International Journal of Wavelets, Multiresolution and Information Processing

The Heisenberg uncertainty principle and the uncertainty principle for self-adjoint operators have been known and applied for decades. In this paper, in the framework of Clifford algebra, we establish a stronger Heisenberg–Pauli–Wely type uncertainty principle for the Fourier transform of multivector-valued functions, which generalizes the recent results about uncertainty principles of Clifford–Fourier transform. At the end, we consider another stronger uncertainty principle for the Dunkl transform of multivector-valued functions.

We propose in this paper to study the multifractal properties of a special case of biological series, such as the transmembrane proteins. We prove that such series has a multifractal structure allowing their modeling by means of multifractal models issued from wavelet bases to be efficient. We apply the developed method to a numerical example in order to show its efficiency.

This paper discusses the optimal control problem for a class of linear continuous-time stochastic systems with time-delay based on the preview control method. In the design procedure, the time-delay terms are formally eliminated and the augmented error system method is used. The optimal controller of the time-delay stochastic system is obtained based on dynamic programming principle. In the previous approaches, the preview control method is applied to continuous-time deterministic system. Consequently, compared with the existing results, the main contributions of this paper are: (1) the conclusion of the non-singular transformation is improved and (2) the conclusion in deterministic system is extended by using the methods of designing an assistant system and introducing an integrator when building the stochastic augmented error system. The optimal tracking of the stochastic system with time-delay is realized by solving the optimal regulation problem of the stochastic augmented error system. The simulation shows the effectiveness of the controller designed in this paper.

The main goal of this paper is to study continuous fractional wavelet transform (CFrWT) on generalized Sobolev space $B_{p,k}^{\xi ,\theta }(ℝ)$ and its approximation properties. Convergence of convolution for $L_{\psi }^{\theta }$ in $B_{p,k}^{\xi ,\theta }(ℝ)$ is also discussed.

This study aims to investigate the dynamic correlations (co-movement) in between energy commodities such as WTI Crude Oil (WOIL), Brent Crude Oil (BOIL), Heating Oil and Electricity prices. To achieve this goal, we employed partial wavelet coherence (PWC) and multiple wavelet coherence (MWC). Wavelet analysis constitutes the core of these methodologies and MWC is essential to determine the dynamic correlation (co-movement) of time intervals and scales between the time series. We have developed a software program to compute PWC and MWC for quadruple data set. Coherent time intervals of the time series are determined. Vector ARMA models are shown to give a good fit due to having low mean squared errors compared to the univariate case. This allowed us to have better forecast performance.

This study aimed to determine the best constants in the Wirtinger inequality

Boosting, one of the best off-the-shelf classification methods, has evoked widespread interest in machine learning and statistics. However, the original algorithm was developed for binary classification problems. In this paper, we study multi-class boosting algorithms under the ${\ell }_2$-loss framework, and devise two multi-class ${\ell }_2$-Boost algorithms. These are based on coordinate descent and gradient descent to minimize the multi-class ${\ell }_2$-loss function. We derive a scoring coding scheme using optimal scoring constraints to encode class labels and a simple decoder to recover the true class labels. Our boosting algorithms are easily implemented and their results converge to the global optimum. Experiments with synthetic and real-world datasets show that, compared with several state-of-art methods, our algorithms provide more accurate results.

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.

This study focuses on the multi-scale turbulent structures of triangle cylinder wake flow measured by high-speed particle image velocimetry (PIV). The measured turbulent structures were spatially extracted into large-, intermediate-, relatively small-, and small-scale components using two-dimensional orthogonal wavelet analysis. Then the features of multi-scale turbulent structures were analyzed in terms of two-point auto correlation, power spectra, turbulent kinetic energy and Reynolds shear stress. It is found that the large-scale structure attributed to coherent motions in the wake flow makes largest contribution to the total fluctuating energy and maximum Reynolds shear stress, accounting for 84% and 92% respectively. The intermediate scale structures, which are unidentifiable by PIV measurement, are found to shed into the separated shear layer due to Kelvin–Helmholtz instability, and the central frequency of intermediate scale vortex is approximately twice of Kármán vortex shedding frequency. Besides, two symmetric regions of intensive velocity fluctuations having relatively small-scale are observed near the cylinder surface and the boundary of separation region, and they are not dominant.

