Visibility Polygon Research Articles

On colourability of polygon visibility graphs

We study the problem of colouring visibility graphs of polygons. In particular, for visibility graphs of simple polygons, we provide a polynomial algorithm for 4-colouring, and prove that the 5-colourability question is already NP-complete for them. For visibility graphs of polygons with holes, we prove that the 4-colourability question is NP-complete.

Service coverage optimization for facility location: considering line-of-sight coverage in continuous demand space

The reliable service coverage of many facilities or sensors used in smart city infrastructure is highly susceptible to obstructions in urban environments. Optimizing the line-of-sight (LOS) service coverage is essential to locating these facilities for smarter city services. Despite progression in the maximal coverage location problem (MCLP) model for locating facilities, maximizing the LOS service coverage in continuous demand space for facility location problems remains challenging. This study defined the LOS-constrained MCLPs (LOS-MCLPs) and proposed a service coverage optimization model to solve these LOS-MCLPs. We employed a computational geometry algorithm named the visibility polygon (VP) algorithm to simulate the LOS coverage in two-dimensional (2D) continuous demand space. We then coupled this algorithm with a robust heuristic algorithm to search for the optimal solutions to maximize effective LOS service coverage. An experiment applied the developed model to a Wi-Fi hotspot planning problem. The experimental results demonstrated that the proposed model can obtain optimal solutions for LOS-MCLPs according to the distribution of obstacles. Comparative results show that ignoring the LOS effect in the optimization of LOS-MCLPs might lead to large areas of service dead zones.

Coloring polygon visibility graphs and their generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3⋅4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a coloring with the claimed number of colors can be computed in polynomial time.

Computational morphology of debris and alluvial fans on irregular terrain using the visibility polygon

A computational method is proposed to simulate the morphology of debris and alluvial fans formed on irregular terrain, where they may abut or bend around steeper valley sides or obstacles. Fan elevation and slope are assumed to depend only on the distance from the apex, measured along straight or winding paths. When there are no obstacles, the resulting fan morphologies are simply surfaces of revolution, with a straight (for constant slope fans) or curved generatrix (for concave fans). When obstacles are present, however, fan surfaces of great variety can be produced. To capture the resulting morphology, we exploit a powerful tool of computational geometry: the visibility polygon algorithm. Starting from one or more initial fan apexes, the proposed algorithm iteratively generates fan sectors with new apexes along their margins, until the fan complex has attained its maximal extent. To validate the method, we first apply it to several cases with analytical solutions, such as single fans on faceted topography, and multiple coalescing fans with weld lines. We then apply the method to real valleys, and check that it can reproduce observed fan morphologies even in complex cases involving diffluences and confluences.

Visibility polygons and visibility graphs among dynamic polygonal obstacles in the plane

We devise an algorithm for maintaining the visibility polygon of any query point in a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update the data structures that store the visibility polygon of the query point. After preprocessing the initial input polygonal domain to build a few data structures, our algorithm takes $$O(k(\lg {|VP_{{\mathcal {P}}'}(q)|})+(\lg {n'})^{2}+h)$$ (resp. $$O(k(\lg n')^2+(\lg |VP_{{\mathcal {P}}'}(q)|)+h)$$ ) worst-case time to update data structures that store visibility polygon $$VP_{\mathcal P'}(q)$$ of a query point q when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain $${\mathcal {P}}'$$ . Here, $$n'$$ is the number of vertices of $${\mathcal {P}}'$$ , h is the number of obstacles in $${\mathcal {P}}'$$ , $$VP_{{\mathcal {P}}'}(q)$$ is the visibility polygon of q in $${\mathcal {P}}'$$ ( $$|VP_{{\mathcal {P}}'}(q)|$$ is the number of vertices of $$VP_{{\mathcal {P}}'}(q)$$ ), and k is the number of combinatorial changes in $$VP_{{\mathcal {P}}'}(q)$$ due to the insertion (resp. deletion) of v. As an application of the above algorithm, we also devise an algorithm for maintaining the visibility graph of a dynamic polygonal domain, i.e., as the polygonal domain is modified with vertex insertions and deletions to its obstacles, we update data structures that store the visibility graph of the polygonal domain. After preprocessing the initial input polygonal domain, our dynamic algorithm takes $$O(k(\lg {n'})^{2}+h)$$ (resp. $$O(k(\lg {n'})^{2}+h)$$ ) worst-case time to update data structures that store the visibility graph when any vertex v is inserted to (resp. deleted from) any obstacle of the current polygonal domain $${\mathcal {P}}'$$ . Here, $$n'$$ is the number of vertices of $${\mathcal {P}}'$$ , h is the number of obstacles in $${\mathcal {P}}'$$ , and k is the number of combinatorial changes in the visibility graph of $${\mathcal {P}}'$$ due to the insertion (resp. deletion) of v.

