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The footwear industry is moving towards automation and intellectualization. To overcome the drawbacks of the high-cost and low-efficiency traditional manual shoe sole gluing process, automatic methods were utilized for generating spray trajectories. Currently, most of the reported automatic methods for generating spray trajectories mainly rely on the outer contour bias method. However, the glue is only applied to the area near the edge/contour of shoe soles and the fixed offset distance in the outer contour bias method cannot adapt to the immense amount of shoe styles with high precision and achieve applicability for irregular and unique sole designs. An intuitive yet logical approach to fulfill the requirements is to utilize the deconvolutional neural network for generating shoe sole spray trajectories. In this work, we treated the glue trajectory prediction as an image-to-image prediction and established a novel deconvolutional neural network to generate shoe sole spray trajectories. The as-proposed deconvolutional neural network for generating spray trajectory offered significant advantages over the existing bias-based methods, including: (1) based on the novel deconvolutional neural network, the proposed method for generating shoe sole spray trajectories exhibits greater applicability to irregular shoe soles, which improves the spray accuracy without compromising the spray efficiency; (2) we discard all the pooling layers, which only consist of convolutional and deconvolutional layers, to preserve more spatial information and achieve higher spray accuracy through end-to-end mapping from shoe sole images to shoe sole spray trajectories, resulting in an improved spray accuracy without sacrificing spray efficiency. The Dice similarity coefficient and Hausdorff distance were used as the evaluation metrics to assess the performance of our approach. Our proposed method showed an ultra-high accuracy and precision with a Dice similarity coefficient over 99.25% and a Hausdorff distance less than 1.2 mm, which are ~10% higher than the spray accuracy of other reported traditional methods. Our findings would bring significant improvements to the field of automatic shoe sole spray trajectory generation, which has the potential to promote the utilization of intelligent technologies in the footwear industry. [ABSTRACT FROM AUTHOR]

This study proposed two algorithms for reconstructing jigsaw puzzles by using a color compatibility feature. Two realistic application cases were examined: one involved using the original image, while the other did not. We also calculated the transformation matrix to obtain the real positions of each puzzle piece and transmitted the positional information to the robotic arm, which then put each puzzle piece in its correct position. The algorithms were tested on 35-piece and 70-piece puzzles, achieving an average success rate of 87.1%. Compared with the human visual system, the proposed methods demonstrated enhanced accuracy when handling more complex textural images. [ABSTRACT FROM AUTHOR]

The reach of a set M ⊂ R d , also known as condition number when M is a manifold, was introduced by Federer in 1959. The reach is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. Under an additional assumption we provide rates of convergence. We also show that it is not possible to determine, based on a finite sample, if the reach of the support of a density is zero or not. We provide a small simulation study and a bias correction method for the case when M is a manifold. • A consistent reach estimator is proposed assuming only positive reach. • Rates of convergence are provided under very general shape restrictions. • It is not possible to consistently decide from a sample if the set has positive reach or not. [ABSTRACT FROM AUTHOR]

The research community considers handwritten word recognition (HWR) as an open research problem to date. The reasons behind this are variations in intra-/interpersonal writing style, overlapping and/or touching characters in a word, degraded scanned document images, etc. Two major approaches, namely holistic and analytical, are followed by the researchers while designing an HWR system. In this work, we have followed the holistic approach as it works well on limited and pre-defined lexicon as compared to the analytical approach. As observed in the literature related to handwritten word recognition, irrespective of the approaches, researchers generally extract various local features from hypothetically partitioned segments of a word image while dealing with the said problem. However, no such work has been found which has considered inter-segment similarity that might carry some distinct information about different patterns (here, word segments). To this end, in the present work, we have used Hausdorff and Fréchet distances to quantize the similarity among all possible word segments taking two at a time. Along with this, conventional chain code histogram (a shape-based feature descriptor) and modified negative refraction-based shape transformation features have been used. Finally, a majority voting schema is used to combine outputs from six different classifiers. The model has been evaluated on two standard databases, namely IAM and IFN/ENIT, and the results obtained are promising in comparison with state-of-the-art holistic word recognition methods. Moreover, a performance comparison of the present method with some deep learning models confirms the usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we propose a new triangular mesh simplification algorithm, in which the optimal contraction vertex is obtained by direct optimizing the Hausdorff distance between the simplified mesh and the input mesh with neural network. The neural network uses an attention module to efficiently extract the geometric features of the meshes. At the same time, we adopt a global strategy to reduce the unnecessary computations in the process of optimizing the Hausdorff distance. The correspondence between the one neighborhood triangular patch of the vertices to be merged and the triangular patch of the input mesh can be obtained via this global strategy, based on which we can replace the Hausdorff distance between the simplified mesh and the input mesh with the Hausdorff distance between another two small-size triangular meshes. This algorithm can deal with any triangular mesh and preserve the geometric characteristics of the input mesh well, and US effectiveness and superiority are demonstrated by the experimental results. [ABSTRACT FROM AUTHOR]