A novel energy functional based on the Mumford–Shah model is established for performing automatic image segmentation. And in order to optimize the global model using graph-based methods, we develop a localized formula. Then, we propose a merging predicate for determining whether an edge connecting two neighboring pixels or regions merge. The dynamic graph merging (DGM) method is applied based on this merging predicate. That is, those edges with large energy merge and the edges with low energy are remained, such that the energy functional is minimized. Compared with other graph-based segmentation methods, our algorithm based on DGM has an important characteristic which is its ability to produce good segmentation on some complex texture images. Another characteristic is that this segmentation algorithm can avoid the “shrinking bias” problem. We also apply DGM to interactive image segmentation and find the results to be encouraging too.

Multi-agent reinforcement learning (MARL) provides a useful and flexible framework for multi-agent coordination in uncertain dynamic environments. However, the generalization ability and scalability of algorithms to large problem sizes, already problematic in single-agent RL, is an even more formidable obstacle in MARL applications. In this paper, a new MARL method based on ordinal action selection and approximate policy iteration called OAPI (Ordinal Approximate Policy Iteration), is presented to address the scalability issue of MARL algorithms in common-interest Markov Games. In OAPI, an ordinal action selection and learning strategy is integrated with distributed approximate policy iteration not only to simplify the policy space and eliminate the conflicts in multi-agent coordination, but also to realize the approximation of near-optimal policies for Markov Games with large state spaces. Based on the simplified policy space using ordinal action selection, the OAPI algorithm implements distributed approximate policy iteration utilizing online least-squares policy iteration (LSPI). This resulted in multi-agent coordination with good convergence properties with reduced computational complexity. The simulation results of a coordinated multi-robot navigation task illustrate the feasibility and effectiveness of the proposed approach.

In this paper, we consider estimation of a power spectrum from noise corrupted spectrum samples on uniform grids of frequencies with missing values. We propose two schemes based on the regularized nuclear norm minimization in combination with a recent subspace identification algorithm. The proposed schemes estimate the model order and the missing spectrum values in one step and are robust to large amplitude noise over short data records. Although this estimation problem can be cast as a spectrum estimation problem from nonuniformly spaced measurements and the algorithms developed for this type of data can be used, the identification example of this paper shows that the incomplete data formulation yields more accurate results. The properties of one of the proposed schemes are illustrated in an application example concerned with low-pass modeling of transformer current.

In digital signal and image processing one can only process discrete signals of finite length, and the space ${ℂ}^L$ is the preferred setting. Recently, Kutyniok and Strohmer constructed orthonormal Wilson bases for ${ℂ}^L$ with general lattices of volume $\frac{L}{2}$ ($L$ even). In this paper, we extend this construction to Wilson frames for ${ℂ}^L$ with general lattices of volume $\frac{L}{2K}$, where $K\in \mathbb{N}$ and $L\in 2K\mathbb{N}$. We obtain a necessary and sufficient condition for two sequences having Wilson structure to be dual frames for ${ℂ}^L$. When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for ${ℂ}^L$, show that a Wilson frame for ${ℂ}^L$ can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

This paper has three main contributions. The first is the construction of wavelet transforms from B-spline scaling functions defined on a grid of non-equispaced knots. The new construction extends the equispaced, biorthogonal, compactly supported Cohen–Daubechies–Feauveau wavelets. The new construction is based on the factorization of wavelet transforms into lifting steps. The second and third contributions are new insights on how to use these and other wavelets in statistical applications. The second contribution is related to the bias of a wavelet representation. It is investigated how the fine scaling coefficients should be derived from the observations. In the context of equispaced data, it is common practice to simply take the observations as fine scale coefficients. It is argued in this paper that this is not acceptable for non-interpolating wavelets on non-equidistant data. Finally, the third contribution is the study of the variance in a non-orthogonal wavelet transform in a new framework, replacing the numerical condition as a measure for non-orthogonality. By controlling the variances of the reconstruction from the wavelet coefficients, the new framework allows us to design wavelet transforms on irregular point sets with a focus on their use for smoothing or other applications in statistics.