Shift-invariant Spaces Research Articles

Distortion-Free Intelligent Sampling of Sparse Surfaces Via Locally Refined T-Spline Metamodelling

Automatic design of the number of sample points and sample locations when measuring surfaces with different geometries is of critical importance to enable autonomous manufacturing. Uniform sampling has been widely used for simple geometry measurement, e.g. planes and spheres. However, there is a lack of appropriate sampling techniques that can be applied to complex freeform surfaces, especially those with sparse topographical features, e.g. cutting edges and other high-curvature features. In this paper, a distortion-free intelligent sampling and reconstruction method with improved efficiency for sparse surfaces is proposed. In this method, a locally-refined T-spline approximation is firstly applied which maps a surface to a simplified T-spline space; then a shift-invariant space sampling method and corresponding reconstruction are applied for the surface measurement. This sampling strategy provides a cost-effective sampling design and guarantees the surface reconstruction without information loss in a T-spline space. Theoretical demonstrations and case studies show that this sampling strategy can provide up to an order of magnitude improvement in accuracy or efficiency over state-of-the-art methods, for the measurement of sparse surfaces, from macro- to nano-scales.

A note on continuous stable sampling

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

Sampling and Average Sampling in Quasi Shift-Invariant Spaces

In this paper, we study the sampling and average sampling problem in a quasi shift-invariant space where X is a discrete subset of and is a continuously differentiable positive definite function satisfying certain decay conditions. We show that any f belonging to can be uniquely and stably reconstructed from its samples as well as from its average samples provided sampling points are close enough. Further, iterative reconstruction algorithms for reconstruction of a function f belonging to from its samples as well as from its average samples are also provided.

Spaces Generated by Orbits of Unitary Representations: A Tribute to Guido Weiss

We present a survey of research on orbits of unitary representations developed under the guidance and leadership of Guido Weiss in the last 20 years. This line of research started via the study of wavelets and other reproducing function systems. It emphasizes the role of translations in such systems and leads naturally to the study of shift invariant spaces. The algebraic properties of integers influence many results in such spaces and suggest a more abstract nature of the entire approach. Unitary representations of various discrete groups provide far reaching generalizations of basic results. This topic is currently actively pursued by numerous authors, many of them being Guido’s collaborators for years.

Regular and Irregular Sampling Linear Transforms in Series of Shift-Invariant Spaces

Regular and Irregular Sampling Linear Transforms in Series of Shift-Invariant Spaces

Shift invariant spaces in $$L^2({\mathbb {R}},{\mathbb {C}}^m)$$ with m generators

The paper deals with sampling and reconstruction of vector valued functions in a shift invariant space with multiple generators. Unlike the case of a shift invariant space with multiple generators in $$L^{2}(\mathbb {R})$$, when the dimension of the vectors is the same as the number of generators, $$\mathbb {Z}$$ turns out to be a stable set of sampling. A sampling formula for reconstructing a function from its samples at integer points is derived and the problem of sampling on a perturbed set of integers is discussed. An illustration of sampling and reconstruction of a function in $$L^{2}(\mathbb {R},\mathbb {R}^{2})$$ on a finite interval is given using Matlab.

Random Phaseless Sampling for Causal Signals in Shift-Invariant Spaces: A Zero Distribution Perspective

We proved that the phaseless sampling (PLS) in the linear-phase modulated shift-invariant space (SIS) $V(e^{\textbf{i}\alpha \cdot}\varphi), \alpha\neq0,$ is impossible even though the real-valued function $\varphi$ enjoys the full spark property (so does $e^{\textbf{i}\alpha \cdot}\varphi$). Stated another way, the PLS in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued generator $\phi$ such that the PLS of nonseparable causal (NC) signals in $V(\phi)$ can be achieved by random sampling. The condition is established from the generalized Haar condition (GHC) perspective. Based on the proposed reconstruction approach, it is proved that if the GHC holds then with probability $1$, the random sampling density (SD) $=3$ is sufficient for the PLS of NC signals in the complex-generated SISs. For the real-valued case we also prove that, if the GHC holds then with probability $1$, the random SD $=2$ is sufficient for the PLS of real-valued NC signals in the real-generated SISs. For the local reconstruction of highly oscillatory signals such as chirps, a great number of deterministic samples are required. Compared with deterministic sampling, the proposed random approach enjoys not only the greater sampling flexibility but the much smaller number of samples. To verify our results, numerical simulations were conducted to reconstruct highly oscillatory NC signals in the chirp-modulated SISs.

The structure of finitely generated shift-invariant spaces in mixed Lebesgue spaces $$L^{p,q}\left( {\mathbb {R}}^{d+1}\right) $$

In this paper, we investigate the structure of finitely generated shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$. We characterize shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ in terms of the semi-discrete convolutions of their generators with sequences in $$\ell ^{p,q}\left( \mathbb {Z}^{d+1}\right) $$. In addition, if the generators of shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ are compactly supported, we prove that shift-invariant spaces in $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ are the intersection of $$L^{p,q}\left( \mathbb {R}^{d+1}\right) $$ and the linear space of functions formed with the semi-discrete convolutions of their generators with sequences on $$\mathbb {Z}^{d+1}$$.

