Wavelet Moments Research Articles

Image and non-image parameters of atmospheric Cherenkov events: a comparative study of their γ-ray/hadron classification potential in ultrahigh energy regime

In this exploratory simulation study, we compare the event-progenitor classification potential of a variety of measurable parameters of atmospheric Cherenkov pulses which are produced by ultrahigh energy γ-ray and hadron progenitors and are likely to be recorded by the TACTIC (TeV atmospheric Cherenkov telescope with imaging camera) array of atmospheric Cherenkov telescopes. The parameters derived from Cherenkov images include Hillas, fractal and wavelet moments, while those obtained from non-image Cherenkov data consist of pulse profile rise time and base width and the relative ultraviolet to visible light content of the Cherenkov event. It is shown by a neural-net approach that these parameters, when used in suitable combinations, can bring about a proper segregation of the two event types, even with modest sized data samples of progenitor particles.

An exact method for computing the area moments of wavelet and spline curves

We present a method for the exact computation of the moments of a region bounded by a curve represented by a scaling function or wavelet basis. Using Green's theorem, we show that the computation of the area moments is equivalent to applying a suitable multidimensional filter on the coefficients of the curve and thereafter computing a scalar product. The multidimensional filter coefficients are precomputed exactly as the solution of a two-scale relation. To demonstrate the performance improvement of the new method, we compare it with existing methods such as pixel-based approaches and approximation of the region by a polygon. We also propose an alternate scheme when the scaling function is sinc(x).

A new algorithm for translating psycho-acoustic information to the wavelet domain

Among the characteristics of a filter bank to be used in subband audio coding, it is desirable that the filter bank and its inverting process maintain a high-frequency selectivity. This design goal is not always met when the audio coder is based on the wavelet transform, making the translation of psycho-acoustic information from the Fourier to the wavelet domain difficult. In this paper, we present a new method for translating the psycho-acoustic information to the wavelet domain, which can be applied to subband audio coders based on the orthonormal wavelet transform, when the subband decomposition approximates the frequency decomposition of sounds in the inner ear. A simple and improved psycho-acoustic model is also described. Both the psycho-acoustic model and the translation algorithm have been included in a wavelet-packet-based audio coder. This uses CD-quality signals and allows transparent coding with binary rates of about 1.5 bits/sample. The inverse relation between the number of vanishing moments of the mother wavelet and the minimum binary rate to ensure transparent coding is shown. A decrease in the number of vanishing moments increases the binary rate slightly, without loss to the subjective quality of the decoded audio signals.

Dispersion of time domain wavelet Galerkin method based on Daubechies' compactly supported scaling functions with three and four vanishing moments

The wavelet-Galerkin method for time-domain electromagnetic field modeling based on Daubechies' compactly supported wavelets proposed by Cheong et al. (see ibid., vol. 9, no. 8, p. 297-299, 1999) has been extended to the use of the scaling functions with three and four vanishing wavelet moments together with the approximate shifted interpolation property. The numerical dispersion properties of the methods are precisely investigated and compared with those of other wavelet-based and finite-difference methods. It was found that Daubechies' scaling functions with larger number of vanishing moments generally give higher accuracy while maintaining the comparable computational expenditure.

On a choice of wavelet bases in the wavelet transform approach

The Daubechies orthogonal wavelet (DOW) is compared with the nonorthogonal cardinal spline wavelet (NCSW) in the wavelet transform approach and it is shown that the DOW is better than the NCSW in view of the computation cost. First, the computation cost required for the wavelet transform based on the DOW is less than that based on the NCSW because the DOW has smaller support provided the same number of vanishing moments of wavelets is used. Second, in contrast with the fact that the wavelet transform based on the DOW does not affect the condition number of the impedance matrix, that, based on the NCSW, has an effect to make it very large. As a result, even though the NCSW results in a sparser impedance matrix, it requires more computation cost for solving the resultant matrix equation in comparison with the DOW because the cost depends not only on the sparsity, but also on the condition number of the matrix.

First results on characterization of Cherenkov images through combined use of Hillas, fractal and wavelet parameters

Based on Monte Carlo simulations using the CORSIKA code, it is shown that Cherenkov images produced by ultrahigh energy γ-rays and cosmic ray nuclei (proton, Neon and Iron) are fractal in nature. The resulting multifractal and wavelet moments when employed in association with the conventional Hillas parameters as inputs to a properly-trained artificial neural network are found to provide a more efficient primary characterization scheme than the one based on the use of Hillas or fractal parameters alone.

