Inverse problems with time-frequency dictionaries and non-white Gaussian noise

Sparse regressions to solve ill-posed inverse problems have been massively investigated over the last decade. Yet, when noise is present in the model, it is almost exclusively considered as Gaussian and white. While this assumption can hold in practice, it rarely holds when observations are time series as they are corrupted by auto-correlated and colored noise. In this work we study sparse regression under the assumption of non white Gaussian noise and explain how to run the inference using proximal gradient methods. We investigate an application in brain imaging: the problem of source localization using magneto- and electroencephalography which allow functional brain imaging with high temporal resolution. We use a time-frequency representation of the source waveforms and a sparse regularization which promotes focal sources with smooth and transient activations. Our approach is evaluated using simulations comparing it to strategies that assume the noise is white or to simple prewhitening.