Reconstruction from convolution random sampling in local shift invariant spaces

In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the available data are the sampled convolution of the original function on the sampling set. We obtain an explicit reconstruction formula. This reconstruction formula succeeds with overwhelming probability when the sampling size is sufficiently large.

Left translates of a square integrable function on the Heisenberg group

The aim of this paper is to study some properties of left translates of a square integrable function on the Heisenberg group. First, a necessary and sufficient condition for the existence of the canonical dual to a function \(\varphi \in L^{2}(\mathbb {R}^{2n})\) is obtained in the case of twisted shift-invariant spaces. Further, characterizations of \(\ell ^{2}\)-linear independence and the Hilbertian property of the twisted translates of a function \(\varphi \in L^{2}(\mathbb {R}^{2n})\) are obtained. Later these results are shown in the case of the Heisenberg group.

Invertibility of Laurent operators and shift invariant spaces with finitely many generators

In this paper, it is shown that for a fixed is a stable set of sampling for if and only if there exists such that a.e. on , where is a symbol matrix of a block Laurent operator associated with the sample set of . A similar result is proved when where and is a stable set of sampling for (single generator) as well as (multi generators ). A perturbation of stable set of sampling is also discussed.

Sharp Results on Sampling with Derivatives in Shift-Invariant Spaces and Multi-Window Gabor Frames

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.

Random sampling and reconstruction in multiply generated shift-invariant spaces

Shift-invariant spaces play an important role in approximation theory, wavelet analysis, finite elements, etc. In this paper, we consider the stability and reconstruction algorithm of random sampling in multiply generated shift-invariant spaces [Formula: see text]. Under some decay conditions of the generator [Formula: see text], we approximate [Formula: see text] with finite-dimensional subspaces and prove that with overwhelming probability, the stability of sampling set conditions holds uniformly for all functions in certain compact subsets of [Formula: see text] when the sampling size is sufficiently large. Moreover, we show that this stability problem is connected with properties of the random matrix generated by [Formula: see text]. In the end, we give a reconstruction algorithm for the random sampling of functions in [Formula: see text].

Multivariate Dynamical Sampling in and Shift-Invariant Spaces

In this article, we mainly study the multivariate dynamical sampling in and shift-invariant spaces. We first provide a sufficient and necessary condition for which can be reconstructed by its spatial and temporal samples. Then we give a concrete example to show that the sufficient and necessary condition is feasible. Finally, we discuss the multivariate dynamical sampling problem in shift-invariant spaces.

Time-frequency shift invariance of Gabor spaces generated by integer lattices

We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. The extreme cases of full translation invariance, full modulation invariance, and full time-frequency shift invariance are also considered. The results show a close analogy with the extra translation invariance of shift-invariant spaces.

Recovery of Compressed Sampling in Shift-Invariant Spaces Base on L1 Norm

Compressed sampling in shift-invariant spaces (SI) is an effective method for sampling of sparse signals. But, reconstruction of compressed sampling may be unstable. In the paper, the possibility of stable reconstruction under a sufficient sparsity is proven. Further, we consider the situation where the minimal L1 norm is used to recover sparse signals from the noisy data. The result shows that they are stable. Finally, we show that the minimal L1 norm through the simulation, and explain the applicability of our algorithm to sampling systems.

Phase retrieval of real-valued signals in a shift-invariant space

Phase retrieval arises in various fields of science and engineering and it is well studied in a finite-dimensional setting. In this paper, we consider an infinite-dimensional phase retrieval problem to reconstruct real-valued signals living in a shift-invariant space from its phaseless samples taken either on the whole line or on a set with finite sampling rate. We find the equivalence between nonseparability of signals in a linear space and its phase retrievability with phaseless samples taken on the whole line. For a spline signal of order $N$, we show that it can be well approximated, up to a sign, from its noisy phaseless samples taken on a set with sampling rate $2N-1$. We propose an algorithm to reconstruct nonseparable signals in a shift-invariant space generated by a compactly supported continuous function. The proposed algorithm is robust against bounded sampling noise and it could be implemented in a distributed manner.

Relevant sampling in finitely generated shift-invariant spaces

We consider random sampling in finitely generated shift-invariant spaces V(Φ)⊂L2(Rn) generated by a vector Φ=(φ1,…,φr)∈(L2(Rn))r. Following the approach introduced by Bass and Gröchenig, we consider certain relatively compact subsets VR,δ(Φ) of such a space, defined in terms of a concentration inequality with respect to a cube with side lengths R. Under very mild assumptions on the generators, we show that for R sufficiently large, taking O(Rnlog(R)) many random samples (taken independently uniformly distributed within CR) yields a sampling set for VR,δ(Φ) with high probability. We give explicit estimates of all involved constants in terms of the generators φ1,…,φr.

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We introduce an undirected graph to a signal and use connectivity of the graph to characterize whether the signal can be determined, up to a sign, from its magnitude measurements on the whole Euclidean space. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in the shift-invariant space, that are determined from their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. In this paper, we also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robust reconstruction of box spline signals from their noisy phaseless samples.

Sampling and Reconstruction in a Shift Invariant Space with Multiple Generators

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that cannot be a stable set of sampling for where when Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation.