Gradient Vector Field Research Articles

Extremal Area of Polygons, sliding along a circle

We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as ‘zigzag trains’.

An Infrared Small Target Detection Method Based on Gradient Correlation Measure

To overcome the interference of complex background and improve the detection ability of infrared small target under low signal-to-clutter ratio (SCR) scenes, a novel detection method based on gradient correlation measure (GCM) is proposed in this letter. Initially, the infrared gradient vector field (IGVF) of the original image is constructed based on the facet model. Then, a gradient correlation template is designed to distinguish the difference of local gradient between small targets and background. Finally, an adaptive threshold is adopted to extract small targets from background clutter. The proposed GCM method can identify the unique gradient characteristics of small targets. Experimental evaluations prove that the proposed method can achieve higher SCR scores in complex backgrounds. Especially in the scene where the gray contrast of small targets is low, the proposed GCM method shows a more robust detection performance.

Laplace-Beltrami Operator on the Orthogonal Group in Cartesian Coordinates

Using the embedded gradient vector field method (see P. Birtea, D. Comanescu, Hessian operators on constraint manifolds, J. Nonlinear Science 25, 2015), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of $n\times n$ matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions.

Geometric solitons in a D-homothetically deformed Kenmotsu manifold

We consider almost Riemann and almost Ricci solitons in a D-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure, and explicitly obtain the Ricci and scalar curvatures for some cases. We also provide a lower bound for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton.

Connected components of the space of proper gradient vector fields

We show that there exist two proper gradient vector fields on mathbb {R}^n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

Three dimensional contact metric manifolds with Cotton solitons

In this article we study a three dimensional contact metric manifold $M^3$ with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with $Q\xi = \rho\xi$, where $\rho$ is a smooth function on $M$ constant along Reeb vector field $\xi$ and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with $\xi$ or is a gradient vector field. Moreover, if $\rho$ is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: $SU(2)$ or $SO(3)$ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field $\xi$. Secondly, it is proved that a $(\kappa,\mu,\nu)$-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that $M$ is Sasakian, flat, a contact metric $(0,−4)$-space or a contact metric $(\kappa,0)$-space with $\kappa \lt 1$ and $\kappa\neq0$. For the potential vector field being orthogonal to $\xi$, if $\nu$ is constant we prove that $M$ is either Sasakian, or a $(\kappa,\mu)$-contact metric space.

On the automorphism group of the Morse complex

Let K be a finite, connected, abstract simplicial complex. The Morse complex of K, first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient vector fields on K. We show that if K is neither the boundary of the n-simplex nor a cycle, then Aut(M(K))≅Aut(K). In the case where K=Cn, a cycle of length n, we show that Aut(M(Cn))≅Aut(C2n). When K=∂Δn, we prove that Aut(M(∂Δn))≅Aut(∂Δn)×Z2. These results are based on recent work of Capitelli and Minian.

Modeling the gravitational field by using CFD techniques

A harmonic scalar field has a Laplacian (i.e., both source-free and curl-free) gradient vector field and vice versa. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, which is known as the complex-boundary value problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a Laplacian vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of the comet 67P/Churyumov–Gerasimenko which has an irregular shape. The method is validated in a closed-loop simulation by comparing the result with a direct integration of Newton’s formula. Both methods are consistent with a relative magnitude discrepancy at the percentage level and with a small directional difference root-mean-square value of 0.78^{circ }. Moreover, the Laplacian property of the potential flow’s velocity field is proved mathematically. From both theoretical and practical points of view, the new numerical method is able to overcome the divergence problem and, hence, has a good potential for solving CBVPs.

A note on the structure of prescribed gradient–like domains of non–integrable vector fields

Given a geometric structure on with (e.g., Euclidean, symplectic, Minkowski, and pseudo‐Euclidean), we analyze the set of points inside the domain of definition of an arbitrary given vector field, where the value of the vector field equals the value of the left/right gradient–like vector field of some fixed potential function, although a natural non‐integrability condition holds at each such a point. More precisely, we prove that if not empty, this is a Borel set which admits a cover consisting of at most regularly embedded m−dimensional submanifolds of , and consequently, its Hausdorff dimension is less or equal to m. Particular examples of gradient–like vector fields include the class of gradient vector fields with respect to Euclidean or pseudo‐Euclidean inner products, and the class of Hamiltonian vector fields associated to symplectic structures on . The main result of this article provides a geometric version of the similar result for the classical gradient vector field.

