International Journal of Wavelets, Multiresolution and Information Processing

A simple constructive proof of polynomial matrix spectral factorization theorem is presented in the rank-deficient case. It is then used to provide an elementary solution to the wavelets completion problem.

In this paper, we propose a construction of a new cubic spline-wavelet basis on the hypercube satisfying homogeneous Dirichlet boundary conditions. Wavelets have two vanishing moments. Stiffness matrices arising from discretization of elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and we show that these condition numbers are small. We present quantitative properties of the constructed basis and we provide a numerical example to show the efficiency of the Galerkin method using the constructed basis.

Near exact fusion Banach frames have been discussed with the help of examples. Further, it has been proved that if a Banach space has a fusion Banach frame then it has a normalized tight and exact fusion Banach frame. In the sequel, we consider block perturbation of fusion Banach frames and proved that a block perturbation of a fusion Banach frame is also a fusion Banach frame. Some stability results for fusion Banach frames have also been obtained. Finally, we give an application of near exact fusion Banach frame.

Phonocardiograms (PCGs), recording of heart sounds, have many advantages over traditional auscultation in that they may be replayed and analyzed for spectral and frequency information. PCG is not a widely used diagnostic tool as it could be. One of the major problems with PCG is noise corruption. Many sources of noise may pollute a PCG signal including lung and breath sounds, environmental noise and blood flow noises which are known as murmurs. These murmurs contain many information on heart hemodynamic which can be used particularly in detecting of heart valve diseases. An automated system for heart murmurs processing can be an important tool in diagnostic of heart diseases using a simple electronic stethoscope. However, the first step before developing any automated system is the segmentation of the PCG signals from which the murmurs can be separated. A robust segmentation algorithm must have a robust denoising technique, where, wavelet transform (WT) is among the ones which exhibits very high satisfactory results in such situations. However, the selection of level of decomposition and the mother wavelet are the major challenges. This paper proposes a novel approach for an automatic selection of mother wavelet and level of decomposition that can be used in heart sounds denoising. The obtained results on both simulative and real PCG signals showed that the proposed approach can successfully select the best level of decomposition with the best mother wavelet that can be used in extraction operation of main PCG sound components (S1 and S2) from various types of murmurs.

The system of totally interpolating wavelet frames is discussed in this paper, in which both the scaling function and one of wavelet functions are interpolating. It will be shown that corresponding filter banks possess the special structure, and the parametrization of filter banks is present. Moreover, we show that when considering tight frame systems with two generators, the Ron–Shen's continuous-linear-spline-based tight frame is the only one with totally interpolating property and symmetry. But in the dual frame context, more good examples of bi-frames with symmetric/antisymmetric property can be obtained and constructed, which in particular, include frames with the uniform symmetry.

Digital watermarking consists in embedding imperceptible information into a host signal. It has been proposed to solve problems as varied as the protection of the copyright, content authentication, fingerprinting and broadcast monitoring. This paper presents a new approach for audio watermarking using the QR factorization in wavelet domain. This approach is based on embedding a watermark binary image in the R matrices of low frequency blocks DWT coefficients of audio signal. In this algorithm, the watermark is embedded by applying a Quantization Index Modulation (QIM) process on the determined optimal sample for each matrix R. The watermark can be blindly extracted without the knowledge of the original audio signal. Experimental results show that the proposed audio watermarking scheme maintains high quality of the audio signal. Signal to Noise Ratio (SNR), Log Spectral Distortion (LSD) and Mean Opinion Score (MOS) are about 40 dB, 0.37 dB and 4.84, respectively. Moreover, the scheme is quite robust against common signal processing attacks such as noise addition, filtering and MP3 compression. In addition, this method ensures a secure extraction process by using a private key, making it suitable for secure applications such as copyright protection.