Narrow Results

It is well known that orthonormal bases for a separable Hilbert space $H$ are precisely collections of the form ${\{\mathrm{\Theta }{e}_k\}}_{k\in 𝕀}$, where $\mathrm{\Theta }$ is a linear unitary operator acting on $H$ and ${\{e_k\}}_{k\in 𝕀}$ is a given orthonormal basis for $H$. We show that this is not true for the matrix-valued signal space $L^2(G,{ℂ}^{s\times r})$, $G$ is a locally compact abelian group which is $\sigma$-compact and metrizable, and $s$ and $r$ are positive integers. This problem is related to the adjointability of bounded linear operators on $L^2(G,{ℂ}^{s\times r})$. We show that any orthonormal basis of the space $L^2(G,{ℂ}^{s\times r})$ is precisely of the form ${\{{𝒰}{E}_k\}}_{k\in 𝕀}$, where ${𝒰}$ is a linear unitary operator acting on $L^2(G,{ℂ}^{s\times r})$ which is adjointable with respect to the matrix-valued inner product and ${\{E_k\}}_{k\in 𝕀}$ is a matrix-valued orthonormal basis for $L^2(G,{ℂ}^{s\times r})$.

In the first part of the paper, we prove that a twisted shift-invariant subspace of $L^2({ℝ}^{2n})$ can be decomposed as a direct sum of mutually orthogonal principal twisted shift-invariant spaces such that the respective system of twisted translates forms a Parseval frame sequence. Later, we introduce twisted shift preserving operators and the corresponding range operators. We establish that the twisted shift preserving operators and the corresponding range operators simultaneously share some properties in common, namely, self-adjoint, unitary, range of the spectrum and bounded below properties. Finally, we prove that the frame operator and its inverse associated with a system of twisted translates of ${\{{\varphi }_s\}}_{s\in \mathbb{Z}}$ are shift preserving and investigate their corresponding range operators.

Due to their adaptive nature, empirical wavelets (EWs) had several successes in many fields from engineering, science, medical signal/image processing. Recently, a general theoretical framework has been developed in the one-dimensional case, showing the possibility to build EWs from any classic mother wavelets. Given extensive literature both in theory and applications of classic wavelet frames, it is legitimate to ask about the feasibility of building EW frames. We address this question in this paper. We prove several results which provide conditions on the existence of EW frames taking into account the above-mentioned adaptability.

In this study, a novel hybrid deep clustering approach is proposed for the effective moving object segmentation. Initially, the data is collected, and the keyframe selection is performed using the threshold-based Kennard–Stone method. Then, the preprocessing step involves noise filtering using bilateral wavelet thresholding and binary color conversion. The blob detection is performed using normalized Laplacian of Gaussian. Finally, the segmentation of moving objects is performed using a hybrid clustering approach called improved fuzzy C-mean (IFCM) clustering with chaotic salp swarm (CSS) optimization algorithm (Hybrid IFCM-CSS). The overall evaluation is done in MATLAB. The performance of the hybrid IFCM-CSS is compared to other approaches based on some measures. The proposed Hybrid IFCM-CSS achieves the highest precision of 0.971, using the SBM-RGBD dataset.

In this paper, we study the problem of recovering approximate shape from the shading of a three-dimensional object in a single image when knowledge about the object is available. The application of knowledge-based methods to low-level image processing tasks will help overcome problems that arise from processing images using a pixel-based approach. Shape-from-shading has generally been approached by precognitive vision methods where a standard operator is applied to the image based on assumptions about the imaging process and generic properties of what appears. This paper explores some advantages of applying knowledge and hypotheses about what appears in the image. The knowledge and hypotheses used here come from domain knowledge and edge-matching. Specifically, we are able to find solutions to some problems that cannot be solved by other methods and gain advantages in terms of computation speed over similar approaches. Further, we can fully automate the derivation of the approximate shape of an object. This paper demonstrates the efficacy of using knowledge in the basic operation of an early vision operator, and so introduces a new paradigm for computer vision that may be applied to other early vision operators.

Generally, cochlear implant system consists of one or more electrodes which directly activate the auditory nerve. The microphone transforms external speech processor input into stimuli for each electrode and transcutaneous the connection between the electrodes and processor. As a result, cochlear implants continue to improve the performance of speech processor. After the implantation, preservation gains importance in acoustic hearing to prevent the neural degradation from the loss of cells. Eventually, the possibility of preserving low-frequency and high-frequency hearing using modified surgical technique has been explored for hearing in cochlear implantation. The primary focus on the architecture is to reduce the major potential thread performance and power trade-off. So, frequency mapping function with modified FIR filter has been proposed. For this implementation, the values relative to the frequency range defines upper and lower stimulus limits of each electrode. The proposed FIR filter is designed with distributed arithmetic function where the reconfigurable FIR filter can change the filter input coefficients. By scaling both the frequency ranges and by assessing the deriving frequency parameters, the mapping function will be yielded. Mapping function is manipulated to rectify any mapping deviations. This results in favor of the development of frequency specific mapping function with flexible system. From the proposed results, it is evident that the speech recognition performance is improved significantly by better minimization of overall energy consumption based on effective and pipelined FFT architecture. This implementation also reduces the power and improves the accuracy by 90% and above for different frequency ranges.