Redirected Walking in Static and Dynamic Scenes Using Visibility Polygons.

We present a new approach for redirected walking in static and dynamic scenes that uses techniques from robot motion planning to compute the redirection gains that steer the user on collision-free paths in the physical space. Our first contribution is a mathematical framework for redirected walking using concepts from motion planning and configuration spaces. This framework highlights various geometric and perceptual constraints that tend to make collision-free redirected walking difficult. We use our framework to propose an efficient solution to the redirection problem that uses the notion of visibility polygons to compute the free spaces in the physical environment and the virtual environment. The visibility polygon provides a concise representation of the entire space that is visible, and therefore walkable, to the user from their position within an environment. Using this representation of walkable space, we apply redirected walking to steer the user to regions of the visibility polygon in the physical environment that closely match the region that the user occupies in the visibility polygon in the virtual environment. We show that our algorithm is able to steer the user along paths that result in significantly fewer resets than existing state-of-the-art algorithms in both static and dynamic scenes. Our project website is available at https://ganuna.umd.edu/vis.poly/.

Space decomposition based on visible objects in an indoor environment

When a person moves, the set of objects in their visual range changes. Hence, the set of objects perceived from a specific range of locations may be considered as a signature (possibly non-unique) of this region and be used for the localization of this person. In the case of fixed objects, the number of regions with a set of specific visible objects is limited. A verbal description containing references to elements of this set of visible objects could then be used to localize a person in the space. This paper proposes an approach for decomposing a space into regions that are characterized by such sets of visible objects. In our approach, at least a portion of partial surfaces of an object must be visible (beyond single points) to make part of the signature. Our method calculates two-dimensional visibility polygons for a portion of an object’s surface. Overlaying these polygons, we partition the space in regions of visibility signatures. The approach has been implemented, and we demonstrate how to represent space by qualitative locations using these visibility signatures. We further show how this representation can be used to locate a person within a space by a set of visible objects.

Attraction-convexity and normal visibility

Beacon attraction, or simply attraction, is a movement system whereby a point moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (also a point). This results in the point moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle or being stuck (unable to move). When the point can reach the activated beacon by this method, we say that the beacon attracts the point. In this paper, we study attraction-convex polygons, which are those where every point in the polygon attracts every other point. We find that these polygons are a subclass of weakly externally visible polygons, which are those where every point on the boundary is visible from some point arbitrarily distant (or at infinity on the projective plane). We propose a new class of polygons called normally visible, and show that this is exactly the class of attraction-convex polygons. This alternative characterization of attraction-convex polygons leads to a simple linear-time attraction-convex polygon recognition algorithm. We also give a Helly-type characterization of inverse-attraction star-shaped polygons.

A Fast Shortest Path Algorithm on Terrain-like Graphs

Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications in areas such as physics, geography, and medical sciences. Computing shortest paths in visibility graphs is a common task, for example, in line-of-sight communication. For time series analysis, graph characteristics involving shortest paths lengths (such as centrality measures) have proven useful. In this paper, we devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visibility graphs). Our algorithm runs in O(d^*log varDelta ) time, where d^* is the length (that is, the number of edges) of the shortest path and varDelta is the maximum vertex degree. Alternatively, with an O(n^2)-time preprocessing our algorithm runs in O(d^*) time.

Dynamic Algorithms for Visibility Polygons in Simple Polygons

We devise the following dynamic algorithms for both maintaining as well as querying for the visibility and weak visibility polygons amid vertex insertions and deletions to the simple polygon. A fully-dynamic algorithm for maintaining the visibility polygon of a fixed point located interior to the simple polygon amid vertex insertions and deletions to the simple polygon. The time complexity to update the visibility polygon of a point [Formula: see text] due to the insertion (resp. deletion) of vertex [Formula: see text] to (resp. from) the current simple polygon is expressed in terms of the number of combinatorial changes needed to the visibility polygon of [Formula: see text] due to the insertion (resp. deletion) of [Formula: see text]. An output-sensitive query algorithm to answer the visibility polygon query corresponding to any point [Formula: see text] in [Formula: see text] amid vertex insertions and deletions to the simple polygon. If [Formula: see text] is not exterior to the current simple polygon, then the visibility polygon of [Formula: see text] is computed. Otherwise, our algorithm outputs the visibility polygon corresponding to the exterior visibility of [Formula: see text]. An incremental algorithm to maintain the weak visibility polygon of a fixed-line segment located interior to the simple polygon amid vertex insertions to the simple polygon. The time complexity to update the weak visibility polygon of a line segment [Formula: see text] due to the insertion of vertex [Formula: see text] to the current simple polygon is expressed in terms of the sum of the number of combinatorial updates needed to the geodesic shortest path trees rooted at [Formula: see text] and [Formula: see text] due to the insertion of [Formula: see text]. An output-sensitive algorithm to compute the weak visibility polygon corresponding to any query line segment located interior to the simple polygon amid both the vertex insertions and deletions to the simple polygon. Each of these algorithms requires preprocessing the initial simple polygon. And, the algorithms that maintain the visibility polygon (resp. weak visibility polygon) compute the visibility polygon (resp. weak visibility polygon) with respect to the initial simple polygon during the preprocessing phase.

Optimal Art Gallery Localization is NP-hard

Art Gallery Localization (AGL) is the problem of placing a set T of broadcast towers in a simple polygon P in order for a point to locate itself in the interior. For any point p∈P: for each tower t∈T∩V(p) (where V(p) denotes the visibility polygon of p) the point p receives the coordinates of t and the Euclidean distance between t and p. From this information p can determine its coordinates. We study the computational complexity of AGL problem. We show that the problem of determining the minimum number of broadcast towers that can localize a point anywhere in a simple polygon P is NP-hard. We show a reduction from Boolean Three Satisfiability problem to our problem and give a proof that the reduction takes polynomial time.

Human Visual Acuity in Analysis of Intervention in the Landscape of Coxilha Rica/Brazil

The Drovers’ Paths are remnants of important land access roads from Rio Grande do Sul to São Paulo at the time of Colonial Brazil. They were built and used between the 18th and 20th centuries, particularly in the region of Coxilha Rica. The main objective of this research is to develop a method for decision-making applied to the territorial landscape management in the Coxilha Rica. The method consisted of generating criteria to map the visibility spot reached from the main selected points; define the human visual acuity, realize bibliographic research, use cartographic and historical documents, inter-views, as well as field surveys that enabled the identification, characterization and mapping of historical farms and drovers’ paths. After data processing, the information was entered into the cartographic database; the data were cross-checked and analysis was made of the visibility of the surrounding farms and stone-walled corridors. Quality assessments showed that, with the visibility polygons, and through the use of cartographic tools, we could cross-check between different levels of information and analyze landscape intervention alternatives in order to minimize environmental impacts. When applying the method in the Coxilha Rica it was possible mapping the visibility polygon, taking human visual acuity into consideration, based on historical farms and stone-walled corridors; and making spatial analyses to explore alternatives to intervention (installation of power transmission systems) in order to preserve the scenic environment of the region. In the end, the decision was by does not construct the system.

A space–time trade-off for computing the visibility polygon in the multi-pass model

We study the problem of computing the visibility polygon of a simple polygon $${\mathcal {P}}$$ from a viewpoint inside it, when $${\mathcal {P}}$$ resides in a read-only memory without random accessibility. This model of stored data is called the multi-pass model in the literature. We present an algorithm to compute the visibility polygon in this model in $$O(nr/t + nt)$$ time, where n is the size of input data, r is the number of reflex vertices of input data, and t is the number of additional variables which we use to compute the visibility polygon where $$2<t<r$$.

Quickest Visibility Queries in Polygonal Domains

Let s be a point in a polygonal domain $${\mathcal {P}}$$ of $$h-1$$ holes and n vertices. We consider a quickest visibility query problem. Given a query point q in $${\mathcal {P}}$$ , the goal is to find a shortest path in $${\mathcal {P}}$$ to move from s to seeq as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size $$O(n^22^{\alpha (n)}\log n)$$ that can answer each query in $$O(K\log ^2 n)$$ time, where $$\alpha (n)$$ is the inverse Ackermann function and K is the size of the visibility polygon of q in $${\mathcal {P}}$$ (and K can be $$\varTheta (n)$$ in the worst case). In this paper, we present a new data structure of size $$O(n\log h + h^2)$$ that can answer each query in $$O(h\log h\log n)$$ time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., $$h=1$$ ), which is optimal. As a by-product, we also have a new algorithm for a shortest-path-to-segment query problem. Given a query line segment $$\tau $$ in $${\mathcal {P}}$$ , the query seeks a shortest path from s to all points of $$\tau $$ . Previously, Arkin et al. gave a data structure of size $$O(n^22^{\alpha (n)}\log n)$$ that can answer each query in $$O(\log ^2 n)$$ time, and another data structure of size $$O(n^3\log n)$$ with $$O(\log n)$$ query time. We present a data structure of size O(n) with query time $$O\big (h\log \frac{n}{h}\big )$$ , which also favors small values of h and is optimal when $$h=O(1)$$ .

Visibility extension via mirror-edges to cover invisible segments

Given a simple polygon P with n vertices, the visibility polygon (VP) of a point q, or a segment pq‾ inside P can be computed in linear time. We propose a linear time algorithm to extend the VP of a viewer (point or segment), by converting some edges of P into mirrors, such that a given non-visible segment uw‾ can also be seen from the viewer. Various definitions for the visibility of a segment, such as weak, strong, or complete visibility are considered. Our algorithm finds every edge that, when converted to a mirror, makes uw‾ visible to our viewer. We find out exactly which interval of uw‾ becomes visible, by every edge middling as a mirror, all in linear time. In other words, in this article, we present an algorithm that, in linear time, for every edge e of P reveals precisely which part of uw‾ is mirror-visible through e.

