International Journal of Wavelets, Multiresolution and Information Processing

Tight wavelet frames (TWFs) are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with TWF construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to TWFs with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline TWF constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity.

In this paper, we obtain the unified error estimate for some weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces by using some kind of K-functional. From this estimate, we derive the sufficient conditions for the convergence and the convergence rates on sparse classes induced by the K-functional. The results on convergence and the convergence rates are sharp.

A multiscaling equation in the Fourier domain accommodates a trigonometric matrix polynomial. This trigonometric matrix polynomial is known as the symbol function. The existence and properties of a multiscaling function, which is the solution of a multiscaling equation, depend on the symbol function. It is possible to construct symbol functions corresponding to compactly supported and symmetric multiscaling functions from standard pairs. A standard pair carries the spectral information about the symbol function. In this paper, we briefly explain the construction of compactly supported and symmetric multiscaling functions and the corresponding mulitwavelets using standard pairs. We derive the necessary as well as sufficient condition, on the eigenspace of the square matrix in the standard pair, for the existence of a symbol function corresponding to a multiscaling equation with a compactly supported solution. We create a pseudo bi-orthogonal pair of symmetric and compactly supported multiscaling functions and the corresponding multiwavelets using standard pairs.

This paper studies a kind of weighted $l_1$-minimization that is motivated by function interpolation. Combining with the weighted robust null space property, we first propose a new sufficient condition for robust recovery via the weighted $l_1$-minimization when the measurements are corrupted by arbitrary noise without requiring the proper estimation of noise level. Second, we investigate the instance optimality of the weighted $l_1$-minimization decoder according to the weighted quotient property and the weighted restricted isometry property (RIP). In addition, to give a better error estimation in a general problem to recover noisy compressible signals, we improve the $\omega$-RIP constant ${\delta }_{\omega ,3s}$ from $1/3$ to $(\sqrt{17}-1)/8$ by using the weighted sparsity and Stechkin-type estimate. Our results show that the weighted $l_1$-minimization remains not only stable but also robust to reconstruct signals with noisy observations.

The magnetized Jeffery–Hamel flow due to a point sink or source in convergent and divergent channels is studied. Such viscous flow plays an important role in understanding rivers and canals, human anatomy, and the connection between capillaries and arteries. The simplified governing equation ruled by the Reynolds number, the Hartmann number and the divergent–convergent angle with appropriate boundary conditions is solved by the newly proposed Coiflet wavelet-homotopy analysis method (CWHAM). A highly accurate solution is obtained, whose accuracy is rigidly checked. Our proposed method combines the advantages of the homotopy analysis method for strong nonlinearity and the Coiflet wavelet method for excellent local expression capability so that it holds higher computational efficiency, larger applicable range of physical parameters, and better nonlinear processing capability. Different from the previously published research on the CWHAM, we focus on establishing the solution process of nonlinear flow problems with extreme physical parameters, which is not considered before.

Advanced driver assistance systems improve driving safety and comfort by applying onboard sensors to collect environmental data, analyze environmental data and decision making. Therefore, advanced driver assistance systems have high requirements for distance perception of the environment. Perceptual sensors commonly used in traditional solutions include stereo vision sensors and the Light Detection and Ranging (LiDAR) sensors. This paper proposes a multi-sensing sensor fusion method for disparity estimation, which combines the perceptual data density characteristics of stereo vision sensors and the measurement accuracy characteristics of LiDAR sensors. The method enhances the sensing accuracy by ensuring high-density sense, which is suitable for distance sensing tasks in complex environments. This paper demonstrates with experimental results on real data that our proposed disparity estimation method performs well and is robust in different scenarios.

