In this paper, we investigate the spherical t-designs with large value of t for function approximation, construction of spherical framelets, and the important task of spherical signal processing. Based on the spherical framelet systems and the fast framelet transform algorithms, we propose an effective denoising scheme for spherical signal denoising that utilizes the nice properties of spherical t-designs with large t value. We provide numerical results of signal/image denoising on several data sets.
Shannon’s sampling theorem plays a central role in the discrete-time processing of bandlimited signals. However, the infinite precision assumed by Shannon’s theorem is impractical because of the ADC clipping effect that limits the signal’s dynamic range. Moreover, the power consumption of an analog-to-digital converter (ADC) increases linearly with the sampling frequency and may be prohibitively high for a wide bandwidth signal. Recently, unlimited and one-bit sampling frameworks have been proposed to address these shortcomings. The former is a high-resolution technique that employs self-reset ADCs to achieve an unlimited dynamic range. The latter achieves relatively low cost and reduced power consumption at an elevated sampling rate. In this paper, we examine jointly exploiting the appealing attributes of both techniques. We propose unlimited one-bit (UNO) sampling, which entails a judicious design of one-bit sampling thresholds. This enables storing the distance between the input signal value and the threshold. We then utilize this information to accurately reconstruct the signal from its one-bit samples via a randomized Kaczmarz algorithm (RKA) which is considered to be a strong linear feasibility solver that selects a random linear equation in each iteration. The numerical results illustrate the effectiveness of RKA-based UNO over the state-of-the-art.
In this work, we consider the problem of recovering a sparse signal consisting of a sum of filtered spikes from the output of a time encoding machine (TEM). This problem was addressed before with recovery methods designed for filters with specific shapes, mostly relying on Prony’s method for recovery. Here we propose a new recovery method for sparse inputs from TEM samples. Compared to existing approaches, the new method relaxes significantly the assumption on the filters. The method is associated by a theoretically guaranteed algorithm. We provide numerical examples to evaluate the new method, including an example with filters that are not compatible with previous methods.
It has been observed by several authors (see [8], [17], [26]) that well-known periodization strategies like tent or Chebyshev transforms lead to remarkable results for the recovery of multivariate functions from few samples. So far, theoretical guarantees are missing. The goal of this paper is twofold. On the one hand, we give such guarantees and briefly describe the difficulties of the involved proof. On the other hand, we combine these periodization strategies with recent novel constructive methods for the efficient subsampling of finite frames in ${\mathbb{C}^m}$. As a result we are able to reconstruct non-periodic multivariate functions from very few samples. The used sampling nodes are the result of a two-step procedure. Firstly, a random draw with respect to the Chebyshev measure provides an initial node set. A further sparsification technique selects a significantly smaller subset of these nodes with equal approximation properties. This set of sampling nodes scales linearly in the dimension of the subspace on which we project and works universally for the whole class of functions. The method is based on principles developed by Batson, Spielman, and Srivastava [4] and can be numerically implemented. Samples on these nodes are then used in a (plain) least-squares sampling recovery step on a suitable hyperbolic cross subspace of functions resulting in a near-optimal behavior of the sampling error. Numerical experiments indicate the applicability of our results.
We develop connections between some of the most powerful theories in analysis, tying the Shannon sampling formula to Cauchy’s integral and residue formulae, Jacobi interpolation, and Levin’s sine-type functions. The techniques use tools from complex analysis, and in particular, the Cauchy theory and the theory of entire functions, to realize sampling sets Λ as zero sets of well-chosen entire functions (sampling set generating functions). We then reconstruct the signal from the set of samples using the Cauchy-Jacobi machinery. These methods give us powerful tools for creating a variety of general sampling formulae, e.g., allowing us to derive Shannon sampling and Papoulis generalized sampling via Cauchy theory and sampling in radial domains.
We provide an example for the generating matrix A of a two-dimensional lattice Γ = Aℤ 2 , such that the following holds: For any sufficiently smooth and localized mother wavelet ψ, there is a constant β(A,ψ) > 0, such that βΓ∩(ℝ×ℝ + ) is a set of stable sampling for the wavelet system generated by ψ, for all 0 < β ≤ β(A,ψ). The result and choice of the generating matrix are loosely inspired by the studies of low discrepancy sequences and uniform distribution modulo 1. In particular, we estimate the number of lattice points contained in any axis parallel rectangle of fixed area. This estimate is combined with a recent sampling result for continuous wavelet systems, obtained via the oscillation method of general coorbit theory.
