PENDANTSS: PEnalized Norm-Ratios Disentangling Additive Noise, Trend and Sparse Spikes

Denoising, detrending, deconvolution: usual restoration tasks, traditionally decoupled. Coupled formulations entail complex ill-posed inverse problems. We propose PENDANTSS for joint trend removal and blind deconvolution of sparse peak-like signals. It blends a parsimonious prior with the hypothesis that smooth trend and noise can somewhat be separated by low-pass filtering. We combine the generalized quasi-norm ratio Smoothed One-Over-Two/Smoothed
$p$
-Over-
$q$
(SOOT/SPOQ) sparse penalties
$\ell _{p}/\ell _{q}$
with the Baseline Estimation And Denoising with Sparsity (BEADS) ternary-assisted source separation algorithm. This results in a both convergent and efficient tool, with a novel Trust-Region block alternating variable metric forward-backward approach. It outperforms comparable methods, when applied to typically peaked analytical chemistry signals. Reproducible code is provided.