Blind Hierarchical Deconvolution

Abstract

Deconvolution is a fundamental inverse problem in signal processing and the prototypical model for recovering a signal from its noisy measurement. Nevertheless, the majority of model-based inversion techniques require knowledge on the convolution kernel to recover an accurate reconstruction and additionally prior assumptions on the regularity of the signal are needed. To overcome these limitations, we parametrise the convolution kernel and prior length-scales, which are then jointly estimated in the inversion procedure. The proposed framework of blind hierarchical deconvolution enables accurate reconstructions of functions with varying regularity and unknown kernel size and can be solved efficiently with an empirical Bayes two-step procedure, where hyperparameters are first estimated by optimisation and other unknowns then by an analytical formula.

Hierarchical Deconvolution for Incoherent Scatter Radar Data

“…3.1) of P and therefore must be estimated. Following Arjas et al (2020b), we solve the inverse problem of computing P in equation ( 7) in two parts (described in more detail in Sect. 4):…”

“…This could be done for instance by available optimisation algorithms, such as the limited memory BFGS algorithm (Liu and Nocedal, 1989) implemented in the R function optim(), which computes the derivatives automatically by finite differences. This approach was taken in Arjas et al (2020b), but creates a computational bottleneck. In this study, to improve reconstruction times we specify the gradient of the target function explicitly for a more efficient computation, instead of time consuming finite-difference approximations of the gradients.…”

“…An automatic system to optimize the length-scales is thus desirable. In this paper we use hierarchical prior models following from Arjas et al (2020b) in deconvolution of ISR power profiles in presence of strong, narrow layers. Our aim is to study the performance of these prior models in deconvolution of ISR data and to show that they are a potential solution to the problem of variable length-scales, without the need to explicit control the weighting, that is, the length-scale estimate is 2 https://doi.org/10.5194/amt-2021-287 Preprint.…”

Hierarchical deconvolution for incoherent scatter radar data

“…An automatic system to optimise the length scales is thus desirable. In this paper we use hierarchical prior models following from Arjas et al (2020b) in deconvolu-tion of ISR power profiles in the presence of strong, narrow layers. Our aim is to study the performance of these prior models in deconvolution of ISR data and to show that they are a potential solution to the problem of variable length scales, without the need to explicitly control the weighting; that is, the length-scale estimate is adaptively estimated from the profiles during an efficient optimisation approach.…”

Comment on amt-2021-287

“…An automatic system to optimize the length-scales is thus desirable. In this paper we use hierarchical prior models following from Arjas et al (2020b) in deconvolution of ISR power profiles in presence of strong, narrow layers. Our aim is to study the performance of these prior models in deconvolution of ISR data and to show that they are a potential solution to the problem of variable length-scales, without the need to explicit control the weighting, that is, the length-scale estimate is adaptively estimated from the profiles during an efficient optimisation approach.…”