scholarly journals On the asymptotic equivalence between the radon and the hough transforms of digital images

2019 ◽  
Author(s):  
Riccardo Aramini ◽  
Fabrice Delbary ◽  
Mauro C Beltrametti ◽  
Claudio Estatico ◽  
Michele Piana ◽  
...  
Keyword(s):  
Parameter Space ◽  
Asymptotic Equivalence ◽  
Reconstruction Problem ◽  
Dirac Delta ◽  
Family Of Curves ◽  
Hough Transforms ◽  
Discretization Step ◽  
Analytical Proof ◽  
Delta Functions ◽  
Mathematical Definitions

ABSTRACTAlthough characterized by different mathematical definitions, both the Radon and the Hough transforms ultimately take an image as input and provide, as output, functions defined on a preassigned parameter space, i.e., the so-called either Radon or Hough sinograms. The parameters in these two spaces describe a family of curves, which represent either the integration domains considered in the Radon transform, or the kind of curves to be detected by the Hough transform.It is heuristically known that the Hough sinogram converges to the corresponding Radon sinogram when the discretization step in the parameter space tends to zero. By considering generalized functions in multi-dimensional setting, in this paper we give an analytical proof of this heuristic rationale when the input grayscale digital image is described as a set of grayscale points, that is, as a sum of weighted Dirac delta functions. On these grounds, we also show that this asymptotic equivalence may have a valuable impact on the image reconstruction problem of inverting the Radon sinogram recorded by a medical imaging scanner.

Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

1969 ◽  
Vol 51 (6) ◽  
pp. 2359-2362 ◽  
Author(s):  
Kenneth G. Kay ◽  
H. David Todd ◽  
Harris J. Silverstone
Keyword(s):  
Dirac Delta ◽  
Delta Functions ◽  
Laplace Type

scholarly journals Electrostatic and magnetostatic fields of point dipoles revisited

2019 ◽  
Vol 65 (1) ◽  
pp. 71 ◽  
Author(s):  
Y. Muniz ◽  
Anderson José Fonseca ◽  
C. Farina
Keyword(s):  
Magnetic Dipole ◽  
Electric Dipole ◽  
Alternative Procedure ◽  
Dirac Delta ◽  
Delta Functions ◽  
Point Electric Dipole

After reviewing how the Dirac delta contributions to the electrostatic and magnetostatic fields of a point electric dipole and a point magnetic dipole are usually introduced, we present an alternative procedure for obtaining these terms based on a regularization prescription similar to that used in the computation of the transverse and longitudinal delta functions. We think this method may be useful for the students in other analogous calculations.

Review of basic quantum physics

Quantum 20/20 ◽  
2019 ◽  
pp. 1-20
Author(s):  
Ian R. Kenyon
Keyword(s):  
Quantum Physics ◽  
Particle Density ◽  
Electromagnetic Coupling ◽  
Black Body ◽  
Black Body Radiation ◽  
Gauge Transformations ◽  
Dirac Delta ◽  
Wave Particle Duality ◽  
Delta Functions ◽  
Lorentz Invariants

Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.

Variational Modal Identification of Conservative Nongyroscopic Systems

1989 ◽  
Vol 111 (2) ◽  
pp. 160-171 ◽  
Author(s):  
L. Silverberg ◽  
S. Kang
Keyword(s):  
Longitudinal Vibration ◽  
Modal Identification ◽  
System Response ◽  
Bending Vibration ◽  
Free System ◽  
Dirac Delta ◽  
Vibration Problem ◽  
Identification Method ◽  
Delta Functions ◽  
Admissible Functions

A new modal identification method for Conservative Nongyroscopic Systems is proposed. The modal identification method is formulated as a variational problem in which stationary values of a functional quotient are sought. The computation of the functional quotient is carried out using a set of admissible functions defined over the spatial domain of the system. Measurements of the free system response at discrete points are carried out using any combination of displacements, velocities, and/or accelerations. Three types of admissible functions have been considered—global functions, spatial Dirac-delta functions, and finite element interpolation functions. The variational modal identification method is applied to a pure bending vibration problem, to a pure longitudinal vibration problem, and to a combined bending and longitudinal vibration problem. The effectiveness of the variational modal identification method using different sets of admissible functions is examined.

Dirac delta functions

2015 ◽  
pp. 489-492
Author(s):  
J. M. Carpenter ◽  
C.-K. Loong ◽  
Marie-Louise Saboungi
Keyword(s):  
Dirac Delta ◽  
Delta Functions

Tutorial/Artical didactique: A high-accuracy mathematical and numerical method for Fourier transform, integral, derivatives, and polynomial splines of any order

1998 ◽  
Vol 76 (9) ◽  
pp. 659-677 ◽  
Author(s):  
N Beaudoin
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Computation Time ◽  
Polynomial Splines ◽  
Dirac Delta ◽  
Optimum Parameters ◽  
Complex Functions ◽  
Discrete Counterpart ◽  
Delta Functions ◽  
Fifth Order

From few simple and relatively well-known mathematical tools, an easily understandable, though powerful, method has been devised that gives many useful results about numerical functions. With mere Taylor expansions, Dirac delta functions and Fourier transform with its discrete counterpart, the DFT, we can obtain, from a digitized function, its integral between any limits, its Fourier transform without band limitations and its derivatives of any order. The same method intrinsically produces polynomial splines of any order and automatically generates the best possible end conditions. For a given digitized function, procedures to determine the optimum parameters of the method are presented. The way the method is structured makes it easy to estimate fairly accurately the error for any result obtained. Tests conducted on nontrivial numerical functions show that relative as well as absolute errors can be much smaller than 10-100, and there is no indication that even better results could not be obtained. The method works with real or complex functions as well; hence, it can be used for inverse Fourier transforms too. Implementing the method is an easy task, particularly if one uses symbolic mathematical software to establish the formulas. Once formulas are worked out, they can be efficiently implemented in a fast compiled program. The method is relatively fast; comparisons between computation time for fast Fourier transform and Fourier transform computed at different orders are presented. Accuracy increases exponentially while computation time increases quadratically with the order. So, as long as one can afford it, the trade-off is beneficial. As an example, for the fifth order, computation time is only ten times greater than that of the FFT while accuracy is 108 times better. Comparisons with other methods are presented.PACS Nos.: 02.00 and 02.60

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