The Euclidean k-matching problem is NP-hard

Self-organizing feature maps for solving location–allocation problems with rectilinear distances

Contents volume 108

In this article, we give the construction of new four-dimensional signal constellations in the Euclidean space, which represent a certain combination of binary frequency-shift keying (BFSK) and M-ary amplitude-phase-shift keying (MAPSK). Description of such signals and the formulas for calculating the minimum squared Euclidean distance are presented. We have developed an analytic building method for even and odd values of M. Hence, no computer search and no heuristic methods are required. The new optimized BFSK-MAPSK (M = 5,6,···,16) signal constructions are built for the values of modulation indexes h =0.1,0.15,···,0.5 and their parameters are given. The results of computer simulations are also provided. Based on the obtained results we can conclude, that BFSK-MAPSK systems outperform similar four-dimensional systems both in terms of minimum squared Euclidean distance and simulated symbol error rate.

In 2003, Maurer <i>at al</i>. [7] published a paper describing an algorithm that computes the exact distance transform in a linear time (with respect to image size) for the rectangular binary images in the k-dimensional space Rk and distance measured with respect to <i>L</i>_p-metric for 1 &le; p &le; &infin;, which includes Euclidean distance <i>L</i>₂. In this paper we discuss this algorithm from theoretical and practical points of view. On the practical side, we concentrate on its Euclidean distance version, discuss the possible ways of implementing it as signed distance transform, and experimentally compare implemented algorithms. We also describe the parallelization of these algorithms and the computation time savings associated with such an implementation. The discussed implementations will be made available as a part of the CAVASS software system developed and maintained in our group [5]. On the theoretical side, we prove that our version of the signed distance transform algorithm, GBDT, returns, in a linear time, the exact value of the distance from the geometrically defined object boundary. We notice that, actually, the precise form of the algorithm from [7] is not well defined for <i>L</i>₁ and <i>L</i>&infin; metrics and point to our complete proof (not given in [7]) that all these algorithms work correctly for the Lp-metric with 1 &lt; p &lt; &infin;.

In 2003, Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) published a paper describing an algorithm that computes the exact distance transform in linear time (with respect to image size) for the rectangular binary images in the k-dimensional space ? k and distance measured with respect to L p -metric for 1?p??, which includes Euclidean distance L 2. In this paper we discuss this algorithm from theoretical and practical points of view. On the practical side, we concentrate on its Euclidean distance version, discuss the possible ways of implementing it as signed distance transform, and experimentally compare implemented algorithms. We also describe the parallelization of these algorithms and discuss the computational time savings associated with them. All these implementations will be made available as a part of the CAVASS software system developed and maintained in our group (Grevera et al. in J. Digit. Imaging 20:101---118, 2007). On the theoretical side, we prove that our version of the signed distance transform algorithm, GBDT, returns the exact value of the distance from the geometrically defined object boundary. We provide a complete proof (which was not given of Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) that all these algorithms work correctly for L p -metric with 1<p<?. We also point out that the precise form of the algorithm from Maurer et al. (IEEE Trans. Pattern Anal. Mach. Intell. 25:265---270, 2003) is not well defined for L 1 and L ? metrics. In addition, we show that the algorithm can be used to find, in linear time, the exact value of the diameter of an object, that is, the largest possible distance between any two of its elements.

