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Image segmentation is a crucial task in artificial intelligence fields such as computer vision and medical imaging. While convolutional neural networks (CNNs) have achieved notable success by learning representative features from large datasets, they often lack geometric priors and global object information, limiting their accuracy in complex scenarios. Variational methods like active contours provide geometric priors and theoretical interpretability but require manual initialization and are sensitive to hyper-parameters. To overcome these challenges, we propose a novel segmentation approach, named PolarVoting, which combines the minimal path encoding rich geometric features and CNNs which can provide efficient initialization. The introduced model involves two main steps: firstly, we leverage the PolarMask model to extract multiple source points for initialization, and secondly, we construct a voting score map which implicitly contains the segmentation mask via a modified circular geometric voting (CGV) scheme. This map embeds global geometric information for finding accurate segmentation. By integrating neural network representation with geometric priors, the PolarVoting model enhances segmentation accuracy and robustness. Extensive experiments on various datasets demonstrate that the proposed PolarVoting method outperforms both PolarMask and traditional single-source CGV models. It excels in challenging imaging scenarios characterized by intensity inhomogeneity, noise, and complex backgrounds, accurately delineating object boundaries and advancing the state of image segmentation. [ABSTRACT FROM AUTHOR]

We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three‐dimensional Euclidean space Cl(3,0)$$ Cl\left(3,0\right) $$. We apply a new form of dimensionally minimal embedding of octonions in geometric algebra that expresses octonion multiplication nonassociativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas. [ABSTRACT FROM AUTHOR]

Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2 n Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals. [ABSTRACT FROM AUTHOR]

Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion. [ABSTRACT FROM AUTHOR]

Reducing the complexity of the pipeline of instance segmentation is crucial for real-world applications. This work addresses this issue by introducing an anchor-box free and single-shot instance segmentation framework, termed PolarMask, which reformulates the instance segmentation problem as predicting the contours of objects in the polar coordinate, with several appealing benefits. (1) The polar representation unifies instance segmentation (masks) and object detection (bounding boxes) into a single framework, reducing the design and computational complexity. (2) Two modules are carefully designed (i.e., soft polar centerness and polar IoU loss) to sample high-quality center examples and optimize polar contour regression, making the performance of PolarMask does not depend on the bounding box prediction results and thus becomes more efficient in training. (3) PolarMask is fully convolutional and can be easily embedded into most off-the-shelf detection methods. To further improve the accuracy of the framework, a Refined Feature Pyramid is introduced to further improve the feature representation at different scales, termed PolarMask++. Extensive experiments demonstrate the effectiveness of both PolarMask and PolarMask++, which achieve competitive results on instance segmentation in the challenging COCO dataset with single-model and single-scale training and testing, as well as new state-of-the-art results on rotate text detection and cell segmentation. We hope the proposed polar representation can provide a new perspective for designing algorithms to solve single-shot instance segmentation. The codes and models are available at: github.com/xieenze/PolarMask . [ABSTRACT FROM AUTHOR]

[Display omitted] • A novel polar representation is introduced for video object segmentation. • An asymmetric Siamese network is developed using similar backbones with different depths. • We design the repeated cross correlation and semi-FPN based on peeling convolutions. Video object segmentation (VOS) is an essential part of autonomous vehicle navigation. Besides the accuracy metric, the real-time speed is very important for the algorithms of autonomous vehicles. In this paper, we propose a semi-supervised real-time method based on the Siamese network using a new polar representation. The input of bounding boxes are initialized rather than the object masks, which are applied to the video object detection tasks. The polar representation could reduce the parameters for encoding masks with subtle accuracy loss, so that the algorithm speed can be improved significantly. An asymmetric siamese network is also developed to extract the features from different spatial scales. Moreover, the peeling convolution is proposed to reduce the antagonism among the branches of the polar head. The repeated cross-correlation and semi-FPN are designed based on this idea. The experimental results on the DAVIS-2016 dataset and other public datasets demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in sl 2 corresponds to the KdV hierarchy, and the triple (f , 0 , e θ) , where f is the sum of negative simple root vectors and e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy. [ABSTRACT FROM AUTHOR]

