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.