Digital image segmentation is a thrilling but challenging open problem that has been well researched in the fields of computer vision, and image processing. It has many practical applications like biometric identification, ship detection, building extraction, road marking recognition, deoxyribonucleic acid matching, welding inspection, pedestrian re-identification, object tracking, image editing, pest monitoring, and shopping items recommendation. In recent years, image segmentation has come to rely heavily on superpixel methods to circumvent the computational complexity inherent in pixel processing. The superpixel approach is generally used to group similar pixels into a semantic cluster of fewer pixels to increase the processing speed and simplify computational intricacy. However, the reliance on the existing superpixel based segmentation methods on the Euclidean distance metric as a measure of similarity between two pixels in an image presents an inherent challenge. The Euclidean distance has a real-world advantage because of its assumption of non-uniformity that most image colour distributions generally follow. This assumption states that real data will occupy a small clustered subset of the entire space, but not necessarily distributed evenly in a higherdimensional space. However, since it cannot deal with illumination change in images, it is limited in compactly measuring similarity in the context of an application that complies with the human perception of similarity. The human eyes can recognise similar or irrelevant image colours under the illumination change for which the Euclidean distance does not perform well. This study aimed to investigate the performance of an attribute concurrence influence distance metric on image compactness in a superpixel segmentation algorithm. It is hypothesized that superpixel segmentation based on attribute cooccurrence similarity measure is likely to achieve better results than Euclidean distance in terms of the performance metrics of under segmentation error, achievable segmentation accuracy, compactness, boundary recall, and contour density. Superpixel segmentation experiments were performed using two widely used colour models which are hue, saturation, value (HSV), and lightness, redness, yellowness (LAB) with the strong attribute concurrence influence distance (SAID) and Euclidean distance in a superpixel segmentation algorithm. The results presented for the LAB colour model showed that SAID outperformed the Euclidean distance for images reflecting overlapping and complex objects with regular compactness. However, the Euclidean distance performed better than the SAID for images with multiple, centre, and low contrast objects with regular compactness across the under segmentation error, achievable segmentation accuracy, boundary recall and contour density performance evaluation metrics. Consequently, for irregular compactness, SAID further outperformed the Euclidean distance for images with overlapping, complex, multiple, Centred and low contrast objects for boundary recall. However, the Euclidean distance performed better than SAID for under segmentation error, achievable segmentation accuracy, and contour density. Furthermore, the compactness performance for SAID and Euclidean distance gave the same compactness value for both regular and irregular compactness. Consequently, based on the analysis of the results for the HSV colour model, it was observed that performances of SAID and Euclidean with regular compactness were at par across all the performance metrics used for images with overlapping, complex, multiple, centre, and low contrast objects. However, the Euclidean distance outperformed SAID with irregular compactness for images with overlapping, complex, multiple, centre, and low contrast objects.

Image-to-class distance ratio: A feature filtering metric for image classification

Incrementally finding the k nearest neighbors (kNN) in a spatial network is an important problem in location-based services. One method (INE) simply applies Dijkstra's algorithm. Another method (IER) computes the k nearest neighbors using Euclidean distance followed by computing their corresponding network distances, and then incrementally finds the next nearest neighbors in order of increasing Euclidean distance until finding one whose Euclidean distance is greater than the current k nearest neighbor in terms of network distance. The LBC method improves on INE by avoiding the visit of nodes that cannot possibly lead to the k nearest neighbors by using a Euclidean heuristic estimator, and on IER by avoiding the repeated visits to nodes in the spatial network that appear on the shortest paths to different members of the k nearest neighbors by performing multiple instances of heuristic search using a Euclidean heuristic estimator on candidate objects around the query point. LBC's drawback is that the maintenance of multiple instances of heuristic search (called wavefronts) requires k priority queues and the queue operations required to maintain them incur a high in-memory processing cost. A method (SWH) is proposed that utilizes a novel heuristic function which considers objects surrounding the query point together as a single unit, instead of as one destination at a time as in LBC, thereby eliminating the need for multiple wavefronts and needs just one priority queue. These results in a significant reduction in the in-memory processing cost components while having the same reduced cost of the access to the spatial network as LBC. SWH is also extended to support the incremental distance semi-join (IDSJ) query, which is a multiple query point generalization of the kNN query. In addition, SWH is shown to support landmark-based heuristic functions, thereby enabling it to be applied to non-spatial networks/graphs such as social networks. Comparisons of experiments on SWH for kNN queries with INE, the best single-wavefront method, show that SWH is 2.5 times faster, and with LBC, the best existing heuristic search method, show that SWH is 3.5 times faster. For IDSJ queries, SWH-IDSJ is 5 times faster than INE-IDSJ, and 4 times faster than LBC-IDSJ.

