Orthogonal Convex Research Articles

Rectilinear convex hull of points in 3D and applications

Let P be a set of n points in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} in general position, and let RCH(P) be the rectilinear convex hull of P. In this paper we obtain an optimal O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} time and O(n) space algorithm to compute RCH(P). We also obtain an efficient O(nlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log ^2 n)$$\\end{document} time and O(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n\\log n)$$\\end{document} space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of P as we rotate R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document} around the Z-axis. We study some combinatorial properties of the rectilinear convex hulls of point sets in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. Finally, as an application of the obtained results, we show an approximation algorithm to an optimization fitting problem in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}.

Fixed Point Theorems via Orthogonal Convex Contraction in Orthogonal ♭-Metric Spaces and Applications

In this paper, we introduce the concept of orthogonal convex structure contraction mapping and prove some fixed point theorems on orthogonal ♭-metric spaces. We adopt an example to highlight the utility of our main result. Finally, we apply our result to examine the existence and uniqueness of the solution for the spring-mass system via an integral equation with a numerical example.

Separating bichromatic point sets in the plane by restricted orientation convex hulls

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and mathcal {O} be a set of kge 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of mathcal {O} for which the mathcal {O}-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlog n) time and O(n) space, n = vert R vert + vert B vert , we compute the set of rotation angles such that, after simultaneously rotating the lines of mathcal {O} around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where mathcal {O} is formed by k ge 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of mathcal {O}, let alpha _i be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O({1}/{Theta }cdot N log N) time and O({1}/{Theta }cdot N) space, where Theta = min { alpha _1,ldots ,alpha _k } and N=max {k,vert R vert + vert B vert }. We finally consider the case in which mathcal {O} is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to pi . We show that this last case can also be solved in optimal O(nlog n) time and O(n) space, where n = vert R vert + vert B vert .

A fast and efficient algorithm for determining the connected orthogonal convex hulls

The Quickhull algorithm for determining the convex hull of a finite set of points was independently conducted by Eddy in 1977 and Bykat in 1978. Inspired by the idea of this algorithm, we present a new efficient algorithm, for determining the connected orthogonal convex hull of a finite set of points through extreme points of the hull, that still keeps advantages of the Quickhull algorithm. Consequently, our algorithm runs faster than the others (the algorithms introduced by Montuno and Fournier in 1982 and by An, Huyen and Le in 2020). We also show that the expected complexity of the algorithm is O(nlogn), where n is the number of points.

Geometric separability using orthogonal objects

Given a bichromatic point set P=R∪B of red and blue points, a separator is an object of a certain type that separates R and B. We study the geometric separability problem when the separator is a) rectangular annulus of fixed orientation b) rectangular annulus of arbitrary orientation c) square annulus of fixed orientation d) orthogonal convex polygon. In this paper, we give polynomial time algorithms to construct separators of each of the above type that also optimizes a given parameter.

A modified Graham’s convex hull algorithm for finding the connected orthogonal convex hull of a finite planar point set

Graham’s convex hull algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull (Allison and Noga, 1984). To use this algorithm for finding an orthogonal convex hull of a finite planar point set, we introduce the concept of extreme points of a connected orthogonal convex hull of the set, and show that these points belong to the set. Then we prove that the connected orthogonal convex hull of a finite set of points is an orthogonal (x,y)-polygon where its convex vertices are its connected orthogonal convex hull’s extreme points. As a result, an efficient algorithm, based on the idea of Graham’s convex hull algorithm, for finding the connected orthogonal convex hull of a finite planar point set is presented. We also show that the lower bound of computational complexity of such algorithms is O(nlogn). Some numerical results for finding the connected orthogonal convex hulls of random sets are given.

A linear time combinatorial algorithm to compute the relative orthogonal convex hull of digital objects

A linear time combinatorial algorithm to construct the relative orthogonal convex hull of the inner isothetic cover of a digital object (hole-free) with respect to its outer isothetic cover is presented in this paper. The algorithm first constructs the inner and the outer isothetic covers of the digital object which is the input to the algorithm and then constructs the relative orthogonal convex hull from the inner isothetic cover and the outer isothetic cover. The algorithm can also be used to construct the relative orthogonal convex hull of one orthogonal polygon with respect to another orthogonal polygon which encloses the first polygon entirely where the input will be two orthogonal polygons one containing the other instead of the digital object. The proposed algorithm first finds the concavities of the inner polygon and checks for intruded concavities of the outer polygon. A combinatorial technique is used to obtain the relative orthogonal convex hull. The efficiency and the correctness of the algorithm are verified as it is tested on various types of digital objects.

Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations

Let P be a set of n points in the plane. We compute the value of $$\theta \in [0,2\pi )$$ for which the rectilinear convex hull of P, denoted by $$\mathcal {RH}_{P}({\theta })$$ , has minimum (or maximum) area in optimal $$O(n\log n)$$ time and O(n) space, improving the previous $$O(n^2)$$ bound. Let $$\mathcal {O}$$ be a set of k lines through the origin sorted by slope and let $$\alpha _i$$ be the sizes of the 2k angles defined by pairs of two consecutive lines, $$i=1, \ldots , 2k$$ . Let $$\Theta _{i}=\pi -\alpha _i$$ and $$\Theta =\min \{\Theta _i :i=1,\ldots ,2k\}$$ . We obtain: (1) Given a set $$\mathcal {O}$$ such that $$\Theta \ge \frac{\pi }{2}$$ , we provide an algorithm to compute the $$\mathcal {O}$$ -convex hull of P in optimal $$O(n\log n)$$ time and O(n) space; If $$\Theta < \frac{\pi }{2}$$ , the time and space complexities are $$O(\frac{n}{\Theta }\log n)$$ and $$O(\frac{n}{\Theta })$$ respectively. (2) Given a set $$\mathcal {O}$$ such that $$\Theta \ge \frac{\pi }{2}$$ , we compute and maintain the boundary of the $${\mathcal {O}}_{\theta }$$ -convex hull of P for $$\theta \in [0,2\pi )$$ in $$O(kn\log n)$$ time and O(kn) space, or if $$\Theta < \frac{\pi }{2}$$ , in $$O(k\frac{n}{\Theta }\log n)$$ time and $$O(k\frac{n}{\Theta })$$ space. (3) Finally, given a set $$\mathcal {O}$$ such that $$\Theta \ge \frac{\pi }{2}$$ , we compute, in $$O(kn\log n)$$ time and O(kn) space, the angle $$\theta \in [0,2\pi )$$ such that the $${\mathcal {O}}_{\theta }$$ -convex hull of P has minimum (or maximum) area over all $$\theta \in [0,2\pi )$$ .

Pattern Synthesis of Time-Modulated Sparse Array by an OPM-CVX Algorithm

This paper addresses the constrained multiobjective optimization problem of time-modulated sparse arrays. The synthesis objective is to find an optimal element arrangement and associated excitation strategy of sparse arrays, which realize the balance of radiation power and sideband suppression performance with minimum number of elements, and suppress side lobe level simultaneously. A novel hybrid algorithm based on orthogonal perturbation method and convex optimization (OPM-CVX) for the synthesis of time-modulated sparse antenna array is presented in this paper. In order to satisfy the main lobe beamforming and side lobe suppression of sparse arrays, the proposed method optimizes element positions with minimum array numbers by orthogonal perturbation method and optimizes excitations of array element with dynamic range ratio constraint by convex optimization. Furthermore, a trapezoidal pulse time-modulated switching function is proposed to find the balance of radiation power and sideband suppression performance. The numerical results indicate that the proposed algorithm can be an effective approach for synthesis problems of time-modulated sparse arrays.

A hybrid algorithm of orthogonal perturbation method and convex optimization for beamforming of sparse antenna array

ABSTRACT In order to realize the beamforming and suppress the side lobe of the sparse antenna array, a hybrid sparse antenna array optimization algorithm based on orthogonal perturbation method (OPM) and convex (CVX) optimization is proposed. The algorithm aims at the expected beam response of main lobe and the suppression of peak side lobe level (PSLL). The OPM is used to optimize the array layout, and it can improve the time performance of pattern synthesis with its deterministic numerical characteristic. Because it is a local optimization algorithm, its optimal effect depends on the initial array layout and element excitation coefficient. Therefore, considering that the local optimal solution of the CVX algorithm is the global optimal solution, the CVX algorithm is used to obtain the optimal element excitation coefficient to achieve the expected beam response of the main lobe. The simulation results show that the hybrid algorithm can achieve a good trade-off between the optimization time, beam response, and the suppression of the PSLL.

