International Journal of Wavelets, Multiresolution and Information Processing

We investigate the efficiency of weak orthogonal super greedy algorithm (WOSGA) for $n$-term approximation with respect to dictionaries which are $(n,D)$-unconditional in arbitrary Hilbert space ${\mathscr{H}}.$ For an element $f\in {\mathscr{H}}$, let $f_{An}$ be the output of WOSGA after $An$ steps for some constant $A$. We show that the residual $\parallel f-f_{An}\parallel$ can be bounded by a constant multiplying the error of best $n$-term approximation to $f.$ Moreover, we get an element $f_n^{\ast }$, through a simple postprocessing of $f_{An}$ by retaining its $n$ largest components in absolute value, which realizes near best $n$-term approximation for $f.$ Our results are obtained for dictionaries in ${\mathscr{H}}$ which satisfies the weaker assumption than the RIP condition.

In this paper, the relation between Bessel wavelet convolution product and Hankel convolution product is obtained by using the Bessel wavelet transform and the Hankel transform. Approximation results of the Bessel wavelet convolution product are investigated by exploiting the Hankel transformation tool. Motivated from the results of Pinsky, heuristic treatment of the Bessel wavelet transform is introduced and other properties of the Bessel wavelet transform are studied.

The Hilbert transform is tightly associated with the Fourier transform. As the offset linear canonical transform (OLCT) has been shown to be useful and powerful in signal processing and optics, the concept of generalized Hilbert transform associated with the OLCT has been proposed in the literature. However, some basic results for the generalized Hilbert transform still remain unknown. Therefore, in this paper, theories and properties of the generalized Hilbert transform have been considered. First, we introduce some basic properties of the generalized Hilbert transform. Then, an important theorem for the generalized analytic signal is presented. Subsequently, the generalized Bedrosian theorem for the generalized Hilbert transform is formulated. In addition, a generalized secure single-sideband (SSB) modulation system is also presented. Finally, the simulations are carried out to verify the validity and correctness of the proposed results.

This paper first discusses the relationship between the rank null space property (NSP) and the nuclear norm minimization. Several versions of the rank NSP, i.e. the stable rank NSP, robust rank NSP and Frobenius robust rank NSP are proposed, and their equivalent forms are derived. At the same time, it is shown that the stable rank NSP is weaker than the rank restricted isometry property (RIP) to recover the low-rank matrices via the nuclear norm minimization. Finally, the rank NSP is extended to the case of Schatten-$q$ NSP for $0<q<1$, and the solutions to the Schatten-$q$ quasi-norm minimization are characterized by the different types of Schatten-$q$ NSP.

Differential privacy becomes a standard for evaluating the privacy protection performance for an algorithm these years. However, the definition of differential privacy seems not so easy to understand as the classical k-anonymity and etc. In this paper, we propose a new measure which is more comprehensible. Some properties of such measure are investigated and the relationship between our new definition and differential privacy is studied.

In this paper, the linear B-spline scaling functions and wavelets operational matrix of fractional integration are derived. A new approach implementing the linear B-spline scaling functions and wavelets operational matrices combining with the spectral tau method is introduced for approximating the numerical solutions of time-space fractional partial differential equations with initial-boundary conditions. They are utilized to reduce the main problem to a system of algebraic equations. The uniform convergence analysis for the linear B-spline scaling functions and wavelets expansion and an efficient error estimation of the presented method are also introduced. Illustrative examples are given and numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Special attention is given to a comparison between the numerical results obtained by our new technique and those found by other known methods.

This paper deals with the approximation properties of Riesz means in $L_p(\mathrm{\text{SO}}(3))$. The convergence rate of the Riesz means is obtained. The equivalence of Riesz means and the Peetre $K$-functional on the rotation group $\mathrm{\text{SO}}(3)$ is established.

Multiwindow discrete Gabor transform (M-DGT) is applied to present the Gabor time–frequency representation for transient signals (exponentially damped sinusoidal signals) with high time–frequency resolution. Due to the limitation of the constrained time–frequency localization governed by the Heisenberg uncertainty principle, using a wider analysis window in time domain will lead to the Gabor time–frequency spectrum (or representation) with higher frequency resolution but poor time resolution for the transient signals, and using a narrower analysis window in time domain will result in the Gabor time–frequency spectrum (or representation) with higher time resolution but poor frequency resolution for the transient signals. To obtain the Gabor time–frequency representation with both higher frequency resolution and higher time resolution, the above two spectra can be combined by geometric average. The experimental results show that the combined Gabor time–frequency representation for the transient signals has higher time–frequency resolution than that obtained when only the single analysis window is used in the traditional discrete Gabor transform.

The aim of multi-focus image fusion is to create a synthetic all-in-focus image from several images each of which is obtained with different focus settings. However, if the resolution of source images is low, the fused images with traditional fusion method would be also in low-quality, which hinders further image analysis even the fused image is all-in-focus. This paper presents a novel joint multi-focus image fusion and super-resolution method via convolutional neural network (CNN). The first level network features of different source images are fused with the guidance of the local clarity calculated from the source images. The final high-resolution fused image is obtained with the reconstruction network filters which act like averaging filters. The experimental results demonstrate that the proposed approach can generate the fused images with better visual quality and acceptable computation efficiency as compared to other state-of-the-art works.

In this paper, we introduce controlled frames in Hilbert $C^{\ast }$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^{\ast }$-module. Also multiplier operators for controlled frames in Hilbert $C^{\ast }$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^{\ast }$-modules and verify their relations to controlled frames and multiplier operators.

In this paper, we present an efficient and simple shadow detection algorithm for indoor environments, as well as give a brief description on the advantages of this method. In this method, we use three types of approaches: image enhancement, chromaticity consistency, and gradient features. Multiple shadow direction is becoming an increasingly challenging task for many moving shadow detection algorithms because some objects have large self-shadows. Our system is able to achieve good performance solving spread shadow problems in indoor scenes, leading to improved foreground segmentation in surveillance scenarios. The image enhancement approach is first employed to input images to generate high-quality images for artificial light source indoor areas. Afterwards, the chromaticity information is utilized to create a mask of possible candidate shadow pixels. Subsequently, gradient features are applied to remove foreground pixels that have been incorrectly included in the mask. In comparison with existing algorithms, the proposed method can correctly detect and remove shadow pixels to identify original foreground shapes without distortion for delivering object recognition and tracking tasks.

In the framework of wave packet analysis, finite wavelet systems are particular classes of finite wave packet systems. In this paper, using a scaling matrix on a permuted version of the discrete Fourier transform (DFT) of system generator, we derive a locally-scaled version of the DFT of system generator and obtain a finite equal-norm Parseval wavelet frame over prime fields. We also give a characterization of all multiplicative subgroups of the cyclic multiplicative group, for which the associated wavelet systems form frames. Finally, we present some concrete examples as applications of our results.