Fast and Stable Signal Deconvolution via Compressible State-Space Models

Abstract

Objective-Common biological measurements are in the form of noisy convolutions of signals of interest with possibly unknown and transient blurring kernels. Examples include EEG and calcium imaging data. Thus, signal deconvolution of these measurements is crucial in understanding the underlying biological processes. The objective of this paper is to develop fast and stable solutions for signal deconvolution from noisy, blurred and undersampled data, where the signals are in the form of discrete events distributed in time and space.Methods-We introduce compressible state-space models as a framework to model and estimate such discrete events. These state-space models admit abrupt changes in the states and have a convergent transition matrix, and are coupled with compressive linear measurements. We consider a dynamic compressive sensing optimization problem and develop a fast solution, using two nested Expectation Maximization algorithms, to jointly estimate the states as well as their transition matrices. Under suitable sparsity assumptions on the dynamics, we prove optimal HHS Public Access Author Manuscript Author ManuscriptAuthor ManuscriptAuthor Manuscript stability guarantees for the recovery of the states and present a method for the identification of the underlying discrete events with precise confidence bounds.Results-We present simulation studies as well as application to calcium deconvolution and sleep spindle detection, which verify our theoretical results and show significant improvement over existing techniques.Conclusion-Our results show that by explicitly modeling the dynamics of the underlying signals, it is possible to construct signal deconvolution solutions that are scalable, statistically robust, and achieve high temporal resolution.Significance-Our proposed methodology provides a framework for modeling and deconvolution of noisy, blurred, and undersampled measurements in a fast and stable fashion, with potential application to a wide range of biological data.

Multiplicative Updates for Optimization Problems with Dynamics

“…Here the baseline is assumed to have been estimated and subtracted separately, but can be estimated similarly. We refer to [5] for details on estimation of the unknown parameters σ 2 and θ and a list of methods used for calcium deconvolution. These approaches require solving convex optimization problems, which do not scale well with the temporal dimension of the data.…”

“…One can use spatial regularization on elements of w t in this setup as well as compressive sensing regimes for when A satisfies the restricted isometry property RIP [5]. We refer to [5] for a more detailed discussion.…”

“…Such optimization problems arise in many applications of interest. Examples include nonnegative matrix factorization for images of objects [1], Poisson image reconstruction [2], point process smoothing for stimulus-response experiments in neurophysiology [3], nonnegative least squares [4] and nonnegative calcium deconvolution [5]. In this paper we will use the KKT conditions [6] to provide a unified framework for solving such optimization problems with nonnegativity constraints.…”

Statistical Model of Centimeter Range Signals That Have Passed The Surface and Space Propagation Path

“…It is known that the output signals of antennas on the tropospheric and space propagation paths of radio waves can be expressed as the sum of the radio emission source signal and the scattered component due to medium inhomogeneities [1][2][3][4][5]. While scattering of radio waves by small-scale inhomogeneities in the troposphere, the scattered component is described by a stationary normal random process, when the observation interval of the signal does not exceed several seconds and the transmitting points are fixed [3,5].…”

Statistical Model of Centimeter Range Signals That Have Passed Surface and Space Propagation Path

“…It is known that the output signals of antennas on the tropospheric and space propagation paths of radio waves can be expressed as the sum of the radio emission source signal and the scattered component due to medium inhomogeneities [1][2][3][4][5]. While scattering the radio waves by small-scale inhomogeneities in the troposphere, the scattered component is described by a stationary normal random process, when the observation interval of the signal does not exceed several seconds and the transmitting points are fixed [3,5].…”