A Kind of Affine Weighted Moment Invariants

A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of local affine differential invariants and a framework of global integral, they can more effectively extract features of images and help to increase the number of low-order invariants and to decrease the calculating cost. The experimental results show that AWMIs have good stability and distinguishability and achieve better results in image retrieval than traditional moment invariants. An extension to 3D is straightforward

Three-Dimensional Exact Legendre Moment Invariants For Amphetamine-Type Stimulants Molecular Structure Representation

The abuse of amphetamine-type stimulants (ATS) drugs has become a global,harrowing social problem.The technical limitations of the current test kits to detect new brand of ATS drugs present a challenge to national law enforcement authorities and scientific staff of forensic laboratories.Meanwhile,new molecular imaging devices which allowed mankind to characterize the physical three-dimensional (3D) molecular structure have been recently introduced,and it can be used to remedy the limitations of existing drug test kits.Thus,a new type of 3D molecular structure representation technique,or molecular descriptors,should be developed to cater the 3D molecular structure acquired physically using these molecular imaging devices.One of the image processing methods to represent a 3D image is 3D moments and moment invariants. However,there are problems exhibited by the existing 3D moments and moment invariants.Therefore,it is necessary to propose a new 3D moment invariants which is free from these problems.This study compares various 3D moments and identified 3D Legendre moments as the best moments to construct 3D moment invariants,namely 3D exact Legendre moment invariants (3D ELMI),which is used to represent the 3D molecular structure of ATS drugs.Since the 3D molecular structure of ATS drugs dataset obtained using molecular imaging devices are currently unavailable,this study acquired the 3D molecular structure of ATS drugs data from United Nations Office of Drug and Crime (UNODC) and pihkal.info database instead.The proposed technique was compared to the existing 3D moment invariants and molecular descriptors techniques in terms of processing time,memory consumption,single instance invariance,intra- and inter-class variance,and classification accuracy.The comparative study conducted found that 3D ELMI performs better than the existing 3D moment invariants,such as 3D geometric moment invariants (3D GMI),3D Gaussian–Hermite moment invariants (3D GHMI),and 3D Zernike descriptors (3D ZD).The satisfactory performance of 3D ELMI is attributed to numerous factors,such as the quality of the 3D Legendre,exact computation of the 3D Legendre,and the novelty of the proposed invariants techniques.The proposed technique was also compared to existing 3D molecular descriptors,for example weighted holistic invariants molecular (WHIM),geometry,topology,and atom weights assembly (GETAWAY),radial distribution function (RDF),and 3D molecule representation of structure based on electron diffraction (3D-MoRSE) descriptors.Despite 3D ELMI is capable to overcome the limitations of existing 3D molecular descriptors which depends on 3D molecular structure model instead of physical molecular structure obtained from molecular imaging devices,the test reveals 3D ELMI is not as good as these techniques,primarily due to the substantial number of features produced by the proposed technique.Nevertheless,the promising applicability and the unique approach of the proposed technique to represent the 3D molecular structure of ATS drugs has been demonstrated and worth to receive further exploration in the future works

Poincar\'e polynomials for Abelian symplectic quotients of pure rr-qubits via wall-crossings

In this paper, we compute a recursive wall-crossing formula for the Poincar\'e polynomials and Euler characteristics of Abelian symplectic quotients of a complex projective manifold under a special effective action of a torus with non-trivial characters. An analogy can be made with the space of pure states of a composite quantum system containing $r$ quantum bits under action of the maximal torus of Local Unitary operations

Semitoric integrable systems on symplectic 4-manifolds

Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.Comment: To appear in Invent. Mat