International Journal of Wavelets, Multiresolution and Information Processing

This paper provides a new theoretical support for block sparse recovery. By embedding prior support information into the block ${\ell }_p-{\ell }_1$ minimization with $0<p\le 1$, we establish a sufficient condition on block $p$-restricted isometry property (RIP) to guarantee the stable and robust recovery of block sparse signals, as well as obtain a corresponding upper bound estimate of the error. The obtained result not only generalizes the existing ones, but also shows that the block ${\ell }_p-{\ell }_1$ minimization with prior support information behaves robust and stable recovery performance under weaker condition than the analogous one for the block ${\ell }_p-{\ell }_1$ minimization if more than $50\%$ of the support information is accurate.

By applying fundamental mathematical knowledge, this paper proves that the function $y=x^m(m$ is an integer no less than $2),x\in [0,1]$ has the property that the difference between the function value of middle point of arbitrarily two adjacent equidistant distribution nodes on $[0,1]$ and the mean of function values of these two nodes is a constant depending only on the number of nodes if and only if $m=2.$ By them, we establish an important result about deep neural networks that the function $y=x^2,x\in [0,1]$ can be interpolated by a deep Rectified Linear Unit (ReLU) network with depth $n+1$ on the equidistant distribution nodes in interval $[0,1],$ and the error of approximation is $2^{-(2n+2)}.$ Then based on the main result that has just been proven and the Chebyshev orthogonal polynomials, we construct a deep network and give the error estimate of approximation to polynomials and continuous functions, respectively. In addition, this paper constructs one deep network with local sparse connections, shared weights and activation function $y=x^2,$ and discusses its density and complexity.

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions $\phi$ with frequency domain $\text{supp}\widehat{\phi }=G$. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function $\phi$ is convex and completely symmetric about the origin, then it must be weakly invariant and $0\in \text{supp}\widehat{\phi }\subset {[-\frac{4}{3}\pi ,\frac{4}{3}\pi ]}^d$. If $\phi$ has a ball-shaped frequency domain, the ball radius must be bounded by $max\{\frac{4}{3}\pi ,(\sqrt{2+\frac{1}{4d}}-\sqrt{\frac{1}{4d}})\pi \}$. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

Approximation theory constitutes a useful field that is related to quasi all other fields, in both theoretical and applied sciences. In approximation theory, the aim is generally to construct an idea about a function that is usually impossible or difficult to evaluate directly, and which is usually unknown. Such functions appear widely in PDE, probability law distributions, statistical modeling, etc. Some of the most known approximators nowadays are neural networks and wavelets, which constitute good classes of elementary functions permitting as efficiently as possible to describe functions in appropriate spaces. This paper aims to develop combined neural networks and wavelet approximators for functions, based on the involvement of wavelets as activation functions. Some necessary conditions on the activation function to approximate $L^p(\mu )$ and $W^{m,p}(\mu )$-elements are relaxed as well as those on the measure $\mu$. We prove that for a wavelet activation function, any element of $L^p(\mu )$ as well as $W^{m,p}(\mu )$ can be well approximated for arbitrary measures $\mu$. The theoretical results are subject to an experimental application in order to show their effectiveness.

In this paper, the inversion formula of the continuous Bessel wavelet transform of distributions in ${\beta }_{\mu }^{\prime }$-space and its associated results with Bessel wavelet kernel containing non-vanishing moments of even orders weighted by $x^{\mu +\frac{1}{2}}$ are investigated by exploiting the theory of Hankel transform.

In this paper, a nonlinear modulation ${{ℳ}}_{\frac{1}{1+{\omega }^2}}$ and a frequency-varying dilation ${{𝒟}}_{1+{\omega }^2}$ both with Poisson kernel are introduced. Two classes of time-frequency atoms ${\varphi }_j^{t,\omega }(x),x\in ℝ,j=1,2$ are designed from a basic atom $\varphi$ in the Schwartz class $𝕊(ℝ)$ acted upon by three operators: translation, nonlinear modulation and dilation. Two time-frequency transformations ${{𝒮}}_j,j=1,2$ are constructed based on the above designed time-frequency atoms, where ${{𝒮}}_1$ maps $L^2(ℝ)$ into $L^2({ℝ}^2)$ with Lebesgue measure while ${{𝒮}}_2$ maps $L^2(ℝ)$ into $L^2({ℝ}^2)$ with Haar measure. The corresponding inversion formulae are established and the reproducing kernel Hilbert space property of the images of ${{𝒮}}_j,j=1,2$ is proved. This strategy offers a unified understanding of dilation frequency and Fourier (modulation) frequency.