Many techniques in machine learning attempt explicitly or implicitly to infer a low-dimensional manifold structure of an underlying physical phenomenon from measurements without an explicit model of the phenomenon or the measurement apparatus. This paper presents a cautionary tale regarding the discrepancy between the geometry of measurements and the geometry of the underlying phenomenon in a benign setting. The deformation in the metric illustrated in this paper is mathematically straightforward and unavoidable in the general case, and it is only one of several similar effects. While this is not always problematic, we provide an example of an arguably standard and harmless data processing procedure where this effect leads to an incorrect answer to a seemingly simple question. Although we focus on manifold learning, these issues apply broadly to dimensionality reduction and unsupervised learning.
Graph models and graph-based signals are becoming increasingly important in machine learning, natural sciences, and modern signal processing. In this paper, we address the problem of quantizing bandlimited graph signals. We introduce two classes of noise-shaping algorithms for graph signals that differ in their sampling methodologies. We demonstrate that these algorithms can be efficiently used to construct quantized representatives of bandlimited graph-based signals with bounded amplitude. Moreover, for one of the algorithms, we provide theoretical guarantees on the relative error between the quantized representative and the true signal.
When solving an optimization problem over the set of graph Laplacian matrices, one must deal with a large number of constraints as well as the large objective variable size. In this paper we explore first-order methods for optimization over graph Laplacian matrices. These methods include two popular methods for constrained optimization: the mirror descent algorithm and the Frank-Wolfe (conditional gradient) algorithm. We derive efficiently implementable formulations of these algorithms over graph Laplacians, and use existing theory to show their iteration complexity in various regimes. Experiments demonstrate the efficiency of these methods over alternatives like interior point methods.
In this paper, we provide lower bounds on the L 2 -error of approximation of arbitrary functions f : [0, 1] → ℝ by polynomials of degree at most n, with the constraint that the coefficients of these polynomials in the Bernstein basis of order n are bounded by n α for some α ≥ 0. For Lipschitz functions, this lower bound matches, up to a factor of $\sqrt {\log n} $, a previously obtained constructive upper bound for the error of approximation by one-bit polynomials in Bernstein form via Σ∆ quantization where the functions are bounded by 1 and the coefficients of the approximating polynomials are constrained to be in {±1}.
In this paper, we investigate the spherical t-designs with large value of t for function approximation, construction of spherical framelets, and the important task of spherical signal processing. Based on the spherical framelet systems and the fast framelet transform algorithms, we propose an effective denoising scheme for spherical signal denoising that utilizes the nice properties of spherical t-designs with large t value. We provide numerical results of signal/image denoising on several data sets.
Data science research has found great success with algorithms that leverage the structure of the topological space that the high-dimensional data lies on. In particular, low-rank tensor models which represent low-dimensional latent factors in a succinct and parsimonious way have become indispensable tools. These low-rank models have been utilized in a variety of applications including tensor completion from corrupted or missing entries. In the standard tensor completion problem, the different modes of the tensor are assumed to be completely independent of each other. However, in many real-world problems such as those involving spatio-temporal data, there exist relationships between the different modes. This information can be encoded in terms of graphs which can bring additional structure to the tensor completion problem. In this paper, we introduce methods for structured tensor completion where both the low-rank and smoothness of tensor are incorporated into the optimization problem. In particular, we model tensor data as graph signals on Cartesian product graphs and use the Dirichlet energy to quantify the smoothness of tensor data with respect to the graph. We evaluate the performance of this tensor recovery approach for different types of data, i.e. low-rank, smooth and low-rank plus smooth, and compare with existing methods.
This paper presents a characterization of systems of iterations that generate frames of abstract separable Hilbert spaces. The characterization is achieved through a correspondence with a canonical system of iterations that form Parseval frames of certain subspaces of the space of vector-valued functions ${L^2}(\mathbb{T},\mathcal{K})$, where K is a Hardy space with multiplicity. These subspaces possess the property of being invariant under two shift operators with multiplicity.Furthermore, we provide a clear description of the subspaces generated by these canonical systems of iterations.