Oriented object detection is a fundamental and challenging task in remote sensing image analysis that has recently drawn much attention. Currently, mainstream oriented object detectors are based on densely placed predefined anchors. However, the high number of anchors aggravates the positive and negative sample imbalance problem, which may lead to duplicate detections or missed detections. To address the problem, this paper proposes a novel anchor-free two-stage oriented object detector. We propose the Anchor-Free Oriented Region Proposal Network (AFO-RPN) to generate high-quality oriented proposals without enormous predefined anchors. To deal with rotation problems, we also propose a new representation of an oriented box based on a polar coordinate system. To solve the severe appearance ambiguity problems faced by anchor-free methods, we use a Criss-Cross Attention Feature Pyramid Network (CCA-FPN) to exploit the contextual information of each pixel and its neighbors in order to enhance the feature representation. Extensive experiments on three public remote sensing benchmarks—DOTA, DIOR-R, and HRSC2016—demonstrate that our method can achieve very promising detection performance, with a mean average precision (mAP) of 80.68%, 67.15%, and 90.45%, respectively, on the benchmarks.

We consider plane curves of the form (ax)α + (by)α = rα defined on the first quadrant of R², where α > 0 and a, b, r > 0. We summarize the outlines of them by using elementary differential calculus. We will in this note understand that they are classified into three types of curves, convex, straight and concave, depending on α. [ABSTRACT FROM AUTHOR]

The paper presents a comprehensive investigation of fatigue cracking behaviors under various nonproportional multiaxial cycle loading paths based on the critical plane approach. The maximum normal and shear stress/strain fields are presented to analyze the crack orientation and early growth directions in polar diagrams. The experimental observed crack paths and directions were compared with maximum strain loci of each angle to determine multiaxial fatigue failure mode. The results show that crack orientation and growth paths appear in the maximum shear and normal strain plane, respectively. Likelihood cracking regions of various loading paths are predicted according the determined failure mode. Besides, nonproportionality factor is introduced to characterize the degree of multiaxiality on these loading paths. [ABSTRACT FROM AUTHOR]

Stress and strain analyses on eight materials tested under multiaxial stress- and strain-controlled conditions are performed. Stress and/or strain fields are generated using plane stress-strain transformation. Maximum stress and strain loci are plotted in polar diagrams. For low cycle fatigue tests, normal stress and shear strain are normalized with their yield counterparts to allow plotting them in one polar diagram. The roles of shear strain and normal stress in fatigue can be examined using such a representation. For high cycle fatigue, normal and shear stresses are plotted in a single polar diagram. Observed crack planes are superimposed on all polar diagrams. In general, observed cracks are found to be aligned with the plane of both maximum shear strain and normal stress. Polar representations of Fatemi-Socie fatigue damage for strain-controlled tests show that significant numbers of planes experience high damage value especially for 90 ° out-of-phase loading. [ABSTRACT FROM AUTHOR]

We study polar representations in the sense of Dadok and Kac which are symplectic. We show that such representations are coisotropic and use this fact to give a classification. We also study their moment maps and prove that they separate closed orbits. Our work can also be seen as a specialization of some of the results of Knop on multiplicity free symplectic representations to the polar case. [ABSTRACT FROM AUTHOR]

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T ), v(T )), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled. [ABSTRACT FROM AUTHOR]

The multiplicative or polar representation of hyperbolic scator algebra in 1 + n dimensions is introduced. The transformations between additive and multiplicative representations are presented. The addition and product operations are consistently defined in either representation using additive or multiplicative variables. The product is shown to produce a rotation and scaling for equal director components and solely a scaling in the orthogonal components. [ABSTRACT FROM AUTHOR]