The centers of matching craters can be beneficial additions to the control point database. Crater matching on heterogeneous surfaces is helpful in testing its applicability to the entire moon. Therefore, we propose a coarse-to-fine crater matching method for heterogenous surfaces on images acquired from the narrow angle camera (NAC) of the lunar reconnaissance orbiter camera (LROC) and the Chang’e-2 (CE-2) digital orthophoto map (DOM). First, we perform coarse matching based on the Hausdorff distance using the area and coordinates of the crater. Then, the mismatched craters are removed by using the affine transform (AT) model fitted by corresponding points of mutual information (MI) matching. Finally, we use the retained matched crater centers to fit the affine transformation model between the images, predict the corresponding position, and obtain the corresponding crater around it to achieve fine matching. The results show that the proposed method obtains numerous crater matches on images covering different terrains and solar altitude angles compared to the Hausdorff distance-based crater matching method. For the five experimental scenes registered using matched craters, the mean values of the checkpoints are approximately two and three pixels for scenes with small and large differences from CE-2 solar altitude angles, respectively, and the standard deviations (STDs) for both are approximately one pixel. In addition, the highlands have lower accuracy than the maria, with a variance of less than one pixel. Furthermore, the registration accuracy is related to the diameter and number of craters. [ABSTRACT FROM AUTHOR]

In this paper the control system with integral constraint on the control functions is studied where the behavior of the system by the Urysohn type integral equation is described. The admissible control functions are chosen from the closed ball of the space Lp ([a, b]; Rm) (p > 1) centered at the origin with radius r. Dependence of the set of trajectories on r and p is investigated. It is proved that the set of trajectories is Lipschitz continuous with respect to r and is continuous with respect to p. The robustness of the trajectory with respect to the fast consumption of the remaining control resource is established. [ABSTRACT FROM AUTHOR]

The Fermat–Steiner problem is the problem of finding all points of a metric space Y such that the sum of the distances from them to points of a certain fixed finite subset A of the space Y is minimal. In this paper, we examine the Fermat–Steiner problem in the case where Y is the space of compact subsets of the Euclidean plane endowed with the Hausdorff metric, and points of A are finite pairwise disjoint compact sets. [ABSTRACT FROM AUTHOR]

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study the possible behaviors of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the infinite-dimensional inverse problem – the underlying motivations being the stability or instability of infinite-dimensional Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an infinite-dimensional inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. [ABSTRACT FROM AUTHOR]

In this paper we introduce multivalued modified F-contraction on a metric space. This is a multivalued mapping obtained by incorporating the idea of the recently introduced F-contraction which has attracted much attention in contemporary research. We explore the fixed point problem associated with the above contractive mapping. We also investigate the data dependence and stability properties of the fixed point sets associated with these multival- ued contractions. We discuss an illustration of the main result and present an application of the single valued version of the main theorem to a problem of an integral equation of Volterra type. The domain of the study is fixed point theory and set valued analysis. [ABSTRACT FROM AUTHOR]

A large number of DGs(distributed generators) in the ADN (active distribution network) bring great uncertainty to the power systems, making it difficult to quantify fault characteristics. Undoubtedly, the setting of traditional protection becomes more complex and difficult. Therefore, the differential protection is an effective solution to cope with it. However, the differential protection requires high time synchronization, which is difficult to be satisfied based on the current communication conditions of ADNs. What’s more, accurate extraction of fundamental frequency components is also very difficult considering the distortion, the frequency offset, and the instability of fault currents. In view of this, a new differential protection scheme that makes full use of fault information is proposed in this paper. First, the pre-fault current waveforms are taken as reference. Then the similarity characteristics of the current waveform sequences at both terminals of the line are calculated separately. Finally, a new differential protection scheme considering weak synchronization conditions are constructed using Hausdorff distance, which is based on wide frequency components. The proposed scheme only transmits calculation results instead of sampling values, which effectively avoids the impact caused by data synchronization errors. Simulation shows that the proposed scheme can reliably identify internal faults in various points of the protected line and external faults. Compared with the conventional differential protections, it also shows great resistance to the transition resistance up to $30\Omega $ , which is common in the ADN. Moreover, it has strong resistance to time synchronization errors, which can withstand time synchronization errors of up to 20ms. The proposed protection scheme has excellent engineering application value, especially suitable for distribution networks with weak communication conditions.