Effect analysis on performance enhancement of a hydrogen/air non-premixed micro combustor with sudden expansion and contraction structure

Improvement of combustion stability and efficiency is still the main challenge of micro combustors. In this work, a non-premixed combustion model of the hydrogen/air in a dimensionless micro combustor with sudden expansion and contraction structure at the entrance is established. The effects of different micro combustor diameters, inlet velocities and wall materials on non-premixed combustion performance enhancement of the hydrogen/air in the micro combustor is investigated. By comparing the temperature distribution and combustion efficiency of the micro combustor, the combustion characteristics of the hydrogen/air under different influence factors are obtained. The results show that the appropriate diameter of the micro combustor and the inlet velocity are of a great influence on the stable combustion. Furthermore, the field synergy analysis between temperature gradient and velocity vector fields under different inlet velocities of hydrogen/air is carried out, and an enhanced combustion method is proposed in this work. The calculation results show that the inlet velocity decrescence of the hydrogen can significantly improve combustion stability under constant inlet velocity of air condition, which will be useful for stabilizing non-premixed combustion in the micro combustor.

Existence of a complete holomorphic vector field via the Kähler–Einstein metric

In this paper, we study the existence of a complete holomorphic vector field on a strongly pseudoconvex complex manifold admitting a negatively curved complete Kahler–Einstein metric and a discrete sequence of automorphisms. Using the method of potential scaling, we will show that there is a potential function of the Kahler–Einstein metric whose differential has a constant length. Then, we will construct a complete holomorphic vector field from the gradient vector field of the potential function.

Remarks on Einstein Solitons with Certain Types of Vector Field

We consider almost Einstein solitons $(V,\lambda)$ in a Riemannian manifold when $V$ is a gradient, a solenoidal or a concircular vector field. We explicitly express the function $\lambda$ by means of the gradient vector field $V$ and illustrate the result with suitable examples. Moreover, we deduce some geometric properties when the Ricci curvature tensor of the manifold satisfies certain symmetry conditions.

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

Global exponential stabilization of a quadrotor by hybrid control

The main purpose of this paper is to introduce a hybrid controller for global attitude tracking of a quadrotor. This controller globally exponentially stabilizes the desired attitude, a task that is impossible to accomplish with memoryless discontinuous or continuous state feedback owing to topological obstruction. Thereafter, this paper presents a new centrally synergistic potential function to construct hybrid feedback that defeats the topological obstruction. This function induces a gradient vector field to globally asymptotically stabilize the reference attitude and produces the synergy gap to generate a switching control law. The proposed control structure is consisting of two major parts. In the first part, a synergetic controller is designed to cooperate with the hybrid controller, whereas it exponentially stabilizes the origin of the error dynamics. In the second part, a hybrid controller is introduced to globally stabilize the attitude of the quadrotor, where an average dwell constraint is considered with the switching control law to guarantee the exponential stability of the switched system. Finally, the effectiveness and superiority of the proposed control technique are validated by a comparative analysis in simulations.

Remarks on almost Riemann solitons with gradient or torse-forming vector field

We consider almost Riemann solitons \((V,\lambda )\) in a Riemannian manifold and underline their relation to almost Ricci solitons. When V is of gradient type, using Bochner formula, we explicitly express the function \(\lambda \) by means of the gradient vector field V and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.

Self-Adaptive Asymmetrical Artificial Potential Field Approach Dedicated to the Problem of Position Tracking by Nonholonomic UAVs in Windy Enivroments

Abstract Artificial potential fields (APFs) are a popular method of planning and controlling the path of robot movement, including unmanned aerial vehicles (UAVs). However, in the case of nonholonomic robots such as fixed-wing UAVs, the distribution of velocity vectors should be adapted to their limited manoeuvrability to ensure stable and precise position tracking. The previously proposed local asymmetrical potential field resolves this issue, but it is not effective in the case of windy environments, where the UAV is unable to maintain the desired position and drifts due to the wind drift effect. This is reflected in the growth of position error, which, similar to the steady-state error in the best case, is constant. To compensate for it, the asymmetrical potential field approach is modified by extending definitions of potential function gradient and velocity vector field (VVF) with elements based on the integral of position tracking error. In the case of wind drift, the value of this integral increases over time, and lengths and orientations of velocity vectors will also be changed. The work proves that redefining gradient and velocity vector as a function of position tracking error integrals allows for minimisation of the position tracking error caused by wind drift.