A novel three-way decision method in a hybrid information system with images and its application in medical diagnosis

Given a set [Formula: see text] of [Formula: see text] points, a weight function [Formula: see text] to associate a non-negative weight to each point in [Formula: see text], a positive integer [Formula: see text], and a real number [Formula: see text], we present algorithms for computing a spanner network [Formula: see text] for the metric space [Formula: see text] induced by the weighted points in [Formula: see text]. The weighted distance function [Formula: see text] on the set [Formula: see text] of points is defined as follows: for any [Formula: see text], [Formula: see text] is equal to [Formula: see text] if [Formula: see text], otherwise, [Formula: see text] is [Formula: see text]. Here, [Formula: see text] is the Euclidean distance between [Formula: see text] and [Formula: see text] if points in [Formula: see text] are in [Formula: see text], otherwise, it is the geodesic (Euclidean) distance between [Formula: see text] and [Formula: see text]. The following are our results: (1) When the weighted points in [Formula: see text] are located in [Formula: see text], we compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network of size [Formula: see text]. (2) When the weighted points in [Formula: see text] are located in the relative interior of the free space of a polygonal domain [Formula: see text], we detail an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network with [Formula: see text] edges. Here, [Formula: see text] is the number of simple polygonal holes in [Formula: see text]. (3) When the weighted points in [Formula: see text] are located on a polyhedral terrain [Formula: see text], we propose an algorithm to compute a [Formula: see text]-vertex fault-tolerant [Formula: see text]-spanner network, and the number of edges in this network is [Formula: see text].

The classification loss functions used in deep neural network classifiers can be grouped into two categories based on maximizing the margin in either Euclidean or angular spaces. Euclidean distances between sample vectors are used during classification for the methods maximizing the margin in Euclidean spaces whereas the Cosine similarity distance is used during the testing stage for the methods maximizing margin in the angular spaces. This paper introduces a novel classification loss that maximizes the margin in both the Euclidean and angular spaces at the same time. This way, the Euclidean and Cosine distances will produce similar and consistent results and complement each other, which will in turn improve the accuracies. The proposed loss function enforces the samples of classes to cluster around the centers that represent them. The centers approximating classes are chosen from the boundary of a hypersphere, and the pairwise distances between class centers are always equivalent. This restriction corresponds to choosing centers from the vertices of a regular simplex. There is not any hyperparameter that must be set by the user in the proposed loss function, therefore the use of the proposed method is extremely easy for classical classification problems. Moreover, since the class samples are compactly clustered around their corresponding means, the proposed classifier is also very suitable for open set recognition problems where test samples can come from the unknown classes that are not seen in the training phase. Experimental studies show that the proposed method achieves the state-of-the-art accuracies on open set recognition despite its simplicity.