Abstract In the military and civilian surveillance domain, it is of great significance to mine regular behaviours of targets for situation awareness and command decision support. Most of the existing trajectory clustering algorithms only consider the similarity of spatial position of the trajectory, without sufficient multi‐dimensional information such as time, course and velocity. Some approaches based on information fusion take these multi‐dimensional information into account, but the features with different dimensions fused by weight coefficients are not robust and universal for different scenarios. In this paper, a regular behaviour mining method based on spatiotemporal trajectory multi‐dimensional features and density clustering is proposed. Firstly, multi‐dimensional Hausdorff similarity is defined to measure spatiotemporal trajectory from different feature dimensionalities. Different from methods based on information fusion, the proposed method defines trajectory density in feature similarity of different dimensions and adaptively determines parameters according to feature distribution in different dimensions. Experimental results in simulated and radar measured trajectory data show that the proposed method can be accurate and robust in clustering evaluation indexes such as Purity, Precision, Recall and Rand Index from different scenarios, which has a good application prospect in intelligent surveillance tasks.

Background and Purpose: A wide range of quantitative measures are available to facilitate clinical implementation of auto-contouring software, on-going Quality Assurance (QA) and interobserver contouring variation studies. This study aimed to assess the variation in output when applying different implementations of the measures to the same data in order to investigate how consistently such measures are defined and implemented in radiation oncology. Materials and Methods: A survey was conducted to assess if there were any differences in definitions of contouring measures or their implementations that would lead to variation in reported results between institutions. This took two forms: a set of computed tomography (CT) image data with “Test” and “Reference” contours was distributed for participants to process using their preferred tools and report results, and a questionnaire regarding the definition of measures and their implementation was completed by the participants. Results: Thirteen participants completed the survey and submitted results, with one commercial and twelve in-house solutions represented. Excluding outliers, variations of up to 50% in Dice Similarity Coefficient (DSC), 50% in 3D Hausdorff Distance (HD), and 200% in Average Distance (AD) were observed between the participant submitted results. Collaborative investigation with participants revealed a large number of bugs in implementation, confounding the understanding of intentional implementation choices. Conclusion: Care must be taken when comparing quantitative results between different studies. There is a need for a dataset with clearly defined measures and ground truth for validation of such tools prior to their use.

As military technology continues to evolve and the amount of situational information available on the battlefield continues to increase, data-driven deep learning methods are becoming the primary method for air target intention recognition. Deep learning is based on a large amount of high quality data; however, in the field of intention recognition, it often faces key problems such as low data volume and unbalanced datasets due to insufficient real-world scenarios. To address these problems, we propose a new method called time-series conditional generative adversarial network with improved Hausdorff distance (IH-TCGAN). The innovation of the method is mainly reflected in three aspects: (1) Use of a transverter to map real and synthetic data into the same manifold so that they have the same intrinsic dimension; (2) Addition of a restorer and a classifier in the network structure to ensure that the model can generate high-quality multiclass temporal data; (3) An improved Hausdorff distance is proposed that can measure the time order differences between multivariate time-series data and make the generated results more reasonable. We conduct experiments using two time-series datasets, evaluate the results using various performance metrics, and visualize the results using visualization techniques. The experimental results show that IH-TCGAN is able to generate synthetic data similar to the real data and has significant advantages in the generation of time series data. [ABSTRACT FROM AUTHOR]

We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections, convergence to an embedded Lagrangian submanifold implies convergence to it in the Hausdorff metric. This class of metrics includes well-known metrics such as the Lagrangian Hofer metric, the spectral metric and the shadow metrics introduced by Biran et al. [Lagrangian shadows and triangulated categories, Astérisque 426 (2021) 1–128]. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball. [ABSTRACT FROM AUTHOR]