Subspace clustering aims to seek a multi-subspace representation that is best suitable for data points taken from a high-dimensional space. Sparse representation and low-rank approximation-based methods have become one of the main melodies for subspace clustering. In the existing methods, nuclear norm is used to approximate rank minimization. However, the common deficiency still exists for nuclear norm, which always over-penalizes large singular values and results in a biased solution. In this paper, we propose a nonlinear subspace clustering model that exploits sparsity and low-rank of data in high dimensional feature space by using Schatten-p norm surrogate ($p\in (0,1)$) with learned low-rank kernel. By this manner, the model guarantees that the data mapped in the high-dimensional feature spaces is lower rank and self-expressive. And we show the alternating direction method of multipliers (abbreviated as ADMM) for the corresponding problem in a reproducing kernel Hilbert space. Various experiments on motion segmentation and image clustering display that the proposed model has potentiality in outperforming most of state-of-the-art models in current literature.

Matrix completion is critical in a wide range of scientific and engineering applications, such as image restoration and recommendation systems. This topic is commonly expressed as a low-rank matrix optimization framework. In this paper, a universal and effective rank approximation method for matrix completion (RAMC) is provided. Fundamental to this strategy is developing a general function that meets specific conditions in order to directly approach the rank function and subsequently utilizing it to build a RAMC model. The major goal is to investigate a more accurate estimate of the rank function, allowing for more effective acquisition of the low-rank structure of incomplete data. Further, the RAMC model is easily implemented by a viable iterative method that may be successfully used to matrix completion tasks. Extensive experiments using the synthetic data and natural images reveal the excellent applicability of RAMC over the existing methods.

The generalization is one of the main concerns of machine learning method. k-SVCR is an important multi-class classification algorithm, but there is no theoretical analysis on the generalization of k-SVCR algorithm up to now. Therefore, in this paper, we consider multi-class support vector classification and regression (MSVCR) algorithm. We first establish the generalization bounds of MSVCR based on uniformly ergodic Markovian chain (u.e.M.c.) samples, and obtain its fast learning rate of MSVCR. As applications, we estimate the generalization of MSVCR for independent and identically distributed (i.i.d.) observations and strongly mixing observations, respectively. We also propose a new MSVCR algorithm based on q-times Markovian resampling (MSVCR-Mar-q). The experimental studies indicate that compared to the classical k-SVCR and other multi-class SVM algorithms, the proposed algorithm not only has smaller misclassification rate, but also has less sampling and training total time.

In this paper, an algorithm is discussed for constructing the two-direction Armlet multiwavelet. First, the definition of two-direction Armlet multiwavelet is presented in this paper. A two-direction multiwavelet can be changed to a special multi-wavelet. By Two-scale Similar Transform (TST), a transform can be taken on the two-scale matrix symbols of a two-direction multi-wavelets. This transform keeps the orthogonality of the two-direction multi-wavelets. However, the condition is discussed which the two-direction multi-wavelet corresponding to a two-direction multi-scaling function is an Armlet with order n. An approach is given for constructing the transform matrix. Finally, an example is given for discussing the two-direction Armlet multi-wavelet with order 2.

Deep networks have been widely applied in infrared and visible image fusion. However, the current deep networks cannot well extract and fuse multi-scale information and high-frequency texture features of the source images. In this paper, a deep multi-scale pyramid network, termed MSPFNet, is proposed for infrared and visible image fusion by combining image Laplacian pyramid and deep network. Infrared and visible images are first decomposed into their Laplacian pyramids. For each source image, its Laplacian pyramid consists of a low-frequency component and a series of multi-scale high-frequency components containing texture details. Then, the Laplacian pyramid components of two source images in the same level are fused using convolutional neural networks (CNN). Finally, the final fused image is reconstructed on the fused Laplacian pyramid components using inverse Laplacian pyramid transform. The experimental results on publicly available datasets show that MSPFNet can efficiently extract and fuse the multi-scale detail information of source images, and the fused images of MSPFNet preserve more texture details of infrared and visible images than that of the previous state-of-the-art methods.

In this paper, we establish the concept of a two-wavelet in Weinstein setting. Then we introduce and demonstrate the resolution of the identity formula for the continuous Weinstein wavelet transform. In the end, we show few results on Calderón’s-type reproducing kernels in the context of the two-wavelet transform in Weinstein setting.