Recently, a versatile library of quasi-analytic complex-valued wavelet packets (WPs) which originate from splines of arbitrary orders, was designed [1]. The real parts of the quasi-analytic WPs (qWPs) are the regular spline-based orthonormal WPs. The imaginary parts, which are slightly modified Hilbert transforms of the real parts, are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, are antisymmetric. Both regular and complementary WPs are well localized in time domain and their DFT spectra provide a variety of refined splits of the frequency domain. The waveforms can have arbitrary number of vanishing moments.Tensor products of 1D quasi-analytic WPs (qWPs) provide a diversity of 2D waveforms oriented in multiple directions. The designed computational scheme enables us to get fast and easy implementation of the qWP transforms. The shapes of real and imaginary parts of the qWPs can be regarded as directional cosine waves with different frequencies modulated by localized low-frequency signals. For example, the set of the fourth-level WPs comprises waveforms which are oriented in 314 different directions and are oscillating with 256 different frequencies. Various combinations of qWPs form multiple frames in the 2D signal space.The combination of the exceptional properties of the designed qWPs, such as unlimited directionality and oscillating structure of the waveforms, vanishing moments and refined frequency resolution, make them a powerful tool for image processing applications. The algorithms based on the qWPs proved to be competitive with the best existing methods in solving such classical image processing problems as denoising, inpainting and deblurring. The qWP algorithms are especially efficient for capturing edges and fine texture and oscillating patterns even in severely degraded images.Due to the above properties and next to unlimited diversity of testing waveforms, the qWPs have strong capabilities for extraction characteristic features from signals and images, which are utilized in the image classification algorithms in conjunction with Support Vector Machines and Convolutional Neural Networks.
We propose a new sampling theorem, the complete reconstruction of a function from its samples, for the space of variable bandwidth constructed using Wilson expansions. The theorem is based on the maximal gap between consecutive points and it relates the lower sampling rate to the bandwidths that have an influence on the reconstruction on each particular interval of the signal.
The fractional Fourier transform, denoted by F θ , which is a generalization of the Fourier transform, depends on a parameter 0 ≤ θ ≤ π/2, so that when θ = 0, F 0 is the identity transformation and when θ = π/2, F π/2 is the standard Fourier transform. The transform has been extended to higher dimensions by taking tensor products of one-dimensional transforms.In 2018 the author of this article introduced a novel generalization of the fractional Fourier transform to two dimensions, which is called the coupled fractional Fourier transform and is denoted by F α,β . This transform depends on two independent angles α and β, with 0 ≤ α, β ≤ π/2, so that F 0,0 is the identity transformation and F π/2,π/2 , is the two-dimensional Fourier transform. For other values of α and β, we obtain other interesting configurations of the transform. One immediate application of this transform is in time-frequency representation because of its close relationship to the Wigner distribution function.The goal of this article is to extend the transform to a space of generalized functions and then introduce a sampling theorem for signals that are bandlimited in the domain of the transform.
In this paper we prove that all even Schwartz functions f : ℝ → ℝ are uniquely determined by their values at $\left\{ {|\hat f( \pm \sqrt n )|,f( \pm \sqrt {n/2} )} \right\}$ for n ≥ 0 indexing the non-negative integers.
The present article proposes to reconstruct a bandlimited signal from nonuniform samples by a sliding periodization of the nonuniformity and successive approximations. When the nonuniformity consists of bounded deviations of the sampling instants from a uniform grid, the reconstruction can be made arbitrarily accurate either by increasing the period of the periodizations or by increasing the number of successive approximations.
While event-based sampling allows the use of sampling circuits of higher precision and lower power consumption, it faces the difficult problem of signal reconstruction from generalized nonuniform samples. An ideal solution to this problem is to perform the pseudo-inversion of the linear operator that maps the input signals into the sequences of samples. We show in this article that this is possible with all time-encoding schemes based on input integration, using the method of projection onto convex sets (POCS). This includes multi-channel time encoding.
In conventional analog-to-digital (ADC) conversion, the sampling and quantization steps take place on the time and amplitude axes, respectively. In the case of time encoding machines (TEMs), which convert analog signals into a sequence of time events, sampling and quantization interfere with one another since they both operate on the time axis. Here we introduce a new quantization method for TEMs called QTEM that, due to its model-driven nature, limits the interference of sampling and quantization. We show that existing recovery guarantees don’t apply to QTEM. We provide new guarantees for recovering the input of the QTEM and demonstrate numerically its advantage over conventional TEM quantization.
