Largest Bounding Box, Smallest Diameter, and Related Problems on Imprecise Points

In this paper, we introduce the outlier detection problem in a set of uncertain points. We study two variants of the problems based upon the definition of the outlier. For a given positive integer \(k(<n)\) and a set \(\mathfrak {R}\) of n regions as the imprecise points, the first type of the outlier detection problem that we study is to locate \(n-k\) points on distinct regions, such that the size of the smallest axis-aligned bounding box (AABB), the diameter or the smallest enclosing circle (SEC) of the resulting points gets minimized. The uncertainty regions we study are squares or disks, and the excluded k regions are considered as outliers.We also study the covering versions in which the objectives of the SEC and the AABB problems are to find the smallest circle or axis-aligned bounding box, respectively, that covers the area of at least \(n-k\) regions.In the second-type of outliers, the outliers are those k regions that mostly reduce the uncertainty-induced gap between the lower bound and the upper bound on the size of the output. We give polynomial time algorithms for several variants of the mentioned problems, ranging in running time from \(O(n \log n)\) to \(O(n^{5.5}\log n)\).

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible tours and convex hulls, measured by length, and in the latter case also by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n) to O(n 13), and prove NP-hardness for some geometric problems on imprecise points.KeywordsLine SegmentConvex HullExtreme PointMinimum Span TreeTangent PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this short article, a method to obtain a fuzzy regression curve for a set of imprecise locations is proposed. The given imprecise locations are presented by fuzzy points. The studied fuzzy regression curve is obtained with the help of a smooth regression technique for a set of precise locations. We observe the given imprecise points as a bunch of same points with varied membership values. For a set of same points, we obtain a smooth regression curve. The union of all these smooth regression curves, with different membership values, for the same points yields the proposed fuzzy regression curve. The method is demonstrated with a numerical example.

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

It is sometimes necessary to determine the optimal value for a direction dependent quantity. Using a search technique based on Powell's quadratic convergent method such an optimal direction can be approximated. The necessary geometric transformations in n-dimensional space are introduced. As an example we consider the approximation of the minimum bounding box of a set of three-dimensional points. Minimum bounding boxes can significantly improve accuracy and efficiency of the calculations in modern brachytherapy treatment planning of the volumes of objects or the dose distribution inside an object. A covariance matrix based approximation method for the minimum bounding box is compared with the results of the search method. The benefits of the use of optimal oriented bounding boxes in brachytherapy treatment planning systems are demonstrated and discussed.

Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model). Andoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a polynomial method, does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.

We consider the application of optical computational operations to problems that arise in various geometric applications, such as computer graphics, image processing, pattern matching. etc. We show how to implement, using only a constant number of basic optical operations, a variety of algorithms that solve fairly complex tasks of this sort. We discuss the methodology that we propose, list the geometric algorithms that we have developed to date, and illustrate the approach in two more detailed applications, motivated by computer graphics and pattern matching: drawing all possible lines connecting pairs in a given set of points, and computing the closest pair (orkclosest pairs) of points in a given set.

Given a set of n points in some d-dimensional Euclidean space, each point colored with one of k( ≥ 2) colors, a bichromatic closest (resp., farthest) pair is a closest (resp., farthest) pair of points of different colors. We present efficient algorithms to compute a bichromatic closest pair and a bichromatic farthest pair. We consider both static, and dynamic versions with respect to color flips. We also give some combinatorial bounds on the multiplicities of extreme distances in this setting.KeywordsSpan TreeComplete Bipartite GraphGeometric GraphPattern Recognition LetterConvex QuadrilateralThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Point location problems for n points in d-dimensional Euclidean space (and ep spaces more generally) have typically had two kinds of running-time solutions: (Nearly-Linear) less than dpoly(d) · n logO(d) n time, or (Barely-Subquadratic) f(d) · n2−1/Θ(d) time, for various f. For small d and large n, running times are generally feasible, while the times are generally infeasible, requiring essentially quadratic time. For example, in the Euclidean metric, finding a Closest Pair among n points in Rd is nearly-linear, solvable in 2o(d) · n logO(1) n time, while the known algorithms for finding a Furthest Pair (the diameter of the point set) are only barely-subquadratic, requiring Ω(n2--1/Θ(d)) time. Why do these proximity problems have such different time complexities? Is there a barrier to obtaining nearly-linear algorithms for problems which are currently only barely-subquadratic? We give a novel exact and deterministic self-reduction for the Orthogonal Vectors problem on n vectors in {0, 1}d to n vectors in Zω(log d) that runs in 2o(d) time. As a consequence, barely-subquadratic problems such as Euclidean diameter, Euclidean bichromatic closest pair, and incidence detection do not have O(n2−ϵ) time algorithms (in Turing models of computation) for dimensionality d = ω(log log n)2, unless the popular Orthogonal Vectors Conjecture and the Strong Exponential Time Hypothesis are false. That is, while the poly-log-log-dimensional case of Closest Pair is solvable in n1+o(1) time, the poly-log-log-dimensional case of Furthest Pair can encode difficult large-dimensional problems conjectured to require n2−o(1) time. We also show that the All-Nearest Neighbors problem in ω(log n) dimensions requires n2−o(1) time to solve, assuming either of the above conjectures.

