International Journal of Wavelets, Multiresolution and Information Processing

A one-dimensional (1D) point process, if considered as a random measure, can be represented by a sum, at most countable, of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads to the definition of the continuous wavelet transform of the process itself. As a possible choice of the mother wavelet, we propose the Mexican hat and the Morlet wavelet in order to implement the energy density of the process as a function of two wavelet parameters. Such mathematical tool works as a microscope to process an in-depth analysis of some classes of processes, in particular homogeneous, cluster, and locally scaled processes.

As a general means of expression, audio analysis and recognition have attracted much attention for its wide applications in real-life world. Audio emotion recognition (AER) attempts to understand the emotional states of human with the given utterance signals, and has been studied abroad for its further development on friendly human–machine interfaces. Though there have been several the-state-of-the-arts auditory methods devised to audio recognition, most of them focus on discriminative usage of acoustic features, while feedback efficiency of recognition demands is ignored. This makes possible application of AER, and rapid learning of emotion patterns is desired. In order to make predication of audio emotion possible, the speaker-dependent patterns of audio emotions are learned with multiresolution analysis, and fractal dimension (FD) features are calculated for acoustic feature extraction. Furthermore, it is able to efficiently learn the intrinsic characteristics of auditory emotions, while the utterance features are learned from FDs of each sub-band. Experimental results show the proposed method is able to provide comparative performance for AER.

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

In the last two decades, wavelet bases and associated methodologies have become quite important in many domains, such as signal and image processing, harmonic analysis, statistics, and so on. More recently, they also turn out to be quite useful in the probabilistic framework of stochastic processes, in which, among other things, they allow to obtain fine results concerning erratic sample paths behavior. The goal of our paper is to derive a result, related with zeros of continuous compactly supported mother wavelets, which is useful in this probabilistic framework. More precisely, let $\psi$ be any arbitrary such wavelet; we show that being given an arbitrary point $x_0\in ℝ$ there always exists at least one integer $k_{x_0}\in \mathbb{Z}$ such that $\psi (x_0-k_{x_0})\ne 0$.

The Easy-Path Wavelet Transform (EPWT) is an adaptive transform for bivariate functions (in particular natural images) which has been proposed in G. Plonka, The easy path wavelet: A new adaptive wavelet transform for sparse representation of two-dimensional data, Multiscale Model. Simul. 7(3) (2009) 1474–1496.¹³ It provides a sparse representation by finding a path in the domain of the function leveraging the local correlations of the function values. It then applies a 1-dimensional (ID) wavelet transform to the obtained vector, decimates the points and iterates the procedure. The main drawback of such method is the need to store, for each level of the transform, the path which vectorizes the 2-dimensional (2D) data. Here, we propose a variation on the method which consists of firstly applying a segmentation procedure to the function domain, partitioning it into regions where the variation in the function values is low; in a second step, inside each such region, a path is found in some deterministic way, i.e. not data-dependent. This circumvents the need to store the paths at each level, while still obtaining good quality lossy compression. This method is particularly well suited to encode a region of interest in the image with different quality than the rest of the image.

We propose a new image quality degradation assessment method based on the dual-tree complex discrete wavelet transform (DT-CDWT) for evaluating the image quality of watermarked images. The peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are widely used to evaluate image quality degradation resulting from embedding a digital watermark. The majority of digital image watermarking methods embed a digital watermark in the spatial or frequency domain of an original image. They evaluate image quality degradation using only the spatial domain in spite of the fact that the majority of digital image watermarking methods embed a digital watermark in the spatial or frequency domain. As a result, they do not always fairly evaluate the image quality degradation. Therefore, our method evaluates image quality degradation of the watermarked images using features in the spatial and frequency domains. To extract the features, we defined three indices: 1-norm estimation using bit-planes in the spatial domain, the sharpness, and 1-norm estimation based on the DT-CDWT domains. We describe our image quality assessment method in detail and present experimental results demonstrating that there is a strong positive correlation between the result obtained by our method and a subjective evaluation, in comparison with PSNR and SSIM.

In this paper, systolic array-based novel architecture for dual-tree complex wavelet transform (DTCWT) computation is designed and implemented on FPGA. The wavelet filter coefficients of DTCWT are quantized and rounded to nearest integer and the loss in rounding and quantization is limited to 0.5dB as compared with software implementation. The parallel architecture designed computes row elements simultaneously and pipelined architecture is designed to compute column elements. The proposed architecture is modeled using Verilog and implemented on Xilinx Virtex II FPGA. For 2D implementation, the design operates at a maximum frequency of 156MHz and consumes power less than 3W. This is the first design with systolic array architecture on FPGA for DTCWT computation operating at frequencies greater than 100MHz.

We introduce the notion of a Dunkl two-wavelet multiplier, and we give its trace formula as a bounded linear operator in the trace class from $L_k^2({ℝ}^d)$ into $L_k^2({ℝ}^d)$ in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of Dunkl two-wavelet multipliers on $L_k^p({ℝ}^d)$, $1\le p\le \infty$.

The robust feature extraction method for face representation is an important issue in face recognition. In this paper, we extract a new kind of feature through applying the idea of local binary pattern (LBP) into the resulted sub-images of Gabor transform. The new feature, i.e. Gabor-LBP-Like (GLLBP), together with its extension methods (1) overcome the drawback of losing information after Gabor transform’s down-sampling; (2) are insensitive to noise, compared with the LBP feature extracted from the original face image; and (3) are robust to image variation, especially occlusion and illumination changes when compared with other existing features combined LBP and Gabor transform. To validate the effectiveness of these features, we do experiments on the ORL, FERET, Georgia Tech and LFW facial databases. The numerical results show that GLLBP and its extensions are miraculous features for face recognition.

The continuous fractional wavelet transform (CFrWT) is a nontrivial generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FrFT) domain. Firstly, the Riemann–Lebesgue lemma for the FrFT is derived, and secondly, the CFrWT in terms of the FrFT is introduced. Based on the CFrWT, a different proof of the inner product relation and the inversion formula of the CFrWT are provided. Thereafter, a logarithmic uncertainty relation for the CFrWT is investigated and the convolution theorem related to the CFrWT is established using the convolution of the FrFT. The CFrWT on a generalized Sobolev space is introduced and its important properties are presented.

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 $c_k.$

This paper presents a simple and novel approach for robust face recognition. Extended multi-scale Weber local descriptor based on unfixed size patches is proposed to obtain micro and macro patterns. Then the recognition is performed using nearest neighborhood method based on Chi-square distance. Voting strategy is utilized in classification to reduce the impacts of the poor quality regions in face image caused by pose, illumination variations. Experiments on CMU-PIE, FERET and AR databases show that the proposed method has higher recognition accuracy in comparison with traditional methods.