On uniform observability of gradient flows in the vanishing viscosity limit

We consider a transport equation by a gradient vector field with a small viscous perturbation -εΔ g . We study uniform observability (resp. controllability) properties in the (singular) vanishing viscosity limit ε→0 + , that is, the possibility of having a uniformly bounded observation constant (resp. control cost). We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation ε=0. We also prove that the two minimal times coincide for positive solutions. The proofs rely on a semiclassical reformulation of the problem together with (a) Agmon estimates concerning the decay of eigenfunctions in the classically forbidden region [HS84] (b) fine estimates of the kernel of the semiclassical heat equation [LY84].

Active Contour Directed by the Poisson Gradient Vector Field and Edge Tracking

This paper develops a new active contour (AC) model capable of multiple complex objects segmentation in the presence of heavy noise. The model segments images in the framework of two types of partial differential equations (PDEs): the Euler–Lagrange and Poisson PDEs. The former is used to build an evolution algorithm, while the Poisson solution gradient vector field (PGVF) directs the evolution toward the boundaries of all image objects. The AC halts on boundaries and PGVF separatrices, splits on the latter, and leaves at least one segment (called label) on every boundary. Each label tracks its boundary until the corresponding object is enveloped. The advantages of the new method are validated on a number of skin lesions, road, and aircraft images of varying sizes and in the presence of Gaussian noise. The obtained results are compared against results by contemporary and established active contours and neural networks.

Wasserstein Proximal Algorithms for the Schrödinger Bridge Problem: Density Control With Nonlinear Drift

In this article, we study the Schrödinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite-time horizon, subject to a controlled stochastic differential evolution of the state vector. As such, it can be seen as a stochastic optimal control problem in continuous time with endpoint density constraints—A topic that originated in the physics literature in 1930s, and in the recent years, has garnered burgeoning interest in the systems-control community. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the “Schrödinger factors”—so named since their product at any time yields the optimal controlled joint state PDF at that time. We show that if the drift is a gradient vector field, or is of mixed conservative–dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. Combined with a recently proposed fixed point recursion that is contractive in the Hilbert metric, this opens up the possibility to numerically solve the SBPs in these cases by computing the Schrödinger factors via a single IVP solver for the corresponding (uncontrolled) forward Kolmogorov PDE. The flows generated by such forward Kolmogorov PDEs, for the two aforementioned types of drift, in turn, enjoy gradient descent structures on the manifold of joint PDFs with respect to suitable distance functionals. We employ a proximal algorithm developed in our prior work that exploits this geometric viewpoint, to solve these IVPs and compute the Schrödinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples.

적외선 영상을 이용한 Gradient Vector Field 기반의 표적 및 화염 자동인식 연구

본 논문은 공중 혹은 해상배경에 표적과 화염이 동시에 존재할 때, 무인항공기에 장착된 EOTS (Electro-Optical Targeting System; 전자광학 추적장비)가 표적을 추적하기 위해 화염의 영향에 강건하도록 표적을 자동 인식하는 기법을 제안한다. 제안한 기법은 표적과 화염의 적외선 영상을 Gradient Vector Field로 변환하고, 각 Gradient magnitude를 Polynomial Curve Fitting 도구에 적용하여 다항식 계수를 추출 및 얕은 신경망 모델에 학습함으로써, 표적과 화염을 자동으로 인식한다. 확보한 표적 및 화염의 다양한 적외선 영상 DB를 학습데이터, 검증데이터, 시험데이터로 분류하여 제안한 기법의 표적 및 화염 자동 인식 성능을 확인하였다. 본 알고리듬을 활용하여 무인항공기의 자동비행 중 충돌회피, 산불탐지, 공중 및 해상의 목표물을 자동탐지 및 인식하는 분야에 적용될 수 있다.