Joonas Lahtinen

Joonas Lahtinen

In this paper, we expand the theory of depth-unbiased source localization to unbiased parameter estimation and signal reconstruction of an arbitrary number of non-zero parameters to be recovered. The topic touches on the concept of exact reconstructibility, most commonly known in compressed sensing and multisource estimation in various imaging problems. The theoretical results derive upper bounds on the number of recoverable parameters in the noiseless case, and a probability measure is defined to assess the probability of obtaining all non-zero parameters with correct magnitude order. The work provides a mathematical explanation of the open question regarding the noise robustness of standardized and unbiased methods. Also, the paper reveals a trade-off between the number of sensors and the signal-to-noise ratio. Numerical experiments demonstrate the theoretical findings.

Santtu Söderholm, Joonas Lahtinen, Sampsa Pursiainen

Electroencephalography (EEG) source imaging aims to infer brain activity from electrical potentials measured on the scalp. This is a difficult problem because many different source patterns can explain the same measurements. The result depends strongly on two things: the forward model and the inverse method. In this work, we study how these two parts work together. We focus not only on where the activity is located, but also on how the reconstructed activity is distributed in space. We suggest that different source models create different signatures in the reconstructed activity. We use realistic head models and compute forward solutions with the finite element method using Zeffiro Interface and DUNEuro. We test different source models, including 2 implementations of a divergence-conforming model, and one implementation of Local subtraction approach. For inverse methods, we use advanced methods such as standardized hierarchical adaptive L1 regression (sHAL1R), standardized Kalman filtering (SKF), and classical dipole scanning. To understand the complex interplay between the forward and inverse approaches, we analyze the inverse source localization results using distributional quantitative measures, including Earth Mover's Distance and depth bias scatter plot, and qualitatively assess the amplitude distribution and focality. The results show that there is a strong dependence between the choice of source model and the success rate of a given inverse method: a source model that corresponds well with a single point-like source is a good match with an inverse method that presupposes such a source.

Joonas Lahtinen, Alexandra Koulouri

Electroencephalography (EEG) source imaging aims to reconstruct the spatial distribution of neural activity within the brain from non-invasive scalp measurements. This inverse problem is severely ill-posed due to the low spatial resolution of EEG and the presence of measurement noise, necessitating robust regularization techniques. Bayesian approaches provide a principled framework for incorporating prior knowledge into the solution, where regularization naturally arises through prior distributions and their associated hyperparameters. In this work, we provide an overview of key Bayesian methods for EEG source imaging based on Gaussian, Laplace, and group Laplace priors, with particular emphasis on hierarchical models that promote sparsity. We analyze the connections between these hierarchical formulations and classical optimization techniques, and provide an analytical description of their implementation using expectation -maximization and alternating optimization algorithms. To address the issue of depth bias where deeper sources are systematically underestimated or mislocalized - we extend a statistical signal-to-noise ratio (SNR) framework to derive depth-weighted priors that account for differences in how strongly sources at different depths are reflected in the measurements. Finally, we illustrate the behaviour of the considered models through simulation studies involving sources at varying depths. The results highlight the impact of prior selection and depth weighting on reconstruction accuracy and demonstrate the importance of informed model design for depth-sensitive EEG source localization.