Optimization in the construction of cardinal and symmetric wavelets on the line

2021 ◽  
Author(s):  
Neil D. Dizon ◽  
Jeffrey A. Hogan ◽  
Joseph D. Lakey
Keyword(s):  
Objective Function ◽  
Minimization Problem ◽  
Compact Support ◽  
Scaling Function ◽  
Scaling Functions ◽  
Optimization Approach ◽  
Zak Transform ◽  
Quadrature Mirror Filter ◽  
Measure Of Symmetry ◽  
The Scaling Function

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

Scaling functions for the finite-size effect in fractal aggregates

1994 ◽  
Vol 27 (5) ◽  
pp. 782-790 ◽  
Author(s):  
Y. Zang ◽  
S. Meriani
Keyword(s):  
Size Effect ◽  
Shape Function ◽  
Scaling Function ◽  
Finite Size ◽  
Finite Size Effect ◽  
Scaling Functions ◽  
Fractal Aggregates ◽  
Scattering Intensity ◽  
The Scaling Function ◽  
Exact Scaling

An exact scaling function for the finite-sized fractal aggregates sharply bounded by a sphere of radius R has been established by using the convolution square of the shape function of aggregates and the inhomogeneity function, which is introduced to take into account the presence of inhomogeneity in fractal aggregates. The scaling function for an inhomogeneous aggregate is mainly determined by the geometric shape of the aggregate but is also dependent upon the degree of inhomogeneity present in the aggregate. The differences between the scaling function reported in this paper and the commonly used ones, exp (−r/ξ) and exp [−(r/ξ)2], are discussed. The simulating calculations have shown that the use of different scaling functions will not only influence the cross-over behavior between the Guinier regime and the fractal regime, but also make the low-q scattering intensity converge to different values.

FOURIER SUPPORTS OF SCALING FUNCTIONS DETERMINE CARDINALITIES OF WAVELETS

2012 ◽  
Vol 10 (02) ◽  
pp. 1250011
Author(s):  
ZHIHUA ZHANG
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Scaling Function ◽  
Scaling Functions ◽  
The Fourier Transform ◽  
The Scaling Function

It is well-known that the different kinds of multiresolution analysis (MRA) structures generate different wavelets. In this paper, we give two uniform formulas on the number of mother functions for various wavelets associated with MRA structures. These formulas show that the number of mother functions of wavelets is determined by the support of the Fourier transform of the scaling function in MRA structure.

scholarly journals An Efficient Method for Convex Constrained Rank Minimization Problems Based on DC Programming

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wanping Yang ◽  
Jinkai Zhao ◽  
Fengmin Xu
Keyword(s):  
Objective Function ◽  
Minimization Problem ◽  
Dc Programming ◽  
Rank Minimization ◽  
Closed Form Solutions ◽  
Minimization Problems ◽  
Difference Of Convex Functions ◽  
Convex Objective Function ◽  
Difference Of Convex ◽  
Cut Problems

The constrained rank minimization problem has various applications in many fields including machine learning, control, and signal processing. In this paper, we consider the convex constrained rank minimization problem. By introducing a new variable and penalizing an equality constraint to objective function, we reformulate the convex objective function with a rank constraint as a difference of convex functions based on the closed-form solutions, which can be reformulated as DC programming. A stepwise linear approximative algorithm is provided for solving the reformulated model. The performance of our method is tested by applying it to affine rank minimization problems and max-cut problems. Numerical results demonstrate that the method is effective and of high recoverability and results on max-cut show that the method is feasible, which provides better lower bounds and lower rank solutions compared with improved approximation algorithm using semidefinite programming, and they are close to the results of the latest researches.

Assessing Model Uncertainty for the Scaling Function Inversion of Potential Fields

Geophysics ◽  
2021 ◽  
pp. 1-39
Author(s):  
Mahak Singh Chauhan ◽  
Ivano Pierri ◽  
Mrinal K. Sen ◽  
Maurizio FEDI
Keyword(s):  
Simulated Annealing Algorithm ◽  
Scaling Function ◽  
Vertical Section ◽  
Gravity Data ◽  
Geometric Model ◽  
Salt Dome ◽  
The Body ◽  
Gravity Inversion ◽  
Geometrical Parameters ◽  
The Scaling Function

We use the very fast simulated annealing algorithm to invert the scaling function along selected ridges, lying in a vertical section formed by upward continuing gravity data to a set of altitudes. The scaling function is formed by the ratio of the field derivative by the field itself and it is evaluated along the lines formed by the zeroes of the horizontal field derivative at a set of altitudes. We also use the same algorithm to invert gravity anomalies only at the measurement altitude. Our goal is analyzing the different models obtained through the two different inversions and evaluating the relative uncertainties. One main difference is that the scaling function inversion is independent on density and the unknowns are the geometrical parameters of the source. The gravity data are instead inverted for the source geometry and the density simultaneously. A priori information used for both the inversions is that the source has a known depth to the top. We examine the results over the synthetic examples of a salt dome structure generated by Talwani’s approach and real gravity datasets over the Mors salt dome and the Decorah (USA) basin. For all these cases, the scaling function inversion yielded models with a better sensitivity to specific features of the sources, such as the tilt of the body, and reduced uncertainty. We finally analyzed the density, which is one of the unknowns for the gravity inversion and it is estimated from the geometric model for the scaling function inversion. The histograms over the density estimated at many iterations show a very concentrated distribution for the scaling function, while the density contrast retrieved by the gravity inversion, according to the fundamental ambiguity density/volume, is widely dispersed, this making difficult to assess its best estimate.

