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This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn. [ABSTRACT FROM AUTHOR]

We study optimal strategies in two-player stochastic games that are played on a finite graph, equipped with a general payoff function. The existence of optimal strategies that do not make use of memory and randomisation is a desirable property that vastly simplifies the algorithmic analysis of such games. Our main theorem gives a sufficient condition for the maximizer to possess such a simple optimal strategy. The condition is imposed on the payoff function, saying the payoff does not depend on any finite prefix (shift-invariant) and combining two trajectories does not give higher payoff than the payoff of the parts (submixing). The core technical property that enables the proof of the main theorem is that of the existence of ϵ -subgame-perfect strategies when the payoff function is shift-invariant. Furthermore, the same techniques can be used to prove a finite-memory transfer-type theorem: namely that for shift-invariant and submixing payoff functions, the existence of optimal finite-memory strategies in one-player games for the minimizer implies the existence of the same in two-player games. We show that numerous classical payoff functions are submixing and shift-invariant. [ABSTRACT FROM AUTHOR]

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Making machines learn like humans is the ultimate goal of artificial intelligence. Few-shot learning attempts to simulate the learning mechanism of humans, which is a task that can learn novel concepts from very few labeled samples. Due to the lack of sufficient labeled training data, existing methods often increase the risk of over-fitting or cause a considerable gap between clean and augmented data. To solve these problems, we present a novel compare network to perform robust few-shot learning in a meta-learned end-to-end manner. Specifically, we argue that it is desirable to learn a robust encoder that can draw inferences about other cases from one example. To this end, we improve the accuracy of few-shot learning by mining the internal mechanism of deep networks, which can leverage label information more effectively. By introducing shift-invariant blocks and a self-attention block in our architecture, all these components are seamlessly integrated into our framework, which can give feedback to each other without data augmentation. Furthermore, we provide ablative analyses of different blocks to help understand how each term contributes to performance. Extensive experiments demonstrate that our method can provide more robust results and outperform state-of-the-art few-shot learning methods.

Canonical polyadic decomposition (CPD) of multi-subject complex-valued fMRI data can be used to provide spatially and temporally shared components among groups with both magnitude and phase information. However, the CPD model is not well formulated due to the large subject variability in the spatial and temporal modalities, as well as the high noise level in complex-valued fMRI data. Considering that the shift-invariant CPD can model temporal variability across subjects, we propose to further impose a phase sparsity constraint on the shared spatial maps to denoise the complex-valued components and to model the inter-subject spatial variability as well. More precisely, subject-specific time delays are first estimated for the complex-valued shared time courses in the framework of real-valued shift-invariant CPD. Source phase sparsity is then imposed on the complex-valued shared spatial maps. A smoothed l0 norm is specifically used to reduce voxels with large phase values after phase de-ambiguity based on the small phase characteristic of BOLD-related voxels. The results from both the simulated and experimental fMRI data demonstrate improvements of the proposed method over three complex-valued algorithms, namely, tensor-based spatial ICA, shift-invariant CPD and CPD without spatiotemporal constraints. When comparing with a real-valued algorithm combining shift-invariant CPD and ICA, the proposed method detects 178.7% more contiguous task-related activations. [ABSTRACT FROM AUTHOR]

In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input $x(n)$ , we prove that the output of this system is equivalent to computing the first order derivative of $x(n)$. Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation. [ABSTRACT FROM AUTHOR]

In order to improve the accuracy of multi-focus image and infrared visible image fusion， based on the advantages of the good localization and shift invariance of the finite discrete shearlet transform, we propose a new image fusion algorithm，called FDST，which combines contrast with image gradient information correlation factor. Firstly, the FDST decomposes the registration images，and obtain the low frequency sub-band coefficients and high frequency sub-band coefficients of different scales and directions. The fusion principle o£ low frequency sub-band coefficients bases on the method o£ image gradient information correlation factor for weighting. The high frequency sub-band coefficients combine the high frequency and low frequency sub-band coefficients with contrast, and uses contrast as the criterion for choosing metric coefficients which is taken as the fusion rule. Finally，the low frequency information and high frequency information are reconstructed to obtain a fused image by the finite discrete shearlet inverse transform, and both subjective visual evaluation and objective performance assessments of the fusion results are implemented. The results indicate that the algorithm is superior to other fusion algorithms in subjective visual effects and objective evaluation. [ABSTRACT FROM AUTHOR]

We show that every closed shift-invariant subspace of L( K) is generated by the Λ-translates of a countable number of functions, where K is a local field of positive characteristic and Λ is an appropriate translation set. We use this result to provide a characterization of wavelets on such a field. [ABSTRACT FROM AUTHOR]