The aim of this study is to assess the possibility to apply a new control approach dedicated to turbomachinery. The controller is fuzzy based using inputs expressed in polar coordinates. The advantage is that it manages two significant physical quantities, namely tangential and radial velocities that are related to steady state and transient behaviors, respectively. A synchronous filter is associated to the controller in order to enhance the ratio command force/bearing dynamic capacity. The approach was previously applied experimentally with success for the control of an academic test rig. It is adapted here for the control of an industrial compressor whose flexible rotor is supported by active magnetic bearings (AMB). At this stage, only numerical investigations are performed. The controller has to satisfy the standards and the end users requirements. In addition, it should be easy to implement. The behavior of the machine studied is assessed for several configurations of unbalances. A test that corresponds to usual industrial excitations (subsynchronous excitations at nominal speed) is also carried out. Results obtained are satisfactory and give insight into the potential of the approach. In addition, and as the fuzzy controller parameters are independent from the rotor design, the approach is a first step for the standardization of magnetic bearing controller synthesis. [ABSTRACT FROM AUTHOR]

We derive a point-wise estimate for a map u : Ω ⊂ ℝn → ℝm that minimizes JA(v) : ƒA 1/2|∇v|² + U(v) subjected to the Dirichlet condition v = u on ∂Ω for every open smooth and bounded set A ⊂ Ω. We discuss some consequences of this basic estimate. [ABSTRACT FROM AUTHOR]

This article presents an approach for improving the accuracy in the reliability and redundancy assessment of ship cross-sections by using the Bayesian updating method. The vertical bending moments associated with ultimate failure and first failure for a given ship cross-section are evaluated using an optimization-based method and the progressive collapse method, respectively. The prior information on the wave-induced load effects is calculated based on the linear theory. Having extracted the hogging and sagging peaks from the low-frequency structural health monitoring signals, the Bayesian method is used to update the Rayleigh-distributed prior load effects. The original and updated reliability and redundancy indexes of the ship cross-sections are evaluated, and the results are displayed in polar plots. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a framework for the assessment of structural safety of ships under different operational conditions by evaluating performance indicators such as reliability and redundancy. Flexural failure mode for several ship cross-sections is investigated. Reliability and redundancy are based upon the evaluation of the flexural capacities associated with the ultimate hull failure and the failure of the first stiffened panel within a selected cross-section. Furthermore, aging effects due to corrosion phenomena are investigated. Polar representations of reliability and redundancy are obtained for different encountered sea states, ship speeds and headings, covering a large spectrum of operational conditions. Load effects due to waves are computed based on strip theory, while flexural capacity is simulated using Latin Hypercube sampling. The proposed approach is applied to a Joint High Speed Sealift. [Copyright &y& Elsevier]

Abstract: The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions. [Copyright &y& Elsevier]

Oriented object detection is a fundamental and challenging task in remote sensing image analysis that has recently drawn much attention. Currently, mainstream oriented object detectors are based on densely placed predefined anchors. However, the high number of anchors aggravates the positive and negative sample imbalance problem, which may lead to duplicate detections or missed detections. To address the problem, this paper proposes a novel anchor-free two-stage oriented object detector. We propose the Anchor-Free Oriented Region Proposal Network (AFO-RPN) to generate high-quality oriented proposals without enormous predefined anchors. To deal with rotation problems, we also propose a new representation of an oriented box based on a polar coordinate system. To solve the severe appearance ambiguity problems faced by anchor-free methods, we use a Criss-Cross Attention Feature Pyramid Network (CCA-FPN) to exploit the contextual information of each pixel and its neighbors in order to enhance the feature representation. Extensive experiments on three public remote sensing benchmarks—DOTA, DIOR-R, and HRSC2016—demonstrate that our method can achieve very promising detection performance, with a mean average precision (mAP) of 80.68%, 67.15%, and 90.45%, respectively, on the benchmarks. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new family of navigation functions for robot navigation and obstacle avoidance. The method can be used for both path finding and real-time path planning. Each navigation function is composed of three parts: a proportionality term, a deviation function and a deviation constant. Deviation functions are time-varying functions satisfying certain conditions. These functions and parameters are updated in real-time to avoid collision with obstacles. Our strategy uses polar kinematics equations to model the navigation problem in terms of the range and direction between the robot and the goal. The obstacles are mapped to polar planes, and represented by the range and the direction from the robot or the final goal in polar coordinates. This representation gives a certain weight to the obstacles based on their relative position from the robot and facilitates the design of the navigation law. There exists an infinite number of navigation functions obtained by changing the proportionality constant, the deviation constant or the deviation function. This offers an infinite number of possibilities for the robot's path. Our navigation strategy is illustrated using an extensive simulation where different navigation parameters are used. [ABSTRACT FROM AUTHOR]