Set estimation under biconvexity restrictions

A set in the Euclidean plane is said to be biconvex if, for some angle θ ∈ [0, π∕2), all its sections along straight lines with inclination angles θ and θ + π∕2 are convex sets (i.e., empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of “rectilinear convexity”, in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set S from a random sample of points drawn on S. By analogy with the classical convex case, one would like to define the “biconvex hull” of the sample points as a natural estimator for S. However, as previously pointed out by several authors, the notion of “hull” for a given set A (understood as the “minimal” set including A and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of “biconvex hull” (often called “rectilinear convex hull”): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set S and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on S, the biconvexity angle θ is also given.

Latency, Throughput and Power Aware Adaptive NoC Routing on Orthogonal Convex Faulty Region

Reliability of a Network-on-Chip (NoC) relies vastly upon the efficiency of handling faults. Faults those lead to trouble during on-chip communication process are basically of two types namely soft and hard. Here, hard faults are considered. Hard faults may be caused due to failure of links, routers, or other processing units. These are mainly dealt with fault-tolerant routing algorithms or by employing redundant hardware. Multiple faulty nodes are being avoided by acquiring region-based approaches. Most of the fault-tolerant routing techniques are designed on homogeneous faulty regions where some active nodes also act as deactivated nodes to build the region homogeneous. On the other hand, adaptive routing on nonhomogeneous faulty regions increases load on its boundary and most of them does not assure deadlock freeness. This paper proposes a deadlock-free adaptive fault-tolerant NoC routing named F-Route-NoC-Mesh (FRNM) ignoring any virtual channel on orthogonal convex faulty regions. Contributions of this work focus on balancing network traffic by assuming a virtual faulty block boundary and routing packets through this virtual boundary. Destination does not exist within that virtual faulty block regions to reduce load on the boundary of orthogonal faulty regions. Thus, this work is aimed at acquiring proper incorporation of procedures being able to reach fault-tolerant degree, routing efficiency and performance enhancement. Using the proposed algorithm (FRNM), a fault block model-based approach is developed. Significant improvements of average latency (43.37% to 60.44%), average throughput (4.18% to 90.81%) and power consumption (5.93% to 33.28%) are achieved over the state-of-the-art by using a cycle accurate simulator.

Reconstructing Generalized Staircase Polygons with Uniform Step Length

Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). As far as we are aware, the only class of orthogonal polygons that are known to have efficient reconstruction algorithms is the class of orthogonal convex fans (staircase polygons) with uniform step lengths. We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an $O(n^2m)$-time reconstruction algorithm for orthogonally convex polygons, where $n$ and $m$ are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time $O(n^2m)$ under alignment restrictions. As a consequence of our reconstruction techniques, we also get recognition algorithms for visibility graphs of these classes of polygons with the same running times.

On the [formula omitted] of a planar point set

We study the Oβ-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle β. Given a set P of n points in the plane, we show how to maintain the Oβ-hull of P while β runs from 0 to π in Θ(nlog⁡n) time and O(n) space. With the same complexity, we also find the values of β that maximize the area and the perimeter of the Oβ-hull and, furthermore, we find the value of β achieving the best fitting of the point set P with a two-joint chain of alternate interior angle β.

Deep Network Based on Stacked Orthogonal Convex Incremental ELM Autoencoders

Extreme learning machine (ELM) as an emerging technology has recently attracted many researchers’ interest due to its fast learning speed and state-of-the-art generalization ability in the implementation. Meanwhile, the incremental extreme learning machine (I-ELM) based on incremental learning algorithm was proposed which outperforms many popular learning algorithms. However, the incremental algorithms with ELM do not recalculate the output weights of all the existing nodes when a new node is added and cannot obtain the least-squares solution of output weight vectors. In this paper, we propose orthogonal convex incremental learning machine (OCI-ELM) with Gram-Schmidt orthogonalization method and Barron’s convex optimization learning method to solve the nonconvex optimization problem and least-squares solution problem, and then we give the rigorous proofs in theory. Moreover, in this paper, we propose a deep architecture based on stacked OCI-ELM autoencoders according to stacked generalization philosophy for solving large and complex data problems. The experimental results verified with both UCI datasets and large datasets demonstrate that the deep network based on stacked OCI-ELM autoencoders (DOC-IELM-AEs) outperforms the other methods mentioned in the paper with better performance on regression and classification problems.