A Time-Space Trade-off for the Shortest Path Tree in a Simple Polygon

A [Formula: see text]-workspace algorithm may use a workspace of [Formula: see text] words which can be read and written, in addition to their input, which is provided as a read-only array of [Formula: see text] items. We present a [Formula: see text]-workspace algorithm for constructing the shortest path tree of a simple polygon [Formula: see text] with [Formula: see text] vertices, with respect to a given point inside the polygon in [Formula: see text] time, where [Formula: see text] is the time for computing the visibility polygon of a given point inside a simple polygon with [Formula: see text] additional words of space.

The mere exposure effect for visual image.

Mere exposure effect refers to a phenomenon in which repeated stimuli are evaluated more positively than novel stimuli. We investigated whether this effect occurs for internally generated visual representations (i.e., visual images). In an exposure phase, a 5 × 5 dot array was presented, and a pair of dots corresponding to the neighboring vertices of an invisible polygon was sequentially flashed (in red), creating an invisible polygon. In Experiments 1, 2, and 4, participants visualized and memorized the shapes of invisible polygons based on different sequences of flashed dots, whereas in Experiment 3, participants only memorized positions of these dots. In a subsequent rating phase, participants visualized the shape of the invisible polygon from allocations of numerical characters on its vertices, and then rated their preference for invisible polygons (Experiments 1, 2, and 3). In contrast, in Experiment 4, participants rated the preference for visible polygons. Results showed that the mere exposure effect appeared only when participants visualized the shape of invisible polygons in both the exposure and rating phases (Experiments 1 and 2), suggesting that the mere exposure effect occurred for internalized visual images. This implies that the sensory inputs from repeated stimuli play a minor role in the mere exposure effect. Absence of the mere exposure effect in Experiment 4 suggests that the consistency of processing between exposure and rating phases plays an important role in the mere exposure effect.

Weak visibility queries of line segments in simple polygons and polygonal domains

ABSTRACTIn this paper we consider the problem of computing the weak visibility polygon of a query line segment pq (or ) inside a given polygon . Our first algorithm runs in simple polygons and needs time and space in the preprocessing phase to report of any query line segment pq in time . We also give an algorithm to compute the weak visibility polygon of a query line segment in a non-simple polygon with pairwise-disjoint polygonal obstacles with a total of n vertices. Our algorithm needs time and space in the preprocessing phase and in query time of, in which is an output sensitive parameter of at most , and is the output size.

Approximability of guarding weak visibility polygons

The art gallery problem enquires about the least number of guards that are sufficient to ensure that an art gallery, represented by a polygon P, is fully guarded. In 1998, the problems of finding the minimum number of point guards, vertex guards, and edge guards required to guard P were shown to be APX-hard by Eidenbenz, Widmayer and Stamm. In 1987, Ghosh presented approximation algorithms for vertex guards and edge guards that achieved a ratio of O(logn), which was improved up to O(loglogOPT) by King and Kirkpatrick (2011). It has been conjectured that constant-factor approximation algorithms exist for these problems. We settle the conjecture for the special class of polygons that are weakly visible from an edge and contain no holes by presenting a 6-approximation algorithm for finding the minimum number of vertex guards that runs in O(n2) time. On the other hand, for weak visibility polygons with holes, we present a reduction from the Set Cover problem to show that there cannot exist a polynomial time algorithm for the vertex guard problem with an approximation ratio better than ((1−ϵ)/12)lnn for any ϵ>0, unless NP=P. We also show that, for the special class of polygons without holes that are orthogonal as well as weakly visible from an edge, the approximation ratio can be improved to 3. Finally, we consider the point guard problem and show that it is NP-hard in the case of polygons weakly visible from an edge.

두 가시성 다각형의 교집합과 합집합을 구하는 효율적인 RMESH 알고리즘

We can consider the following problems for two given points p and q in a simple polygon P. (1) Compute the set of points of P which are visible from both p and q. (2) Compute the set of points of P which are visible from either p or q. They are corresponding to the problems which are to compute the intersection and the union of two visibility polygons. In this paper, we consider algorithms for solving these problems on a reconfigurable mesh(in short, RMESH). The algorithm in [1] can compute the intersection of two general polygons in constant time on an RMESH with size O(), where n is the total number of vertices of two polygons. In this paper, we construct the planar subdivision graph in constant time on an RMESH with size O() using the properties of the visibility polygon for preprocessing. Then we present O() time algorithms for computing the union as well as the intersection of two visibility polygons, which improve the processor-time product from O() to O().