Qiyu Sun

Sampling and reconstruction of signals in a reproducing kernel subspace of Lp(Rd)

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of Lp(Rd), 1⩽p⩽∞, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of p-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of Lp(Rd). We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation–projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap.

Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2=L2(R) with dilation integer factor M≥2, the standard “matrix extension” approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M=2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M=2 to arbitrary integer M≥2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M−1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M−1 such frame generators. Linear spline examples are given for M=3,4 to demonstrate our constructive approach.

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(ϕ1, ..., ϕN) generated by finitely many compactly supported functions ϕ1, ..., ϕN, we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(ϕ1, ..., ϕN) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(ϕ) generated by a compactly supported refinable function ϕ, we prove that for almost all

A Unified Formulation of Gaussian Versus Sparse Stochastic Processes—Part I: Continuous-Domain Theory

(x_0, x_1)in [0,1]^2

Frames in spaces with finite rate of innovation

, any signal in V(ϕ) can be locally reconstructed from its samples from

Localized nonlinear functional equations and two sampling problems in signal processing

{x_0, x_1}+{mathbb Z}

Left-inverses of fractional Laplacian and sparse stochastic processes

with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(ϕ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.

Function Spaces for Sampling Expansions

We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson, or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern, which is determined by the type of noise excitation. The latter is fully specified by a Lévy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some Lp bound for p ≥ 1. We present a novel operator-based method that yields an explicit characterization of all Lévy-driven processes that are solutions of constant-coefficient stochastic differential equations. When the underlying system is stable, we recover the family of stationary continuous-time autoregressive moving average processes (CARMA), including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Lévy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.

Convolution sampling and reconstruction of signals in a reproducing kernel subspace

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space

Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

V_q(Phi, Lambda)