Traditional progress monitoring can be inaccurate and time-consuming, potentially causing time delay and cost overrun in construction projects. With development in technology, tools such as cameras, laser scanners, and building information modelling (BIM) have been used to overcome existing problems in the traditional approach. However, noise mitigation, extracting objects of interest from laser point clouds, and detailed progress measurement are problems that still exist. In this study a novel method of construction progress monitoring to measure the progress percentage is presented. The study integrates the simultaneous localization and mapping (SLAM) technique with as-built BIM to gather quick and accurate construction site progress information. The Hausdorff distance is utilized to extract objects of interest and filter out noise from site-scan data. As-built and as-planned BIM models are compared using Python and Dynamo, to obtain progress percentage. A case study was conducted on a residential building located in Sydney, Australia, to validate the application of the developed method. The outcome demonstrates that utilizing the SLAM technique and Hausdorff distance are effective in mitigating noise and extracting objects of interest from site-scan data, respectively. In addition, with an accuracy of 94.67 percent in estimation, the progress percentage was obtained based on material quantities. The obtained progress percentage could also be used in updating construction schedules and assisting decision-making. [ABSTRACT FROM AUTHOR]

Combining least absolute deviation（LAD) and least absolute shrinkage with selection operator（LASSO), we studied the multiple change points estimation problem in linear regression models, that is, the LAD-LASSO estimation. Firstly, the multiple change points estimation problem was transformed into a variable selection problem. Secondly, the locations and number of change points were estimated by minimizing the objective function. Thirdly, the simulation study was carried out between the LAD-LASSO method and the traditional LASSO method. Simulation results show that the Hausdorff distance between the change points estimation set and the true value set obtained by the LAD-LASSO is smaller than that of LASSO, which indicates that the LAD-LASSO method has better robustness. Finally, we estimated the variance change points of Shanghai airport stock（600009) yield data, and the results show that the proposed method can be used to deal with practical problems. [ABSTRACT FROM AUTHOR]

The goal of non-rigid point set registration is to estimate the optimal correspondence between points, and then recover the non-rigid deformation between point sets in a specific way, typically by using a set of complex interpolation functions. Many non-rigid matching algorithms have been studied, but only a few algorithms fully exploit the local structure between point sets. To improve the accuracy of point set registration, this paper proposes a new non-rigid registration algorithm that uses both the global structure and the stable local structure of a non-rigid shape to constrain the registration. Specifically, we consider the point set registration problem as a probability assignment problem, with the probability determined by the Gaussian mixture model and the local structure of the point set. In particular, the Hausdorff distance can effectively measure the similarity of the local structure of the point set in the proposed algorithm. The transformation between the two-point sets is determined by the reproducing kernel Hilbert space based on the motion coherence theory once the correspondence is determined. A significant number of experiments show that the proposed technique has higher registration accuracy than several other state-of-the-art algorithms when dealing with non-rigid registration problems, especially when the point set contains outliers and severely missing points. [ABSTRACT FROM AUTHOR]

In the present study, the performance evaluation of the signal decomposition methods; variational mode decomposition, empirical mode decomposition, and ensemble empirical mode decomposition, for the ball bearing fault detection and classification for the experimentally recorded vibration signals has been done. This work proposed a novel hybrid sensitive mode selection method combining three statistical measures (energy-based index, fault correlation-based index, and Hausdorff distance-based index) and investigating the effect of the selected sensitive mode extracted by the decomposition methods for the bearing defect frequency detection. The vibration data have been acquired for the healthy and seeded faults of different sizes for the inner and outer raceway defects. The complete features dataset comprises five time-domain, four spectral-domain, and two non-linear statistical features. The k-Nearest Neighbor, Support Vector Machine, and Naive Bayes classifiers are used for fault classification and predict the results with four performance metrics: accuracy, sensitivity, precision, and F-score. Firstly, the results of signal decomposition employing hybrid sensitive mode functions and statistical analysis of condition indicators (RMS, kurtosis and crest factor) revealed that the VMD outperforms the other two techniques. Secondly, the fault classification results predicted that the k-Nearest Neighbor classifier outperforms the other two classifiers. This proposed novel sensitive mode selection method significantly improves the bearing fault classification performance metrics with the features extracted from the selective mode functions with all three decomposition methods. [ABSTRACT FROM AUTHOR]