Abstract: A new model for describing a three-dimensional (3D) trajectory is proposed in this paper. The studied trajectory is viewed as a linear combination of rotatable 3D patterns. The resulting model is thus 3D rotation invariant (3DRI). Moreover, the temporal patterns are considered as shift-invariant. This paper is divided into two parts based on this model. On the one hand, the 3DRI decomposition estimates the active patterns, their coefficients, their rotations and their shift parameters. Based on sparse approximation, this is carried out by two non-convex optimizations: 3DRI matching pursuit (3DRI-MP) and 3DRI orthogonal matching pursuit (3DRI-OMP). On the other hand, a 3DRI learning method learns the characteristic patterns of a database through a 3DRI dictionary learning algorithm (3DRI-DLA). The proposed algorithms are first applied to simulation data to evaluate their performances and to compare them to other algorithms. Then, they are applied to real motion data of cued speech, to learn the 3D trajectory patterns characteristic of this gestural language. [Copyright &y& Elsevier]

The rotating machinery, as a typical example of large and complex mechanical systems, is prone to diversified sorts of mechanical faults, especially on their rotating components. Although they can be collected via vibration measurements, the critical fault signatures are always masked by overwhelming interfering contents, therefore difficult to be identified. Moreover, owing to the distinguished time-frequency characteristics of the machinery fault signatures, classical dyadic wavelet transforms (DWTs) are not perfect for detecting them in noisy environments. In order to address the deficiencies of DWTs, a pseudo wavelet system (PWS) is proposed based on the filter constructing strategies of wavelet tight frames. The presented PWS is implemented via a specially devised shift-invariant filterbank structure, which generates non-dyadic wavelet subbands as well as dyadic ones. The PWS offers a finer partition of the vibration signal into the frequency-scale plane. In addition, in order to correctly identify the essential transient signatures produced by the faulty mechanical components, a new signal impulsiveness measure, named spatial spectral ensemble kurtosis (SSEK), is put forward. SSEK is used for selecting the optimal analyzing parameters among the decomposed wavelet subbands so that the masked critical fault signatures can be explicitly recognized. The proposed method has been applied to engineering fault diagnosis cases, in which the processing results showed its effectiveness and superiority to some existing methods. [ABSTRACT FROM AUTHOR]

Wavelet transform has introduced innovative changes in different fields of science and engineering. One of its important applications is in the de-noising of signals and images. The wavelet packet (WP) is an efficient de-noising method, which has been used for ultrasonic testing (UT) signal de-noising; WP is a generalisation of the discrete wavelet transform (DWT). This paper proposes the use of undecimated wavelet transform (UWT) in implementing WP to overcome the limitation of the shift variance encountered in DWT. Wavelet transforms and their shift-invariance property are briefly explained. Simulations and experiments were carried out on a flow signal contaminated with a material's grain noise. Various wavelet transform processing parameters were investigated, including the number of decomposition levels, analysing wavelets and threshold settings. The results showed a superior de-noising effect of the proposed method over the conventional one. [ABSTRACT FROM AUTHOR]

Classical dictionary learning algorithms (DLA) allow unicomponent signals to be processed. Due to our interest in two-dimensional (2D) motion signals, we wanted to mix the two components to provide rotation invariance. So, multicomponent frameworks are examined here. In contrast to the well-known multichannel framework, a multivariate framework is first introduced as a tool to easily solve our problem and to preserve the data structure. Within this multivariate framework, we then present sparse coding methods: multivariate orthogonal matching pursuit (M-OMP), which provides sparse approximation for multivariate signals, and multivariate DLA (M-DLA), which empirically learns the characteristic patterns (or features) that are associated to a multivariate signals set, and combines shift-invariance and online learning. Once the multivariate dictionary is learned, any signal of this considered set can be approximated sparsely. This multivariate framework is introduced to simply present the 2D rotation invariant (2DRI) case. By studying 2D motions that are acquired in bivariate real signals, we want the decompositions to be independent of the orientation of the movement execution in the 2D space. The methods are thus specified for the 2DRI case to be robust to any rotation: 2DRI-OMP and 2DRI-DLA. Shift and rotation invariant cases induce a compact learned dictionary and provide robust decomposition. As validation, our methods are applied to 2D handwritten data to extract the elementary features of this signals set, and to provide rotation invariant decomposition. [ABSTRACT FROM PUBLISHER]

novel adaptive and approximate shift-invariant wavelet packet feature extraction scheme for event-related potentials (ERPs) in the electroencephalogram (EEG) is introduced in this paper. In this algorithm, the shift-invariant wavelet packed decomposition is done by integrating a cost function for decimation decision in each sub-band expansion. Additionally, a shape adaptation of the wavelet is implemented to find the best adapted wavelet shape for a given class of ERPs. This scheme is used to analyze the time course of the impact of single-pulse transcranial magnetic stimulation (TMS) to the auditory ERPs. We show that the proposed scheme is able to extract even slightest impacts of TMS, making it a promising tool for the extraction of weak ERPs components, particularly in hybrid TMS-EEG/ERP setups. [ABSTRACT FROM AUTHOR]

