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This paper concerns the modeling of smart health sensors for data monitoring, where the data are eventually noisy, with correlated noise. In this work, we applied Clifford-based wavelets/multiwavelets for correlated noise in multi-sensors health data monitoring by estimating the set of sensor nodes minimizing an eventual error computed by the signal values at the selected nodes mostly caused by the correlated noise. Instead of directly minimizing the estimation error, we focused on evaluating a multi-level scheme based on multiwavelets for the estimation of the error between the parameter vector and its sub-vector of those nodes. Numerical simulations are provided with a comparison to some recent existing works. This model showed a performance and a fast time execution compared to those existing works. This model exceeds these models by the non-necessity to assume a priory structure of the data. Wavelets are capable to detect, localize, and eliminate the noise, even correlated, efficiently via the independent uncorrelated multiwavelets’ components.

This paper deals with the theory of frames for vector-valued Weyl–Heisenberg wavelets (VVWHW). We derive frame and the corresponding frame bounds for VVWHW.

An algorithm is presented for constructing a pair of high approximation order biorthogonal multiscaling function with dilation factor a in terms of any given pair of biorthogonal multiscaling function. The special case that a = 2 is discussed. If the dilation factor a = 2, then a biorthogonal multiwavelet pair is constructed explicitly. Finally, examples are given.

Let $A$ be a dilation matrix, an $n\times n$ expansive matrix that maps ${\mathbb{Z}}^n$ into itself. Let $\mathrm{\Lambda }$ be a finite subset of ${\mathbb{Z}}^n,$ and for $k\in \mathrm{\Lambda }$ let $c_k$ be $r\times r$ complex matrices. The refinement equation corresponding to $A,{\mathbb{Z}}^n,\mathrm{\Lambda }$ and $c={\{c_k\}}_{k\in \mathrm{\Lambda }}$ is $f(x)={\sum }_{k\in \mathrm{\Lambda }}{c}_{k}f(Ax-k).$ A solution $f:{ℝ}^n\to {ℂ}^r,$ if one exists, is called a refinable vector function or a vector scaling function of multiplicity $r.$ This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the $p$-norm joint spectral radius of a finite set of finite matrices determined by the coefficients $c_k.$

Biomedical signal/image processing and analysis are always fascinating tasks in scientific researches, both theoretical and practical. One of the powerful tools in such topics is wavelet theory which has been proved to be challenging since its discovery. One of the best measures of the optimality of reconstruction of signals/images is the well-known Shannon’s entropy. In wavelet theory, this is very well known and researchers are familiar with it. In the present work, a step forward is proposed based on more general wavelet tools. New approach is proposed for the reconstruction of signals/images provided with multiwavelets Shannon-type entropy to evaluate the order/disorder of the reconstructed signals/images. Efficiency and accuracy of the approach is confirmed by a simulation study on several models such as ECG, EEG and DNA/Proteins’ signals.

We present a parallel and linear scaling implementation of the calculation of the electrostatic potential arising from an arbitrary charge distribution. Our approach is making use of the multi-resolution basis of multiwavelets. The potential is obtained as the direct solution of the Poisson equation in its Green's function integral form. In the multiwavelet basis, the formally non local integral operator decays rapidly to negligible values away from the main diagonal, yielding an effectively banded structure where the bandwidth is only dictated by the requested accuracy. This sparse operator structure has been exploited to achieve linear scaling and parallel algorithms. Parallelization has been achieved both through the shared memory (OpenMP) and the message passing interface (MPI) paradigm. Our implementation has been tested by computing the electrostatic potential of the electronic density of long-chain alkanes and diamond fragments showing (sub)linear scaling with the system size and efficent parallelization.

Multiwavelet packet is the extension of multiwavelet. Through the analysis and discussion of classical wavelet packet and multiwavelets, an improved algorithm is proposed for the choice of best multiwavelet packet, and some useful conclusions are drawn in the paper. In the end, the algorithm is applied in signal de-noising, and simulation results show that better effect can be obtained with best multiwavelet packet than with multiwavelets.

The lifting scheme provides a new ideal for constructing multiwavelets. Through the introduction and discussion of constructing wavelets and multiwavelets with lifting scheme. A new constructing multiwavelets method with lifting scheme based on equivalent scalar filter bank is proposed in the paper, and the example has proved its feasibility.

The construction method of a pair of biorthogonal interpolatory multiscaling functions is given. What is more, the corresponding biorthogonal interpolatory multiwavelets can also be constructed. Thereby, multiwavelets sampling theorem is established easily.