Continuous-Domain Signal Reconstruction Using $L_{p}$-Norm Regularization

Abstract

We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some Lp-norm regularization. More specifically, we consider p ≥ 1 and the multi-order derivative regularization operator L = D N 0. We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternatingdirection method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the Lp-norm regularization, namely, sparsity, regularity, and oscillatory behavior.

A joint DOA and polarization estimation method based on the conformal polarization sensitive array from the sparse reconstruction perspective

“…Therefore, the observation matrix X with the snapshots is T can be presented as It is noted that à is called the overcomplete dictionary and the number of à columns is much larger than that of rows. Similar to (8), it can be found that à not only depends spatial parameters but also depends polarization parameters. S is called sparse direction weights, each sparse weight has non-zero value only at the true source directions.…”

Sampling and Reconstruction of Sparse Signals in Shift-Invariant Spaces: Generalized Shannon's Theorem Meets Compressive Sensing

“…the vector of expansion coefficients d k has only a few nonzero entries. Such an assumption is exploited in [39]- [43] where the observed signals lie in the L-spline and B-spline spaces. However, in this paper, we assume the expansion coefficients d k are sparse in a certain transform domain Ψ ∈ R L×L .…”

“…The theorem states that the signal can be recovered from M linear measurements, where M ≥ (Q + N 0 ) and N 0 is a spline order. Papers [40]- [43] rely on the main results of the method proposed in [39] and use grid-based discretization strategies that lead to convex optimization problems.…”

Polarization Direction of Arrival Estimation for Vector Array of Unmanned Aerial Vehicle Swarm

“…Thus, placing electromagnetic vector antennas on multiple UAVs can constitute a new type of UAV swarm vector array. Several attempts, such as the application of p-norm (0 ≤ p ≤ 1) methods [21][22][23][24][25], orthogonal matching pursuit (OMP) methods [26,27], and the sparse Bayesian learning (SBL) methods [28][29][30][31], have aroused a lot of attention in DOA estimation. The essential idea of these algorithms is that the directions of incident sources are substantially sparse in the spatial domain, which is intrinsically different from the subspace-based algorithms.…”