This doctoral thesis develops informetric methods for determining cognitive distance between publication portfolios of evaluators and evaluees in research evaluation. In a discipline specific research evaluation, when an expert panel evaluates research groups, it is an open question how one can determine the extent to which the panel members are in a position to evaluate the research groups. This thesis contributes to the literature by proposing six different informetric approaches to measure the match between evaluators and evaluees using their publications as a representation of their expertise. An expert panel is specifically appointed for the research evaluation. Experts are typically selected in one of two ways: (1) straightforward selection: the person(s) in charge of the research evaluation has access to a list of acknowledged experts in specific fields, and limits its selection process to ensuring the experts’ independence regarding the program under evaluation; and (2) gradual selections: preferred profiles of experts are developed with respect to the specialization under scrutiny in the evaluation. Both ways leave some freedom for an “old boys’ network” to appoint someone without properly evaluating their qualifications. There are also other ways for expert selection, for example, inviting open application or the research groups that will be evaluated can propose their choice of experts. In research evaluation, an expert panel usually comprises independent specialists, each of which is recognized in at least one of the fields addressed by the unit under evaluation. The expertise of the panel members should be congruent with the research groups to ensure the quality and trustworthiness of the evaluation. All things being equal, panel members who are credible experts in the field are also most likely to provide valuable, relevant recommendations and suggestions that should lead to improved research quality. However, there was an absence of methods to determine the cognitive distance between evaluators and evaluees in research evaluation when we started working in July 2013. In this thesis, we develop and test informetric methods to identify the cognitive distances between the (members of) an expert panel on the one hand, and the (whole of the) units of assessment (typically research groups) on the other. More generally, we introduce a number of methods that allow measuring cognitive distances based on publication portfolios. In academia, publications are considered key indicators of expertise that help to identify qualified or similar experts to assign papers for review, and to form an expert panel. Our main objective is to propose informetric methods to identify panel members who have closely related expertise in the research domain of the research groups based on their publications profile. The main factor that we have taken into account is the cognitive distance between an expert panel and research groups. We consider the publication portfolio of the involved researchers to reflect the position of the unit in cognitive space and, hence, to determine cognitive distance. Expressed in general terms we measure cognitive distance between units based on how often they have published in the same or similar journals. Our investigations lead to the development of new methods of expert panel composition for the research evaluation exercises. We explore different ways of quantifying the cognitive distance between panel members and research group's publication profiles. We consider all the publications of the research groups (during the eight years preceding their evaluation) and panel members indexed in Web of Science (WoS). We pursue the investigation at two levels of aggregation: WoS subject categories (SCs) and journals. The aggregated citation relations among SCs or journals provide a matrix. From the matrix, one can construct a similarity matrix. From the similarity matrix, one can construct a global SCs or journal map in which similar SCs or journals are located more closely together. The maps can be visualized using a visualization program. During the visualization process, a multi-dimensional space is reduced to a projection in two dimensions. In this process, similar SCs or journals are positioned closer to each other. We propose three methods, namely the use of barycenters, of similarity-adapted publication vector (SAPV) and of weighted cosine similarity (WCS). We take into account the similarity between WoS SCs and between journals, either by incorporating a similarity matrix (in the case of SAPV and WCS) or a 2-dimensional base map derived from it (in the case of barycenters). We determine the coordinates of barycenters using a 2-dimensional base map based on the publication profiles of research groups and panel members, and calculate the Euclidean distances between the barycenters. We also identify SAPV using the similarity matrix and calculated the Euclidean distances between the SAPVs. Finally, we calculate WCS using the similarity matrix. The SAPV and WCS methods use a square N-dimensional similarity matrix. Here N is equivalent to 224 WoS SCs and 10,675 journals. We used the distance/similarity between panel members and research groups as an indicator of cognitive distance. Small differences in Euclidean distances (both between barycenters and SAPVs) or in cosine similarity values bear little meaning. For this reason, we employ a bootstrapping approach in order to determine a 95% confidence interval (CI) for each distance or similarity value. If two CIs do not overlap, difference between the values is statistically significant at the 0.05 level. Although it is possible for two values to have a statistically significant difference while having overlapping CIs, the difference is less likely to have practical meaning. Two levels of aggregation and three methods lead to six informetric approaches to quantify the cognitive distance. Our proposed approaches hold advantages over a simple comparison of publication portfolios. Our approaches quantify the cognitive distance between a research group and panel members. We also compare our proposed approaches. We examine which of the approaches best reflects the prior assignment of main assessor to each research group, how much influence the level of aggregation (journals and WoS SCs) plays, and how much the dimensionality matters. The results show that, regardless of the method used, the level of aggregation has only a minor influence, whereas the influence of the number of dimensions is substantial. The results also show that the number of dimensions plays a major role in the case of identifying shortest cognitive distance. While the SAPV and WCS methods agree at most of cases at both the levels of aggregation the barycenter approaches yield different results. We find that the barycenter approaches score highest at both levels of aggregation to identify the previously assigned main assessor. When it comes to uniquely identifying the main assessor, all methods score better at the journal level than at the WoS SC level. Our approaches, but of course not the numerical result, are independent of the similarity matrix or map used. All six approaches give the opportunity to assess the composition of the panel in terms of cognitive distance if one or more panel members are replaced and compare the relative contribution of each potential panel member to the panel fit as a whole, by observing the changes to the distance between the panel’s and the groups’. In addition, our approaches allow the panel composition authority to see in advance about the panel’s fit to the research groups that are going to be evaluated. Therefore, the concerned authority will have the opportunity to replace outliers among the panel members to make the panel fit well with the research groups to be evaluated. For example, the authority can find a best-fitting expert panel by replacing a more distant panel member with a potential panel member located closer to the groups.