Precisely identifying the household-transformer relationship is of significant importance for both the stability of the power system and the quality of customer electricity consumption. However, the complex network structures and frequent reconfigurations may lead to inaccurate records of household-transformer relationships. In this paper, a novel data-driven similarity assessment solution is proposed to enhance the accuracy and scalability of identifying household-transformer relationships. Initially, a data processing method based on dynamic temporal regularization with sliding windows is employed to optimize dataset quality as well as enhance the efficiency of data processing. Then, a two-stage solution is proposed for identifying the household-transformer relationship. The first stage involves initial normalized clustering based on the basic information of power distribution substations, while the second stage assesses the similarity between households and transformer operational states based on Hausdorff distance. The superior performance of the proposed method is extensively assessed through real historical datasets, compared to benchmarks.

The mutual identification of spatial objects is a fundamental issue when updating geographic data with other data sets. Representations of spatial objects in different sources may not have the same identifiers, which would unambiguously assign them to each other. Intersections of spatial objects can be used for this purpose, but this does not allow for the detection of possible changes and their quantification. The aim of this paper is to propose a simple, applicable procedure for calculating the shape similarity measure, which should be able to efficiently identify different representations of spatial objects in two data sources, even though they may be changed or generalised. The main result is the aggregated index of shape similarity and instructions for its calculation and implementation. The shape similarity index is based on the calculation of the set similarity, the distance of the boundaries, and the differences in the area, perimeter, and number of the vertices of areal spatial objects. In the case study, the footprints of the building complexes in Dúbravka (part of the city of Bratislava, the capital of Slovakia) are compared using data from OpenStreetMap and INSPIRE (Infrastructure for Spatial Information in Europe) Buildings. A contribution to the quality check of the OpenStreetMap data is then a secondary result. The proposed method can be effectively used in the semi-automatic integration of heterogeneous data sources, updating the data source with other spatial data, or in their quality control.

In view of the fact that the topology and fault characteristics of the traditional distribution network have changed due to the access of distributed new energy, which brings new challenges to the relay protection of the distribution network, an adaptive differential protection scheme based on the current waveform similarity at both ends of the line is proposed. By analyzing the current waveform characteristics at both ends of the line, the Hausdorff distance algorithm is used to calculate the current similarity. On this basis, the braking coefficient of current differential protection is constructed and the fault area is self-adaptive discriminated. A distribution network model based on distributed power is built in PSCAD/EMTDC software, and the proposed protection scheme is simulated and verified. The simulation results show that the scheme can accurately identify the fault area, has strong resistance to transition resistance, and does not require voltage information and increase The additional protection device has the advantages of being compatible with low cost and high reliability, and strong anti-synchronization error ability. [ABSTRACT FROM AUTHOR]

Fuzzy sets (FSs) with belief and plausibility measures in Dempster-Shafer theory (DST) are recognized as different methodology to model imperfect, uncertain, and vague information more accurately than probability. Various generalizations of DST to FSs are suggested in the numerous literatures but the generalization of DST to Pythagorean fuzzy sets (PFSs) has not yet been considered so far. In this paper, we first suggest an intuitive and simple way to develop a generalization of DST to PFSs with the characterization of belief function in terms of membership function and plausibility function in terms of 1-nonmemberhip function respectively. We give the interpretation of belief and plausibility on PFSs and then construct belief-plausibility intervals (BPIs) of PFSs. On the basis of suggested BPIs, we use Hausdorff distance to describe the distance between two BPIs and then construct several similarity measures in the generalized context of DST to PFSs. By utilizing the method of VIekriterijumsko KOmpromisno Rangiranje (VIKOR), the suggested belief and plausibility measures on PFSs in the framework of DST enable us to develop a belief-plausibility VIKOR (BP-VIKOR) to manage multicriteria decision-making (MCDM) problems related to daily life settings. Numerical analysis with examples are given to show the suggested method is reasonable, and suitable in the environment of PFSs in the context of generalization of DST. [ABSTRACT FROM AUTHOR]