Discriminative wavelet shape descriptors for recognition of 2-D patterns

In this paper, we present a set of wavelet moment invariants, together with a discriminative feature selection method, for the classification of seemingly similar objects with subtle differences. These invariant features are selected automatically based on the discrimination measures defined for the invariant features. Using a minimum-distance classifier, our wavelet moment invariants achieved the highest classification rate for all four different sets tested, compared with Zernike’s moment invariants and Li’s moment invariants. For a test set consisting of 26 upper cased English letters, wavelet moment invariants could obtain 100% classification rate when applied to 26×30 randomly generated noisy and scaled letters, whereas Zernike’s moment invariants and Li’s moment invariants obtained only 98.7 and 75.3%, respectively. The theoretical and experimental analyses in this paper prove that the proposed method has the ability to classify many types of image objects, and is particularly suitable for classifying seemingly similar objects with subtle differences.

Wavelet-Galerkin scheme of time-dependent inhomogeneous electromagnetic problems

A wavelet-Galerkin scheme based on the time-dependent Maxwell's equations is presented. Daubechies' wavelet with two vanishing wavelet moments is expanded for basis function in spatial domain, and Yee's leap-frog approach is applied. The shifted interpolation property of Daubechies' wavelet family leads to the simplified formulations for inhomogeneous media without the additional matrices for the integral or material operator. The storage effectiveness, execution time reduction, and accuracy of this scheme are demonstrated by calculating the resonant frequencies of the homogeneous and inhomogeneous cavities.

Coiflet systems and zero moments

The Coifman wavelets created by Daubechies (1992) have more zero moments than imposed by specifications. This results in systems with approximately equal numbers of zero scaling function and wavelet moments and gives a partitioning of the systems into three well-defined classes. The nonunique solutions are more complex than for Daubechies wavelets.

High Accuracy Reconstruction from Wavelet Coefficients

The accuracy of the wavelet approximation at resolution h= 2 −n to a smooth function fis limited by O( h N ), where Nis the number of vanishing moments of the mother wavelet ψ. For any positive integer p, we derive a new approximation formula which allows us to recover a smooth ffrom its wavelet coefficients with accuracy O( h p ). Related formulas for recovering derivatives of fare also given.

Simulated annealing wavelet estimation via fourth‐order cumulant matching

The fourth‐order cumulant matching method has been developed recently for estimating a mixed‐phase wavelet from a convolutional process. Matching between the trace cumulant and the wavelet moment is done in a minimum mean‐squared error sense under the assumption of a non‐Gaussian, stationary, and statistically independent reflectivity series. This leads to a highly nonlinear optimization problem, usually solved by techniques that require a certain degree of linearization, and that invariably converge to the minimum closest to the initial model. Alternatively, we propose a hybrid strategy that makes use of a simulated annealing algorithm to provide reliability of the numerical solutions by reducing the risk of being trapped in local minima. Beyond the numerical aspect, the reliability of the derived wavelets depends strongly on the amount of data available. However, by using a multidimensional taper to smooth the trace cumulant, we show that the method can be used even in a trace‐by‐trace implementation, which is very important from the point of view of stationarity and consistency. We demonstrate the viability of the method under several reflectivity models. Finally, we illustrate the hybrid strategy using marine and field real data examples. The consistency of the results is very encouraging because the improved cumulant matching strategy we describe can be effectively used with a limited amount of data.

Rank M Wavelets with N Vanishing Moments

This work generalizes the rank 2 (scale factor of 2) orthogonal wavelet sequences of Daubechies to the case of a rank $M$ wavelet matrix. Several equivalent definitions of $N$th order vanishing moments for rank $M$ wavelets are developed. These notions are used to find an explicit formula for rank $M$ wavelet scaling sequences with $N$ vanishing wavelet moments (of degree $N$ in our terminology). A full wavelet matrix (scaling sequence and $M-1$ wavelet sequences) is constructed, with explicit examples.

Scaling functions and wavelets with vanishing moments

The authors investigate conditions equivalent to the vanishing of moments of wavelets in a multiresolution analysis.