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

Recent advances in convolutional neural networks (CNN) have achieved remarkable results in locating objects in images. In these networks, the training procedure usually requires providing bounding boxes or the maximum number of expected objects. In this paper, we address the task of estimating object locations without annotated bounding boxes which are typically hand-drawn and time consuming to label. We propose a loss function that can be used in any fully convolutional network (FCN) to estimate object locations. This loss function is a modification of the average Hausdorff distance between two unordered sets of points. The proposed method has no notion of bounding boxes, region proposals, or sliding windows. We evaluate our method with three datasets designed to locate people’s heads, pupil centers and plant centers. We outperform state-of-the-art generic object detectors and methods fine-tuned for pupil tracking.

A new Persian license plate recognition algorithm is presented. These operations are highly susceptible to error, especially where the image consists of large amount of either vehicle's linked components or the other existing objects. Although the proposed character recognition procedure is highly optimized for Persian plates, the localization parts can be employed for all types of vehicles. Minimum rectangle bounding box is replaced the common bounding box methods, compensating normal bounding box's inherent flaws. License plate possibility ratio (LPPR) is a robust method proposed here to localize the plate. New method of finding plate's location out of so many rectangles, considering “Sensitive to angle” criterions for characters has also been presented. It should be noted that the process is regardless of the plate's location. Different approach on thresholding namely: “Dynamic Thresholding” is used to overcome the probable drawbacks caused by inappropriate lighting. From OCR point of view, a graph, consisting of two specifications will be formed and a set of rules will be defined to capture the character's label. An automated harassment section is added as the denoising filter, in order to omit the grinning ramifications. Presenting the best percent accuracy (95.33%) among relevant well-known algorithms in localization procedure with 25ms run time of the program, and also the outstanding results with over 97% of percent accuracy in character recognition of Persian plates with 30ms run time of the program on Linux and also average of 90ms on Android, can be listed as strong proofs of algorithm's efficiency.

Dynamic analysis of many mechanical systems is often involved with contacts among bodies. This paper presents a relative contact formulation for multibody dynamics in the context of the compliance contact model. Many conventional collision detection algorithms are based on the absolute coordinate system. This paper proposes to use the relative coordinate system in detecting a contact. A contact reference frame is defined on the defense body of a contact pair. Since all geometric variables necessary to detect a contact are measured relative to the contact reference frame attached to the defense body, the variables for a defense body are constant, which significantly reduces computation time. Therefore, the contact frame plays a key role in developing an efficient contact search algorithm. Contour of a defense body is approximated by many piecewise straight lines, while contour of a hitting body is represented by hitting nodes along its boundary. Bounding boxes containing each body of a contact pair are defined at a pre-search stage to eliminate the exhaustive contact inspection process when two bodies are in a distance. Domain of the bounding box for a defense body is divided into many sectors each of which has a list of line segments lying inside or on the sector boundary. Post-search for a contact is processed in the sequence of broad and narrow phases. In the broad phase, the bounding boxes of a contact pair are inspected for a contact. If two boxes are in a contact, each node on the hitting boundary is inspected to find out to which sector the node belongs. Since each domain sector of the defense body has a list of line segments, each node on the hitting boundary is tested for a contact only with the line segments in the list. In the narrow phase, actual contact calculation is carried out to find the contact penetration used in calculating the contact force. Since the searching algorithm is coupled with the stepping algorithm of the numerical integration, a strategy for deciding an integration stepsize is proposed. One numerical example is presented to demonstrate the validity of the proposed method.

Two-dimensional closest pair problem: A closer look

An efficient multiple meta-model-based global optimization method for computationally intensive problems