Laser ultrasound signal echoes are easily drowned out by the surrounding environmental noise in industrial field applications, and it is worthwhile to study methods of retaining the weak ultrasound signal during signal processing. To address this problem, this paper proposes to adopt the parameters optimized by the whale optimization algorithm to the variational mode decomposition (VMD) of laser ultrasound signals. The optimized parameters can avoid the frequency mixing and incomplete noise separation caused by the choice of artificial VMD parameters. The Hausdorff distance is applied in the process of reconstructing the signal to help accurately select the relevant modes and improve the signal-to-noise ratio. Simulation and experimental results show that the proposed method is feasible and effective compared with the other three available denoising methods. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n for all n ≥ 2 . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78]. [ABSTRACT FROM AUTHOR]

We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n for all n ≥ 2 . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78]. [ABSTRACT FROM AUTHOR]

We introduce and investigate the orbit-closed C-numerical range, a natural modification of the C-numerical range of an operator introduced for C trace-class by Dirr and vom Ende. Our orbit-closed C-numerical range is a conservative modification of theirs because these two sets have the same closure and even coincide when C is finite rank. Since Dirr and vom Ende's results concerning the C-numerical range depend only on its closure, our orbit-closed C-numerical range inherits these properties, but we also establish more. For C self-adjoint, Dirr and vom Ende were only able to prove that the closure of their C-numerical range is convex and asked whether it is convex without taking the closure. We establish the convexity of the orbit-closed C-numerical range for self-adjoint C without taking the closure by providing a characterization in terms of majorization, unlocking the door to a plethora of results which generalize properties of the C-numerical range known in finite dimensions or when C has finite rank. Under rather special hypotheses on the operators, we also show the C-numerical range is convex, thereby providing a partial answer to the question posed by Dirr and vom Ende. [ABSTRACT FROM AUTHOR]

In the military and civilian surveillance domain, it is of great significance to mine regular behaviours of targets for situation awareness and command decision support. Most of the existing trajectory clustering algorithms only consider the similarity of spatial position of the trajectory, without sufficient multi‐dimensional information such as time, course and velocity. Some approaches based on information fusion take these multi‐dimensional information into account, but the features with different dimensions fused by weight coefficients are not robust and universal for different scenarios. In this paper, a regular behaviour mining method based on spatiotemporal trajectory multi‐dimensional features and density clustering is proposed. Firstly, multi‐dimensional Hausdorff similarity is defined to measure spatiotemporal trajectory from different feature dimensionalities. Different from methods based on information fusion, the proposed method defines trajectory density in feature similarity of different dimensions and adaptively determines parameters according to feature distribution in different dimensions. Experimental results in simulated and radar measured trajectory data show that the proposed method can be accurate and robust in clustering evaluation indexes such as Purity, Precision, Recall and Rand Index from different scenarios, which has a good application prospect in intelligent surveillance tasks. [ABSTRACT FROM AUTHOR]

For a Jordan curve Γ in the complex plane, its constant distance boundary Γλ is an inflated version of Γ. A flatness condition, (1/2,r0)-chordal property, guarantees that Γλ is a Jordan curve when λ is not too large. We prove that Γλ converges to Γ, as λ approaching to 0, in the sense of Hausdorff distance if Γ has the (1/2,r0)-chordal property.

In this paper we establish a coupled fixed point theorem of a coupled multivalued mapping defined on a complete metric space. In our result we use a new contractive inequality. There are rational terms in the expression of the inequality. The contractive condition on the nonlinear map is assumed only to hold on the elements that are related by a binary relation. The main theorem has several corollaries and is illustrated with example. The main result is deduced in metric spaces. Its consequences are discussed for α- admissible mapping as well as in metric spaces with partial order and graph respectively.

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G 2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x -coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G 2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels. • Proposing a framework to G2 approximate planar algebraic curves with correct topology and certified error control. • Proposing a construction method of planar quintic PH curve with invariant convexity based on G2 interpolation. • Proposing the theoretical analysis of the existence condition of the solutions. [ABSTRACT FROM AUTHOR]

We interpolate planar rigid body motions by piecewise rotational and translational motions. The trajectories of all points are then arc splines, i.e., curves composed of circular arcs or line segments. For objects with arc spline boundaries, the boundary of the volume swept by the object and the motion is then again an arc spline. We prove that the distance between the original motion and its approximation as between curves in a suitable kinematic image space converges quadratically to zero. This implies the same rate of convergence between point trajectories under the two motions. [ABSTRACT FROM AUTHOR]

This paper discusses the Topp-Leone-G power series class of distributions. The greatest attention is paid to the investigation of intrinsic characteristic “saturation” to the horizontal asymptote in the Hausdorff sense. Some estimates for the value of the Hausdorff distance are obtained. We present a new family of recurrence generated adaptive functions with corresponding applications. The usefulness of the obtained results is demonstrated in a simulation study of some real data sets from the medical sector and insurance. Some suitable software modules within the programming environment CAS MATHEMATICA are proposed.

