Geometric invariance of mass-like asymptotic invariants

B. Michel

doi:10.1063/1.3579137

Invariant Image Representation Using Novel Fractional-Order Polar Harmonic Fourier Moments

Sensors ◽

10.3390/s21041544 ◽

2021 ◽

Vol 21 (4) ◽

pp. 1544

Author(s):

Chunpeng Wang ◽

Hongling Gao ◽

Meihong Yang ◽

Jian Li ◽

Bin Ma ◽

...

Keyword(s):

Image Reconstruction ◽

Fractional Order ◽

Continuous Functions ◽

Kernel Functions ◽

Superior Performance ◽

Image Description ◽

Orthogonal Moments ◽

Integer Order ◽

Geometric Invariance ◽

Order Continuous

Continuous orthogonal moments, for which continuous functions are used as kernel functions, are invariant to rotation and scaling, and they have been greatly developed over the recent years. Among continuous orthogonal moments, polar harmonic Fourier moments (PHFMs) have superior performance and strong image description ability. In order to improve the performance of PHFMs in noise resistance and image reconstruction, PHFMs, which can only take integer numbers, are extended to fractional-order polar harmonic Fourier moments (FrPHFMs) in this paper. Firstly, the radial polynomials of integer-order PHFMs are modified to obtain fractional-order radial polynomials, and FrPHFMs are constructed based on the fractional-order radial polynomials; subsequently, the strong reconstruction ability, orthogonality, and geometric invariance of the proposed FrPHFMs are proven; and, finally, the performance of the proposed FrPHFMs is compared with that of integer-order PHFMs, fractional-order radial harmonic Fourier moments (FrRHFMs), fractional-order polar harmonic transforms (FrPHTs), and fractional-order Zernike moments (FrZMs). The experimental results show that the FrPHFMs constructed in this paper are superior to integer-order PHFMs and other fractional-order continuous orthogonal moments in terms of performance in image reconstruction and object recognition, as well as that the proposed FrPHFMs have strong image description ability and good stability.

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Cones and asymptotic invariants of multigraded systems of ideals

Journal of Algebra ◽

10.1016/j.jalgebra.2005.03.009 ◽

2008 ◽

Vol 319 (5) ◽

pp. 1851-1869

Author(s):

A. Wolfe

Keyword(s):

Asymptotic Invariants

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The theory of conditional invariance. I

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1968.0124 ◽

1968 ◽

Vol 305 (1482) ◽

pp. 405-427 ◽

Cited By ~ 1

Keyword(s):

Bound States ◽

General Relation ◽

Variable Parameter ◽

Geometric Invariance ◽

Hermitian Conjugate ◽

New Type ◽

Definition Of ◽

Conditional Invariance ◽

Conditional Constant ◽

The Continuum

In order to extend the use of group theoretical arguments to the problem of accidental degeneracy in quantum mechanics, a new type of constant of the motion, known as a conditional constant of the motion, is introduced. Such a quantity, instead of commuting with the Hamiltonian H for the system, satisfies the more general relation H A = A † H , where A † denotes the hermitian conjugate (adjoint) of the conditional constant of the motion A . This expression reduces, if A is hermitian, to the usual definition of a constant of the motion. Otherwise it defines a new type of invariance, and it is this which will be referred to as conditional invariance. A discussion of the difficulties arising from the lack of hermiticity of A , which is of course essential to its definition, is given. In particular it is shown, under fairly general conditions, that the process of introducing a variable parameter in the Hamiltonian enabling it to have simultaneous eigenfunctions with A , gives rise to an eigenvalue equation in this parameter with respect to which A may be chosen to be hermitian. Conditional invariance is contrasted with both dynamical and geometric invariance. It is found to be sometimes replaceable by either of the latter forms of invariance and for such, explicit conditions are given. Some applications of conditional invariance are discussed. These include a study of the crossing of potential energy curves, a new model of symmetry breaking, a possible means of calculating the exact number of bound states for certain potentials and conditions for the existence of bound states near to the continuum.

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