THE CONSTRUCTION OF A CLASS OF TRIVARIATE NONSEPARABLE COMPACTLY SUPPORTED WAVELETS

2009 ◽  
Vol 07 (03) ◽  
pp. 255-267 ◽  
Author(s):  
YONGDONG HUANG ◽  
SHOUZHI YANG ◽  
ZHENGXING CHENG
Keyword(s):  
General Theory ◽  
Mild Condition ◽  
Scaling Function ◽  
Moment Condition ◽  
Compactly Supported ◽  
Vanishing Moment ◽  
The Scaling Function

In this paper, under a mild condition, the construction of compactly supported [Formula: see text]-wavelets is obtained. Wavelets inherit the symmetry of the corresponding scaling function and satisfy the vanishing moment condition originating in the symbols of the scaling function. An example is also given to demonstrate the general theory.

NEW AMY AND DELPHI MULTIPLICITY DATA AND THE LOGNORMAL DISTRIBUTION

1991 ◽  
Vol 06 (03) ◽  
pp. 245-257 ◽  
Author(s):  
R. SZWED ◽  
G. WROCHNA ◽  
A.K. WRÓBLEWSKI
Keyword(s):  
Energy Dependence ◽  
Lognormal Distribution ◽  
Scaling Function ◽  
Average Multiplicity ◽  
Linear Functions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
The Scaling Function ◽  
Skewness And Kurtosis

Multiplicity distributions for e+e−→ hadrons recently reported by the AMY and DELPHI collaborations are compared with the data obtained at lower energies. It is proven that the new data obey the KNO-G scaling and the scaling function can be described by the lognormal distribution. The dispersions are linear functions of the mean as for the data measured at lower energies and the standardized moments (such as skewness and kurtosis) are independent of the energy. The energy dependence of the average multiplicity is described by <nch>=β sα−1.

OFF THE CRITICAL POINT BY PERTURBATION (II): THE ISING MODEL SPIN-SPIN CORRELATOR IN A MAGNETIC FIELD

1990 ◽  
Vol 04 (05) ◽  
pp. 1039-1047 ◽  
Author(s):  
Vl. S. Dotsenko
Keyword(s):  
Magnetic Field ◽  
Correlation Function ◽  
Ising Model ◽  
Scaling Function ◽  
Spin Correlation ◽  
Regularization Technique ◽  
Spin Correlation Function ◽  
Analytic Regularization ◽  
Spin Correlator ◽  
The Scaling Function

An extension of the analytic regularization technique based on the conform 1 theory is suggested for the case of the spin-spin correlation function of the Ising model in a magnetic field, <σ0σR>h=F(t)/(R)1/4, t=hR15/8. Several first terms of the expansion of the scaling function F(t) are given.

Multisingularity and scaling in partial differential approximants. I

1988 ◽  
Vol 419 (1857) ◽  
pp. 181-203 ◽  
Keyword(s):  
Scaling Function ◽  
Linear Partial Differential Equation ◽  
Linear Combinations ◽  
Partial Differential ◽  
Polynomial Coefficients ◽  
Explicit Formulae ◽  
Expansion Coefficients ◽  
Scaling Fields ◽  
The Scaling Function ◽  
Asymptotic Scaling

A partial differential approximant (or PDA), F ( x, y ) approximates a function f ( x, y ), specified by its truncated power series, in terms of a solution of a defining linear partial differential equation with polynomial coefficients. The intrinsic multisingularities of a PDA, which may approximate corresponding singularities of f ( x, y ), are analysed formally and shown to obey, in general, asymptotic scaling (as familiar in the theory of critical phenomena), i. e. F ( x, y ) ≈ C | x͠ | - γ Z ( y͠ /| x͠ | ϕ ) + B , where x͠ and y͠ are linear combinations of the deviations, ( x - x c ) and ( y - y c ), from the multisingular point ( x c , y c ). Explicit formulae, suitable for numerical computation, are derived for the characteristic exponents, γ and ϕ , for the scaling function Z ( • ), for its expansion coefficients and for the related coefficients C and B , in the case when the crossover exponent, ϕ , lies in the interval (½, 2). (Part II extends these results to general values of ϕ , which requires the introduction of the nonlinear scaling fields associated with the PDA.)

Q-factor inversion using reconstructed source spectral consistency

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. R699-R710 ◽  
Author(s):  
Matan Shustak ◽  
Ariel Lellouch
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
Seismic Waves ◽  
Velocity Model ◽  
Physical Principle ◽  
Optimization Approach ◽  
Full Spectrum ◽  
Viscoelastic Wave Propagation ◽  
Attenuation Models ◽  
Basic Physical Principle

Seismic waves propagating in an anelastic medium undergo phase and amplitude distortions. Although these effects may be compensated for during imaging processes, a background [Formula: see text]-model is generally required for their successful application. We have developed a new approach to the [Formula: see text]-estimation problem, which is fundamentally related to the basic physical principle of time reversal. It is based on back-propagating recorded traces to their known source location using the reverse tomographic equation. This equation is a ray approximation of viscoelastic wave propagation. It is applied assuming a known and correct velocity model. We subsequently measure consistency between spectral shapes of traces that were back-propagated using the tomographic equation. We formulate an inverse problem using this consistency as an objective function. In conventional inversion, on the contrary, the discrepancy between modeled and recorded data, or some data characteristics, is minimized. The inverse problem is solved by ant-colony optimization, a global optimization approach, to avoid local minima present in the objective function. This method does not require knowledge of the source function and uses the full spectrum rather than its parametric reduction. Through synthetic and field cross-hole examples, we illustrate its accuracy and sensitivity in inverting for complex attenuation models. In the synthetic case, we also compare reconstructed source consistency with the conventional centroid frequency shift objective function. The latter displays poor resolution when recovering complex [Formula: see text] structures. We determine that the reconstructed source-consistency approach should be used as a part of an iterative workflow, possibly yielding initial models for a joint velocity and [Formula: see text] inversion.

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