International Journal of Wavelets, Multiresolution and Information Processing

Uniform approximation of multivariate periodic functions by Haar polynomials is studied. A general sharp Jackson type inequality and its refinement for certain types of numbers $n$ are discussed. An interesting phenomenon appears for some numbers $n$: a sharp estimate is not unique.

We propose a secure interlaced algorithm for RGB image encryption associated with Arithmetic Random Matrix Affine Cipher (ARMAC) in wavelet packet domain. In this approach, the arrangement of ARMAC parameters along with correct de-interlacing is mandatory for correct decryption. Computer simulation with standard example and result (security analysis) is given to analyze the capability and robustness of the proposed approach and comparison with existing methods are drawn. We have designed a highly secure algorithm wherein an RGB image can be encrypted with varied computational cost depending on the user, enabling real-time image encryption. Moreover, the proposed approach can be used in transmission of images over unsecured network.

In this paper, we adapt the notion of a wavelet induced isomorphism of $[0,1)$ associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an ${\mathscr{H}}$-wavelet set, where ${\mathscr{H}}$ is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric ${\mathscr{H}}$-wavelet sets. Examples when ${\mathscr{H}}$ is the classical Hardy space are also considered.

Daubechies posed the following problem in Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992): “It is an open question whether there exist orthonormal wavelet bases (not necessarily associated with a multiresolution analysis), with good time-frequency localization, and with irrational $a$” (that is, for an arbitrary irrational dilation factor $a>1$, with appropriate wavelet function $\psi$ and constant $b>0$, whether can $\{a^{j/2}\psi (a^{j}t-bk):j,k\in \mathbb{Z}\}$ construct an orthonormal wavelet basis with good time-frequency localization?). Our answer is “Yes”. In this paper, we introduce a new type of orthonormal wavelet basis having an arbitrary real dilation factor greater than 1. This orthonormal wavelet basis requires an infinite number of wavelet shapes when its dilation factor is irrational.

In this paper, the concept of a family of local atoms in a Banach space is introduced by using a semi-inner product (s.i.p.). Then this concept is generalized to an atomic system for operators in Banach spaces. We also give some characterizations of atomic systems leading to new frames for operators. In addition, a reconstruction formula is obtained. The characterizations of atomic systems allow us to state some results for sampling theory in s.i.p reproducing kernel Banach spaces. Finally, we define the concept of frame operator for these kinds of frames in Banach spaces and then we establish a perturbation result in this framework.

We prove characterizations of dual multiwavelet frames for $L_2[0,1{)}^s$ constructed from a pair of ${\mathbb{Z}}^s$-periodic multirefinable generators. Based on the notion of the Mixed Fundamental sequence we derive Mixed Extension Principles dictating the construction of dual wavelet masks with respect to given refinement masks and we show that these Principles characterize dual framelets.

This paper studies the convergence rate and consistency of Empirical Risk Minimization algorithm, where the samples need not be independent and identically distributed (i.i.d.) but can come from uniformly ergodic Markov chain (u.e.M.c.). We firstly establish the generalization bounds of Empirical Risk Minimization algorithm with u.e.M.c. samples. Then we deduce that the Empirical Risk Minimization algorithm on the base of u.e.M.c. samples is consistent and owns a fast convergence rate.

This paper aims at adaptive decomposition of signals in terms of nonlinear Fourier atoms. Each nonlinear Fourier atom is analytic and mono-component. The algorithm is considered as an adaptive greedy procedure based on nonlinear Fourier atoms. The convergence results for the proposed algorithms show that it is suitable to approximate a signal by a linear combinations of nonlinear Fourier atoms. Experiments are presented to illustrate the proposed algorithm and theory.

Based on the relationship between the Fourier transform (FT) and linear canonical transform (LCT), a logarithmic uncertainty principle and Hausdorff–Young inequality in the LCT domains are derived. In order to construct the windowed linear canonical transform (WLCT), Gabor filters associated with the LCT is introduced. Using the basic connection between the classical windowed Fourier transform (WFT) and the WLCT, a new proof of inversion formula for the WLCT is provided. This relation allows us to derive Lieb’s uncertainty principle associated with the WLCT. Some useful properties of the WLCT such as bounded, shift, modulation, switching, orthogonality relation, and characterization of range are also investigated in detail. By the Heisenberg uncertainty principle for the LCT and the orthogonality relation property for the WLCT, the Heisenberg uncertainty principle for the WLCT is established. This uncertainty principle gives information how a complex function and its WLCT relate. Lastly, the logarithmic uncertainty principle associated with the WLCT is obtained.

In this paper, we study fixed point sets of wavelet induced isomorphisms for symmetric wavelet sets. Also, introducing the notion of frame induced monomorphism for a frame wavelet set, as a generalization of the wavelet induced isomorphism for a wavelet set, we provide a construction of frame wavelet sets in $ℝ$. We also study fixed point sets of these maps.

Breast cancer is life threatening and dangerous diseases among the women across the world. In this paper, mammogram image classification performed using LS-SVM with various kernels functions namely, Gaussian Radial Basis Function (GRBF) kernel, Polynomial kernel, Quadratic kernel, Linear kernel and MLP kernel. Shearlet transform is a multidimensional version of the composite dilation wavelet transform, and is especially designed to address anisotropic and directional information at various scales and directions, which is used to decompose the regions of interest (ROI) image after preprocessing stage. Initially, mammogram images are transformed into different resolution levels from 2 levels to 4 levels with various directions varying from 2 to 64. The evaluation of the system is carried out on the Mammography Image Analysis Society (MIAS) database. From the experimental analysis, based on classification accuracy and Receiver Operating Characteristics (ROC), it is concluded that LS-SVM with Gaussian RBF kernel function outperforms than Quadratic, polynomial, linear and MLP kernel functions. The classifiers were validated with leave-one-out (training) and cross-validation (testing) modes.

In this paper, a non-stationary time series prediction method based on wavelet transform is proposed. By wavelet decomposition, the non-stationary time series is decomposed into a low frequency signal and several high frequency signals. The high frequency signals are predicted with auto-regressive integrated moving average (ARIMA) models, and the low frequency is predicted with an improved GM(1,1)-Markov chain combined model based on Taylor approximation. Finally, an improved ARIMA-GM(1,1)-Markov chain combined model is constructed by using wavelet reconstruction. As an example, we use the statistical data of the total import and export volume in China from 2001 to 2014 for a validation of the effectiveness of the combined model.