brute force solution is to generate all ellipses that fulfill condition 1 and 2. A voting scheme is then set up to select the covering. In a first step of the pixel set is approximated by a sample * of size about 5$,+ of the original vertex set. This is done heuristically in a way that almost convex boundary segments have denser samples. (Almost convex is defined via the average slope of lines connecting the point with predecessor points and successor points.) The second step in the brute force approach is to discretize the parameter space of possible ellipses. Recall that an axis–parallel ellipse is formed by all points fulfilling the equation ( (. ' ( 5 with parameters . To this end we restrict the ellipses to have centers which are pixel centers and one of the halfaxes, say , has to have multiple pixel side length. Finally, for a fixed we restrict the second halfaxis to values from the set ( $ & . Now, the algorithm simply computes for each center 2 ' and for each the maximal halfaxis such that the ellipse with parameters 2 ' does not intersect ( . For each ellipse we store which points from * it covers and their number. Eventually, in a greedy fashion we choose the covering. Assume a partial covering is already chosen. The next ellipse is selected among all ellipses satisfying condition 3 (with respect to already chosen ellipses) according to the following criteria, ordered as ranked: 1. covers a maximal number of previously uncovered points from * 2. maximizes the length of longest chain of consecutive covered points from * 3. minimizes maximal intersection area with an already chosen ellipse. After selecting a best (ties are broken randomly) we update the scores of all other remaining ellipses by deleting the points covered by . We stop augmenting ellipses when condition 4 is met. The drawback of the brute force approach is obvious, too many ellipses are tested. Moreover by discretizing ellipse parameters we may miss interesting ellipses like the big one in the right hand solution in Figure 3, see 2.3. for a discussion. Remark: Let be the family of ellipses 2 ' as described above, and let be the minimal number of ellipses of , needed to cover all points of * and avoiding the ones of )( . Then the standard "!$# * 3 –approximation (see [5] of the optimal solution by the greedy approach cannot be guaranteed, at least for arbitrary sample sets * in polygons with holes. This is due to the restrictive intersection property. An example that illustrates this observation consists of a set of rather thin ellipses arranged in a grid like fashion. Besides the very recent paper [11] we are not aware of approximation results for set covers with restrictive intersection properties. 2.2 Using an LP Approach to Generate Ellipses Compared to the brute force approach we do not want to restrict the set of ellipses under consideration for covering a region by an apriori parameter discretization and we want to avoid to generate to many explicite ellipses. Observe that each element in the pixel set ( forms a constraint for each ellipse in the cover, an ellipse must not contain such pixels. Since we do not want arbitrary small ellipses in the cover and we do not want degenerated halfaxes ratios as well, we can sample these ‘outer’ constraints by choosing every 4th point. Let us denote this sample by * ( . On the other side we want at least a few points (say, at least three) from *! to Fig. 3. Ellipse covering computed by brute force method (left) and by LP approach (right) be included in an ellipse. Therefore, we start from a randomly chosen triplet of mutually visible points from * and ask whether there is an axis–parallel ellipse containing these points such that it does not violate an outer constraint. Once having the information that for a given triplet there is a feasible solution one can try to extend the covered sample subset. The idea how to make use of linear programming for testing feasibility is based on the observation that the parameters of an ellipse can be transformed into variables of an LP in such a way that each of the three points which has to be covered by adds a linear constraint to the outer constraints. The cons