The carotid artery is the major artery that supplies blood to the brain, neck region, and face. The plaque deposition in these arteries is caused mainly due to the deposition of cholesterol, calcium, and other cellular debris carried along with the bloodstream. Hence identification of plaque is essential to avoid stroke and other diseases related to the heart. This paper proposes a deep learning-based segmentation algorithm for the identification of plaque in carotid artery using Intravascular Ultrasound (IVUS) images. To compare the performance of the proposed algorithm with the existing algorithms, evaluation metrics such as Jaccard Index (JI), Dice Similarity Coefficient (DC), and Hausdorff Distance (HD) are computed. From the results, it is observed that the proposed algorithm exhibited a high value with JI of 0.9562, DC of 0.9587, and HD of 4.8080.

Abstract Objectives In recent years, many studies have reported that the presurgical nasoalveolar molding method improves the nose morphology; however, the reason for its effectiveness after surgery has never been understood. We evaluated the effect of nasoalveolar molding by comparing it with a passive orthopedic method without a nasal stent and focusing on the nostril morphology after primary cheiloplasty using various measurement methods. We then analyzed the essential factors. Materials and methods The patients involved were 31 infants with unilateral complete cleft lip and palate treated with primary cheiloplasty at the University of Tsukuba Hospital from 2004 to 2011. Of the 31 infants, 16 received nasoalveolar molding treatment and 15 received passive orthopedic treatment as controls. Photographic facial measurements were performed for all patients immediately and 7 months after primary cheiloplasty. The esthetics of the nostrils were assessed according to the left–right nostril symmetry, as measured by the Hausdorff distance, area ratio, perimeter ratio, and aspect a/u (the aspect ratio of the affected side)/(the aspect ratio of the unaffected side) ratio. In addition, the inclination of the nasal ridge was assessed using anthropometric measurements (Grc‐Grn∠midline and midline∠columellar axis). Results The area ratio, perimeter ratio, and Grc‐Grn∠midline were significantly greater in the nasoalveolar molding group immediately after surgery (p = 0.00062, 0.016, and 0.048, respectively) than in the control group. However, the Hausdorff distance and aspect a/u ratio were more favorable (p = 0.0018 and 0.0039, respectively) in the nasoalveolar molding group after 7 months. Conclusions The results of our study suggested that using nasoalveolar molding as a presurgical orthopedic treatment could improve the shape of the nasal cartilage with surgeon's corrections.

LetMD.M; be a metric space and let X be a Banach space. Let F be a set-valued mapping fromM into the family Km. of all compact convex subsets of X of dimension at most m. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "finiteness principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of F, i.e., a Lipschitz mapping f WM ! X such that f .x/2 F.x/for every x 2 M. We give new alternative proofs of this result in two special cases. When m D 2, we prove it for X D R2, and when m D 1 we prove it for all choices of X. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping F, i.e., for a mapping GWM ! Km.X/which is Lipschitz with respect to the Hausdorff distance, and such that G.x/- F.x/for all x 2 M. [ABSTRACT FROM AUTHOR]

In this paper, we prove some fixed point results for multivalued mappings in bounded metric spaces via symmetric spaces. Moreover an application to an integral inclusion is given. [ABSTRACT FROM AUTHOR]

The relationship between the Chebyshev center of a closed convex bounded subset and a center of an inscribed ball for some "dual" set (a ball of maximal radius which is contained in the set) is considered in a real Hilbert space. The inscribed ball is unique in the discussed situation. We present an approximate algorithm for calculation of the Chebyshev center (or the center of the inscribed ball for the "dual" set) for a convex compact subset from R n which is given via its supporting function. We reduce the problem to the solution of a linear programming problem and estimate the error between an approximate and the exact solutions in terms of the step of a grid. Few numerical examples are considered. The considered algorithm is efficient in the space of small dimension n. [ABSTRACT FROM AUTHOR]