This paper addresses unsupervised change detection by proposing a proper framework for a formal definition and a theoretical study of the change vector analysis (CVA) technique. This framework, which is based on the representation of the CVA in polar coordinates, aims at: 1) introducing a set of formal definitions in the polar domain (which are linked to the properties of the data) for a better general description (and thus understanding) of the information present in spectral change vectors; 2) analyzing from a theoretical point of view the distributions of changed and unchanged pixels in the polar domain (also according to possible simplifying assumptions); 3) driving the implementation of proper preprocessing procedures to be applied to multitemporal images on the basis of the results of the theoretical study on the distributions; and 4) defining a solid background for the development of advanced and accurate automatic change-detection algorithms in the polar domain. The findings derived from the theoretical analysis on the statistical models of classes have been validated on real multispectral and multitemporal remote sensing images according to both qualitative and quantitative analyses. The results obtained confirm the interest of the proposed framework and the validity of the related theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper the authors propose some new kinds of isotropic laminates, made with identical anisotropic layers. In particular, these laminates satisfy some conditions which generalise the well-known Werren and Norris rules, in order to obtain fully isotropy, that is, isotropy of the three tensors A, B and D. To this purpose, the authors utilise some results found in a preceding research, namely the so-called quasi-trivial solutions. The way to form particular isotropic laminates that do not follow the Werren and Norris rule is also indicated. The paper ends with some numerical examples which illustrate the theoretical results found. [Copyright &y& Elsevier]

Image segmentation is a major area of interest within the field of medical image analysis and processing. In the last few decades, cell nucleus segmentation in non-small lung cancer histopathological images has spawned considerable critical attention. However, most of the previously presented studies have only been concerned with revealing the representation of contours in Cartesian coordinates and suffer from a lack of clarity in connecting the points into a whole contour. Bearing this in mind, we propose a polar representation-based nucleus segmentation model by leveraging the fully convolutional one-stage object detection. In general, center classification and length regression are simultaneously adapted to yield the contour of the nucleus in a polar coordinate. To evaluate the performance of our approach, the comparing experiments are conducted between the state-of-the-art deep learning-based object detection algorithms and the proposed method on one manually collected dataset. Experimental results demonstrate that our model outperformed the state-of-the-art both in efficiency and effectiveness. • A polar representation-based deep learning model by training on collected LUND. • Both polar centeredness and polar IoU loss introduced in the training process. • Experimental results illustrate the effectiveness of the proposed solution over the state-of-the-arts. [ABSTRACT FROM AUTHOR]

In this Letter we study the behavior of the eigenvalues of an operator defined by the action associated to a generic quadratic time-dependent Hamiltonian. This is done using a polar representation of the solutions of the corresponding linear Hamiltonian system. A proof of the Morse index theorem is given. [ABSTRACT FROM AUTHOR]

We study an algebraic structure naturally associated to a standard imbedding of an $$\mathcal{R} $$ -space. This structure determines completely the geometry of an $$\mathcal{R} $$ -space and reduces to a Jordan Triple System if the $$\mathcal{R} $$ -space is symmetric. [ABSTRACT FROM AUTHOR]