Signal enhancement-a composite property mapping algorithm

Abstract

A signal enhancement algorithm is developed which seeks to recover a signal from noise-contaminated distorted measurements made on that signal. This ob,ject is achieved bj utilizing a set of properties which the signal is known o r is hypothesized as possessing. The measured signal is modified to the smallest degree necessary so as to sequentially possess each of the individual properties. Conditions for the algorithm's convergence are established in which the primary requirement is that a composite property mapping be closed. This is a relatively unrestrictive condition in comparison to that required of most existing signal enhancement algorithms. One of the more interesting and important uses of signal enhancement is concerned with applications in which the signal being enhanced is represented in a data or correlation matrix format. I n such applications, the eigencharacterization and structure (e.&, Toeplitz) of the matrix often provides useful signal properties. Several important property mappings for finding a matrix that possesses a specified linear structure or eigencharacterization and which lies closest to a given matrix are developed. It is shown that each of these property mappings is closed and can therefore be used as candidate property mappings in the proposed signal enhancement algorithm.

M-Estimate robust PCA for Seismic Noise Attenuation

“…In these methods [8,9,10,11], data are first transformed to the Fourier domain in the temporal direction (also known as Fx transform [7]) and then for every frequency slice, rank reduction is applied on the Hankel matrix to remove incoherent signals. Cadzow's algorithm [12], popular in array processing, can also be applied for further noise reduction [9]. While these methods perform well under Gaussian noise assumption, the results are highly degraded when there are outliers in the data.…”

“…Here H(D ωi (j)) computes the Hankel matrix of the vector of components of the frequency slice, R K computes the K rank approximation U K × Σ K × V H K , where Σ K is the diagonal matrix containing the singular values up to K, and A computes the Cadzow's algorithm which is simply anti-diagonal averaging of the Hankel matrix [12]. Although both mathematically elegant and popular with practitioners, this method is extremely sensitive to outliers.…”

Singular spectrum analysis based on the minimum variance estimator

“…A general framework for recovering noise free series has been presented in [5]. The method forms the basis for a very general class of subspace-based noise reduction algorithms, is based on the assumption that the original time series exhibits some well-defined properties or obeys a certain model.…”

Robust Structured Low-Rank Approximation on the Grassmannian

“…An obvious drawback of this method is that this projection can destroy the low-rank property established before. The method of Cadzow [6] alternates between these two steps until the algorithm converges to a solution that is indeed low-rank and structured. However, as Chu et al [8] and Markovsky [12] state, this solution can be far away from the initialization with no guarantees of finding an actually meaningful approximation to the data.…”