James quasireflexive space is orthogonally convex

In this paper we prove some norm inequalities in the classical James sequence space J, and use them to prove that orthogonal convexity is a stable property in J.

Dynamic Fault-Tolerant Routing Based on FSA for LEO Satellite Networks

We formalize the construction of fault blocks by a state transition model based on finite state automata. Based on the model, a boundary diffusion method is presented for the rectilinear-monotone orthogonal convex fault model such as the rectangular fault model and minimal-connected-component (MCC) faulty model, whereby an adaptive fault-tolerant routing algorithm, called X-Y boundary routing algorithm (X-YBRA), is presented for deadlock-free fault-tolerant adaptive routing outside the fault blocks. To improve the network resources utilization, we put forward a routing diffusion method in the fault block, which completely solves the routing problem in the fault block. The experiment result shows that the diffusion overhead of our method is far lower than that of the traditional routing algorithms such as distance vector and link state routing algorithms with the light loss in convergence time. For the occurrence and recovery of random faults, the expansion and shrinkage of the fault block are also discussed. Accordingly, the dynamic boundary and routing updating methods are put forward to respond to these cases. Based on these methods, we develop low earth orbit satellite networks into an adaptive fault-tolerant system in routing. Our works can be also applied to other 2D mesh networks such as the interconnect multiprocessor computer systems.

Approximate partitioning of 2D objects into orthogonally convex components

A fast and efficient algorithm to obtain an orthogonally convex decomposition of a digital object is presented. The algorithm reports a sub-optimal solution and runs in O(nlogn) time for a hole-free object whose boundary consists of n pixels. The decomposition algorithm can, in fact, be applied on any hole-free orthogonal polygon; in our work, it is applied on the inner isothetic cover of the concerned digital object. The approximate/rough decomposition of the object is achieved by partitioning the inner cover (an orthogonal polygon) of the object into a set of orthogonal convex components. A set of rules is formulated based on the combinatorial cases and the decomposition is obtained by applying these rules while considering the concavities of the inner cover. The rule formulation is based on certain theoretical properties apropos the arrangement of concavities, which are also explained in this paper. Experimental results on different shapes have been presented to demonstrate the efficacy, elegance, and robustness of the proposed technique.

On finding an orthogonal convex skull of a digital object

AbstractA combinatorial algorithm to compute an orthogonal convex skull of a digital object imposed on the background grid is presented in this paper. The proposed algorithm has the time complexity of O(n log n), which improves the earlier method of O(n2) time complexity for finding the convex skull of a simple orthogonal polygon. A set of rules is formulated first and then an orthogonal convex skull is derived by applying these rules while traversing along the boundary of the inner orthogonal polygon that tightly inscribes the given digital object. The algorithm uses only comparison and addition in the integer domain, which makes it amenable to fast real‐world applications. Experimental results on different shapes have also been presented to demonstrate the efficacy and elegance of the proposed technique. © 2011 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 21, 14–27, 2011.

Fitting a two-joint orthogonal chain to a point set

We study the problem of fitting a two-joint orthogonal polygonal chain to a set S of n points in the plane, where the objective function is to minimize the maximum orthogonal distance from S to the chain. We show that this problem can be solved in Θ ( n ) time if the orientation of the chain is fixed, and in Θ ( n log n ) time when the orientation is not a priori known. Moreover, our algorithm can be used to maintain the rectilinear convex hull of S while rotating the coordinate system in O ( n log n ) time and O ( n ) space, improving on a recent result (Bae et al., 2009 [4]). We also consider some variations of the problem in three-dimensions where a polygonal chain is interpreted as a configuration of orthogonal planes. In this case we obtain O ( n ) , O ( n log n ) , and O ( n 2 ) time algorithms depending on which plane orientations are fixed.