International Journal of Wavelets, Multiresolution and Information Processing

Nonlinear diffusion and wavelet shrinkage are two successfully applied methods for discontinuity preserving denoising of signals and images. Recently, relations between both methods have been established taking into account wavelet shrinkage at one or multiscale. In this paper we show that one step of (stabilized) explicit discretization of nonlinear diffusion can be expressed in terms of tight frame shrinkage on a single spatial level or multiscale. We prove that our scheme permits larger steps while having more choices of shrinkage functions. Numerical examples demonstrate the behavior of our scheme for one or two scales.

The concept of p-adic quincunx Haar MRA was introduced and studied in Ref. 12. In contrast to the real setting, infinitely many different wavelet bases are generated by a p-adic MRA. We give an explicit description for all wavelet functions corresponding to the quincunx Haar MRA. Each one generates an orthogonal basis, one of them was presented in Ref. 12. A connection between quincunx Haar MRA and two-dimensional separable Haar MRA is also found.

A new multi-kernel regression learning algorithm is studied in this paper. In our setting, the hypothesis space is generated by two Mercer kernels, thus it has stronger approximation ability than the single kernel case. We provide the mathematical foundation for this regularized learning algorithm. We obtain satisfying capacity-dependent error bounds and learning rates by the covering number method.

In this paper, we focus on the construction of Wilson frames and their dual frames for general lattices of volume (K even) in the discrete-time setting. We obtain a necessary and sufficient condition for two Bessel sequences having Wilson structure to be dual frames for l²(ℤ). When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for l²(ℤ), show that a Wilson frame for l²(ℤ) 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.

M. M. Hadjiandreou, R. Conejeros and V. S. Vassiliadis (Math. Biosci. Eng. 4 (2007) 489–504) proposed an ODE model of long-term HIV infection dynamics. This model considers all important population compartments including macrophages, latently-infected CD4⁺ T cells and cytotoxic T lymphocytes (CTLs), which in many respects is novel in the field of mathematical modeling of HIV infection. The purpose of the paper is to simplify their model into a five-dimensional HIV dynamics model, and give a detailed classification on the existence of the equilibria of the reduced model by introducing four reproductive numbers. Then the global asymptotic stability of each equilibrium is analyzed by constructing suitable Lyapunov functions. The results in the paper extend and improve the understanding of the dynamics of HIV latency.

This paper, proposes a novel approach for feature extraction based on the segmentation and morphological alteration of handwritten multi-lingual characters. We explored multi-resolution and multi-directional transforms such as wavelet, curvelet and ridgelet transform to extract classifying features of handwritten multi-lingual images. Evaluating the pros and cons of each multi-resolution algorithm has been discussed and resolved that Curvelet-based features extraction is most promising for multi-lingual character recognition. We have also applied some morphological operation such as thinning and thickening then feature level fusion is performed in order to create robust feature vector for classification. The classification is performed with K-nearest neighbor (K-NN) and support vector machine (SVM) classifier with their relative performance. We experiment with our in-house dataset, compiled in our lab by more than 50 personnel.

This paper defines an outlier and a measure that an outlier does not fit the theoretical model in the regression problems, studies the relationship between the theoretical model and the regression model in the regression problems, proposes and verifies an approximate theorem in which one-by-one deletes outlier and constructs SVR to approximate its theoretical model, draws an algorithm of detecting outliers in the SVR problems based on the approximate theorem, and analyzes the convergence and effectiveness of the algorithm in the theory. Next, it combines the step-by-step search algorithm were proposed by authors and the former detecting outlier algorithm to improve it, and brings forward an algorithm detecting outliers in SVR based on large-scale samples. Then, its analysis in theory shows the improved algorithm is convergent and effective too. Finally, its simulation results show that the detecting outlier algorithms are effective and robust, using samples produced by two test functions and in UCI dataset.

Local regularity analysis is useful in many fields, such as financial analysis, fluid mechanics, PDE theory, signal and image processing. Different quantifiers have been proposed to measure the local regularity of a function.

In this paper we present a new quantifier of local regularity of a signal: the pointwise wavelet leaders entropy. We define this new measure of regularity by combining the concept of entropy, coming from the information theory and statistical mechanics, with the wavelet leaders coefficients. Also we establish its inverse relation with one of the well-known regularity exponents, the pointwise Hölder exponent.

Finally, we apply this methodology to the financial data series of the Dow Jones Industrial Average Index, registered in the period 1928–2011, in order to compare the temporal evolution of the pointwise Hölder exponent and the pointwise wavelet leaders entropy. The analysis reveals that temporal variation of these quantifiers reflects the evolution of the Dow Jones Industrial Average Index and identifies historical crisis events.

We propose a new approach to analyze the local regularity variation of a signal and we apply this procedure to a financial data series, attempting to make a contribution to understand the dynamics of financial markets.

Image fusion is a technique of combining information from multiple images of the same scene into an image, so that the fused image contains a more accurate description of the scene than any of the individual source images. SVT (Support Value Transform) method is one of leading multiscale methods for studying image fusion, which achieves better fusion results both in visual inspection and quantitative analysis. One important assumption of SVT method is representing salient feature of image by support value. However, related studies did not give any mathematical explanation of it. Another drawback of this fusion method is ignoring the rich directional information of original image. In this paper, we provide a rational evidence of the physical meaning of support value by introducing Gaussian curvature of image. Also, by combining with DFB (Directional Filter Bank), a multi-directional SVT image fusion method is proposed. Our method can capture different and flexible directional information, which may help obtain the intrinsic geometrical structure of original image. Experimental results on several pairs of multi-focus images show that the proposed method achieves better results than SVT and other traditional common used methods.

This paper considers the problem of recovering a low-rank matrix from a small number of measurements consisting of linear combinations of the matrix entries. We extend the elastic-net regularization in compressive sensing to a more general setting, the matrix recovery setting, and consider the elastic-net regularization scheme for matrix recovery. To investigate on the statistical properties of this scheme and in particular on its convergence properties, we set up a suitable mathematic framework. We characterize some properties of the estimator and construct a natural iterative procedure to compute it. The convergence analysis shows that the sequence of iterates converges, which then underlies successful applications of the matrix elastic-net regularization algorithm. In addition, the error bounds of the proposed algorithm for low-rank matrix and even for full-rank matrix are presented in this paper.