In dynamic scenarios, dynamic participants break the static assumptions of the visual odometry (VO) algorithm. Hence, dynamic participants are typically removed and only static participants are used as motion references. When semantic information is used to eliminate participants with dynamic semantic labels in the scene, the dynamic degree of the scene affects system robustness and poses estimation accuracy. In this paper, we propose a VO system that employs adaptive compensation for semantic information and utilizes an adaptive compensation model for sparse static feature map construction to make the compensation process a flexible, independent thread. Moreover, a layered extraction fusion framework is proposed to uniformly sample equi-probability feature points both globally and locally. This framework combines multi-level weights of scene mapping and spatial-temporal priority information of self-organization. Finally, in the candidate pixel extraction stage, system efficiency is improved through region matching candidate pixel detection, extraction and fusion semantic constraints. Experiments are done by using the TUM RGBD datasets. Comparisons with many conventional visual mileage calculation methods reveal that the absolute and relative trajectory errors of camera motion are significantly reduced in most scenes with different dynamic degrees. Thus, compared to previous algorithms, the proposed algorithm is more robust and precise in dynamic scenes.

Automatic Speech Recognition schemes are the alternating modes in which individuals interrelate with various mobile application applications. The user interactivity needs are a huge vocabulary identification system, high accuracy, energy-efficient solution and time. Although the automatic Speech Recognition system requires power budget and huge memory bandwidth, it is not applicable for many tiny forms of battery-controlled devices. Hence, an effective method is developed using the proposed Multi-kernel Perceptual linear predictive and Stochastic Biogeography-based whale optimization algorithm optimization to adapt non-audible mutter to regular speech. First, the input speech signal is initially given to the pre-processed module. Then, the features, such as spectral centroid, pitch chroma, Taylor amplitude modulation spectrogram (AMS), spectral skewness, and the developed Multi-kernel Perceptual linear predictive, are extracted to determine the appropriate features. After extracting features, the speech recognition is performed based on Deep Convolutional Neural Network, which is trained by the proposed Stochastic Biogeography whale optimization algorithm. The Stochastic Biogeography whale optimization algorithm combines the stochastic gradient descent method, whale optimization algorithm, and biogeography-based optimization. The developed model showed improved results with maximum accuracy of 0.985, minimal FPR of 0.001, maximal TPR of 1, respectively.

With the aim to achieve an efficient time-frequency representations of higher dimensional signals, we introduce the notion of free metaplectic wavelet transform in $L^2({ℝ}^n)$ by employing the convolution structure of the free-metaplectic transforms. The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality and inversion formula. Subsequently, Heisenberg’s and Pitt’s inequalities are also obtained. To extend the scope of the study, a detailed analysis of the double windowed wavelet transform in the free-metaplectic domain is presented at the end.

Hilbert–Schmidt frame (HS-frame) has interested some mathematicians in recent years, which is more general than g-frame. This paper addresses near Riesz and Besselian properties of HS-operator sequences. We characterize HS-frame and Riesz properties of g-operator sequences using their DSI-sequences; prove that an HS-Riesz basis is an exact HS-frame while the converse is not true, and an arbitrary HS-Riesz frame contains an HS-Riesz basis; and present the connection among near HS-Riesz property, Besselian property and the kernel space dimension of synthesis operator of an HS-operator sequence.

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space $L^2({ℝ}^d,{ℂ}^{s\times r})$ from dual pairs of atomic wave packet frames in $L^2({ℝ}^d)$ is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of $L^2({ℝ}^d)$.

Constructing tight wavelet filter banks with prescribed directions is challenging. This paper presents a systematic method for designing a tight wavelet filter bank, given any prescribed directions. There are two types of wavelet filters in our tight wavelet filter bank. One type is entirely determined by the prescribed information about the directionality and makes the wavelet filter bank directional. The other type helps the wavelet filter bank to be tight. In addition to the flexibility in choosing the directions, our construction method has other useful properties. It works for any multidimension, and it allows the user to have any prescribed number of vanishing moments along the chosen directions. Furthermore, our tight wavelet filter banks have fast algorithms for analysis and synthesis. Concrete examples are given to illustrate our construction method and properties of resulting tight wavelet filter banks.