There have been a large number of systems that integrate logic and objects (frames or classes) for knowledge representation and reasoning. Most of those systems give pre-eminence to logic and their objects lack the structure of frames. These choices imply a number of disadvantages, as the inability to represent exceptions and perform default reasoning, and the reduction in the naturalness of representation. In this paper, aspects of knowledge representation and reasoning in SILO, a system integrating logic in objects, are presented. SILO gives pre-eminence to objects. A SILO object comprises elements from both frames and classes. A kind of many-sorted logic is used to express object internal knowledge. Message passing, alongside inheritance, plays a significant role in the reasoning process. Control knowledge, concerning both deduction and inheritance. is separately and explicitly represented via definitions of certain functions, called meta-functions.

In this paper, we first introduce the notion of controlled weaving $K$-$g$-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving $K$-$g$-frames in separable Hilbert spaces. Also, a characterization of controlled weaving $K$-$g$-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled $K$-$g$-woven frames give ordinary weaving $K$-frames and vice-versa.

In this paper, we study Gabor frames in the matrix-valued signal space $L^2(G,{ℂ}^{n\times n})$, where $G$ is a locally compact abelian group which is metrizable and $\sigma$-compact, and $n$ is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space $L^2(G,{ℂ}^{n\times n})$. This generalizes a result due to Aldroubi. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on $L^2(G,{ℂ}^{n\times n})$ constitute a frame for the space $L^2(G,{ℂ}^{n\times n})$ provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.

This paper presents a method for nonlinear dynamic analysis of frames with material and geometric nonlinearities which is based on the semirigid technique. The plastic hinge that accounts for the material nonlinearity is modeled as a pseudo-semirigid connection with nonlinear hysteretic moment-curvature characteristics at the element ends. The stiffness matrix of a frame element with material and geometric nonlinearities is expressed as the sum of products of the standard stiffness matrix and the geometric stiffness matrix of the element, with their corresponding correction matrices based on the plasticity factors developed from the section flexural stiffness at the plastic hinge locations. The combined stress yield condition is used for the force state determination of plastic hinges, and force equilibrium iterations and geometry updating for frames are carried out in every time step. When the key parameters of a structure are updated in a time step, the time step is split up into substeps to ensure accuracy while keeping the computations to a reasonable amount. The plastic rotation history can be calculated directly or in an approximate indirect way. The method is computationally efficient and it needs no additional connection elements, which makes it convenient for incorporation into existing linear dynamic analysis programs. Besides, the method can handle accurately and efficiently the dynamic analysis of nonlinear frames using relatively large time steps in conjunction with time step subdivision to cope with key parameter changes. A portal frame is used to verify the correctness of the proposed method. A more complicated five-story frame is used to illustrate the applicability and performance of the proposed method.

We explore algebraic properties of noncommutative frames. The concept of noncommutative frame is due to Le Bruyn, who introduced it in connection with noncommutative covers of the Connes–Consani arithmetic site.

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L²(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L²(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L²(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.

We study a family of coherent states, called Schrödingerlets, both in the continuous and discrete setting. They are defined in terms of the Schrödinger equation of a free quantum particle and some of its invariant transformations.

We investigate the problems of reconstruction signals’ distinct subcomponents, that are approximately sparse in terms of morphologically different dictionaries, from a few noisy linear measurements. We show that under common assumptions on a restricted isometric property adapted to the composed dictionary and a mutual coherence condition between the two different dictionaries, the two distinct subcomponents can be stably and approximately recovered via the unconstrained $l_1$-split analysis, with an error bound that is in general unimprovable for the Gaussian noises.

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this paper we discuss the construction of frames for L²(ℝⁿ) using the action of closed reductive subgroups H ⊆ GL(n,ℝ) such that H has an open orbit in ℝⁿ under the action (h,ω) ↦ (h^-1)^T(ω). In particular, we show that if is an affine symmetric space, then there is a canonical way to construct frames.

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.

The aim of the paper is to obtain some generalizations of the so-called Plancherel–Polya inequalities which are also known as frame inequalities. By using these inequalities we show that a function f ∈ L_p(R), 1 ≤ p ≤ ∞, which is entire function of exponential type is uniquely determined by a set of numbers {Φ_j(f)}, j ∈ ℕ where {Φ_j}, j ∈ ℕ is a countable sequence of compactly supported distributions. In the case p = 2 we offer two reconstruction methods of a function f from a sequence of samples {Φ_j(f)}, j ∈ ℕ. The first reconstruction algorithm is given in terms of frames. To describe our second algorithm we introduce the so-called average variational splines.

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

We develop sampling expansion formulas on the shift invariant closed subspace V(ϕ) of L²ℝ generated by a frame or a Riesz generator ϕ(t). We find necessary and sufficient conditions under which a regular shifted sampling expansion to hold on V(ϕ) and also introduce a single channel sampling on V(ϕ) together with some illustrating examples.

Many techniques have been devised these last ten years to add an appropriate directionality concept in decompositions of images from the specific transformations of a small set of atomic functions. Let us mention, for instance, works on directional wavelets, steerable filters, dual-tree wavelet transform, curvelets, wave atoms, ridgelet packets, etc. In general, features that are best represented are straight lines or smooth curves as those defining contours of objects (e.g. in curvelets processing) or oriented textures (e.g. wave atoms, ridgelet packets). However, real images present also a set of details less oriented and more isotropic, like corners, spots, texture components, etc. This paper develops an adaptive representation for all these image elements, ranging from highly directional ones to fully isotropic ones. This new tool can indeed be tuned relatively to these image features by decomposing them into a Littlewood–Paley frame of directional wavelets with variable angular selectivity. Within such a decomposition, 2D wavelets inherit some particularities of the biorthogonal circular multiresolution framework in their angular behavior. Our method can therefore be seen as an angular multiselectivity analysis of images. Two applications of the proposed method are given at the end of the paper, namely, in the fields of image denoising and N-term nonlinear approximation.