Abstract: An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function with Fourier transform for some g with . We explain why analysis of this particular generating function can be more likely to provide large jitter bounds ε such that any can be reconstructed from perturbed integer samples whenever . We use this natural deconvolution of to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of φ for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces. [Copyright &y& Elsevier]

The main focus of this paper is to derive a memory efficient approximation to the nonuniform Fourier transform of a support limited sequence. We show that the standard nonuniform fast Fourier transform (NUFFT) scheme is a shift invariant approximation of the exact Fourier transform. Based on the theory of shift-invariant representations, we derive an exact expression for the worst-case mean square approximation error. Using this metric, we evaluate the optimal scale-factors and the interpolator that provides the least approximation error. We also derive the upper-bound for the error component due to the lookup table based evaluation of the interpolator; we use this metric to ensure that this component is not the dominant one. Theoretical and experimental comparisons with standard NUFFT schemes clearly demonstrate the significant improvement in accuracy over conventional schemes, especially when the size of the uniform fast Fourier transform (FFT) is small. Since the memory requirement of the algorithm is dependent on the size of the uniform FFT, the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands. [ABSTRACT FROM AUTHOR]

This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple (A, M, Φ) where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively. The mapping Φ generalizes the concept of a z-transform to bijective linear mappings from a vector space of signal samples into the module M. Common concepts like filtering, spectrum, or Fourier transform have their equivalent counterparts in ASP. Once these concepts and their properties are defined and understood in the context of ASP, they remain true and apply to specific instantiations of the ASP signal model. For example, to develop signal processing theories for infinite and finite discrete time signals, for infinite or finite discrete space signals, or for multidimensional signals, we need only to instantiate the signal model to one that makes sense for that specific class of signals. Filtering, spectrum, Fourier transform, and other notions follow then from the corresponding ASP concepts. Similarly, common assumptions in SP translate into requirements on the ASP signal model. For example, shift-invariance is equivalent to A being commutative. For finite (duration) signals shift invariance then restricts A to polynomial algebras. We explain how to design signal models from the specification of a special filter, the shift. The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time. The latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform (DFT) and its variants as associated Fourier transforms. Finally, ASP provides a systematic methodology to derive fast algorithms for linear transforms. This topic and the application of ASP to space dependent signals and to multidimensional signals are pursued in companion papers. [ABSTRACT FROM AUTHOR]

We extend some of the classical results of the theory of multiresolution analysis (MRA) frames to Euclidean space ℝd; d > 1; and provide relevant examples. In the process, we use the theory of shift-invariant subspaces to bring new insights to the theory of frame multiresolution analysis. [ABSTRACT FROM AUTHOR]

Abstract: We developed a scheme to compute the depths of four idealized magnetized sources, viz., a monopole, a line of monopoles, a dipole and a line of dipoles from the Walsh spectra of the total-field magnetic anomalies. The sequency numbers, l max corresponding to the peaks of the differential energy density spectra over these sources are practically independent of the shape of the observed anomaly and are linearly dependent on the source depths. For each model, we derived a quantitative relation between the depth and the sequency number l max. Analyses of simulated data over idealized isolated sources reveal that (i) a profile length of about 8 times the source depth provides accurate value in computed depth; (ii) data spacing of less than one-fourth the source depth has no significant error in depth computation and (iii) the technique is capable of tolerating random error to the tune of 10% of the peak amplitude of the simulated anomaly. We compared the results of depth estimations from Walsh and Fourier spectra. Analysis of total-field magnetic anomalies over a buried water supply pipe has demonstrated the applicability of the proposed method. [Copyright &y& Elsevier]

Great strides have been taken in the last few years in the development of both approximate and exact reconstruction algorithms for helical cone-beam computed tomography (CT). However, it is hard to achieve a good balance between reconstruction speed, flexibility, and image quality. We propose a new algorithm that combines the advantages of many previously published algorithms. It uses the so-called hybrid convolution, which is the sum of the ramp and Hilbert filters. In this work, we evaluate the new algorithm and compare it to other candidates in terms of spatial resolution, noise, and image artifacts. Our evaluation demonstrated that the proposed algorithm outperforms the helical Feldkamp algorithm in terms of image noise uniformity and the cone beam artifact. We also propose a simplified version for the over-scan reconstruction. [ABSTRACT FROM AUTHOR]

Three-dimensional optical metamaterials based on multilayers typically rely on critical vertical alignment to achieve the desired functionality. Here the conditions under which three-dimensional metamaterials with different functionalities may be realized without constraints on alignment are analyzed and demonstrated experimentally. This study demonstrates that the release of alignment constraints for multilayered metamaterials is allowed, while their anomalous interaction with light is preserved., (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)