We consider 1-local online routing on a special class of geometric graphs called Delaunay triangulations (DTs). A geometric graph G = (V, E) of a point set consists of a set of points in the plane and edges between them, where each edge weighs as the Euclidean distance between it’s end-points. DTs are one of the useful classes of these graphs because of some good properties which can help during the navigation process, therefore over the years DTs have been widely proposed as network topologies for several times.In this paper, we present a new memoryless online routing (MOR) algorithm for DTs which is simple, elegant, and easy to implement, while having an acceptable performance.The set of MOR algorithms are suitable for cases where we want to find a path using only local information, our proposed algorithm is memoryless or 1-local, in k-local routing, we find a path between a source vertex s to a destination vertex t while our knowledge at each step is limited to the locations of s and t, the location of current vertex and it’s k-neighborhood vertices.We also evaluate and compare the perforamnce of our prpopsed algorithm with existing MOR algorithms. Our experimental results implied that our proposed algorithm has an acceptable performance in both Euclidean and link metrics and it outperforms all of the existing MOR algorithms in Euclidean metric, and some of them in the link metric as well. Finally, we pose two open problems to solve in the future.

Epistemological analysis is a research approach that investigates the origin and acquisition of knowledge. This study applies Imre Lakatos's Methodology of Scientific Research Programmes (MSRP) to analyze two bodies of thought: the laws of thermodynamics, i.e., conservation of energy, heat as a form of energy, and entropy increase for irreversible processes; and the classical Traveling Salesman Problem (TSP), evaluating competing theories that seek heuristic or exact combinatorial optimization methods. The analysis is supported by the perspective that mathematical modeling provides tools that facilitate the understanding, analysis, and solution of complex problems, especially in physics and computation. In the first case, thermodynamic cycles were modeled using the functional behavior of the isothermal, adiabatic, isobaric and/or isometric processes involved. In the context of the TSP, modeling can be done in terms of graphs, in which the vertices of the structure represent the cities that the salesman needs to visit and the edges indicate the connections between them. Applying the concurrent approach of the k­nearest neighbors algorithm, the distance between cities is calculated by the Euclidean distance between the points distributed in a network. Determining the value of k, representing the number of nearest neighbors, depends on the size of the data set, the complexity of the problem and the desired accuracy. Tracing the route to a new city involves calculating the distances to all the cities in the training set and identifying the k­nearest neighbors. The process must consider previously defined rules, such as not passing each point more than once and returning to the starting point. The primary output of these studies is a website hosted on Github of the homonymous book "From Micro to Macro: An Introduction to Thermostatistics". It is a free collaborative environment under construction, with user­friendly design providing static and dynamic content. The environment allows virtual evaluation tools to be embedded, which generate a database for validation by users and by a panel of experts. Key-words: Imre Lakatos, Research Program, Thermodynamics Cycles, Traveling Salesman Problem, mathematical modelling, Digital Information and Communication Technologies (DICT)

Most existing radar automatic target recognition (RATR) methods based on high-resolution range profile (HRRP) have been verified to be vulnerable under small sample size conditions, which seriously restricts their real-world promotion and applications. Therefore, enhancing the capability of HRRP few-shot recognition is essential in practical HRRP RATR. In addition, the polarization characteristics of HRRP remain underutilized. Given this, in this article, we propose a full polarimetric radar HRRP (FP-HRRP) few-shot target recognition method from the perspective of optimal matching scatterer components (SCs). Specifically, since heuristic metric methods, such as Euclidean distance are flawed in few-shot classification scenarios, we develop an FP-HRRP few-shot classification method from the perspective of optimal matching between SCs, which transforms the metric problem into solving the optimal solution for a linear programming (LP) model to determine the FP-HRRP relevance. The optimal matching flows between SCs that have the minimum matching cost, which is used to calculate the FP-HRRP distance for classification. To generate the crucial weights of SCs in the LP formulation, we design a cross-similarity weight generation method, which can accurately measure the correlation between SCs and alleviate the adverse impact caused by the intraclass variations arising from azimuth sensitivity and noise. Finally, to integrate intraclass compactness and interclass separation, we proposed a scatterer component cosine loss, further enhancing the separability of feature space. Extensive experiments demonstrate that the proposed method achieves state-of-the-art recognition performance and strong target-aspect robustness for few-shot HRRP recognition with a measured FP-HRRP dataset.