arXiv subjects

Joonas Lahtinen

Publications and source records attributed to Joonas Lahtinen.

12 recordsLinked to original sources

arXiv2026

On Unbiased Parameter Estimation and Signal Reconstruction

In this paper, we extend the theory of depth-unbiased source localization to unbiased parameter estimation and signal reconstruction for an arbitrary number of non-zero parameters. The topic touches on exact reconstructibility, most commonly studied in compressed sensing and multisource estimation across various imaging problems. The theoretical results derive upper bounds on the number of recoverable parameters in the noiseless case, and define a probability measure to assess the likelihood of recovering 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. The paper also reveals a trade-off between the number of sensors and the signal-to-noise ratio. Numerical experiments demonstrate the theoretical findings.

cs.IT↗

arXiv2026

Advanced EEG Source Models from the Perspective of FEM and Inverse Solutions

In this study, we compare forward solutions computed with finite element methods in Zeffiro Interface and DUNEuro, using the different source models they provided. We compared two of the source models from DUNEuro, called Whitney basis and Local subtraction, and the divergence-conforming model of Zeffiro Interface. For source estimation, we applied sparsity-promoting standardized hierarchical adaptive L1 regression (SHAL1R), standardized Kalman filtering (SKF), classical sLORETA, and dipole scanning. Analyses include Earth Mover's Distance, depth bias scatter plots, and qualitative assessments of amplitude distribution and focality. Preliminary experiments with source interpolation for each method revealed that Local subtraction closely matches expectations for the local behavior of the lead field at various depths. The main results reveal that the success of an inverse method depends strongly on the compatibility between its assumptions about the focality of the source and the chosen source model, with point-source models performing best when paired with methods designed for such sources, i.e., sparsity-promoting methods and methods that scan for a single source. Moreover, source models that admit patch sources with a wide distribution are more sensitive to additional noise.

math.NA↗

arXiv2026

Forward--Inverse Interplay in FEM-Based EEG Source Imaging: Distributional Signatures of Advanced Source Models and Inverse Solvers

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.

math.NA↗

arXiv2026

Overview of Bayesian Solvers in EEG Distributed Source Models: Prior Selection, Algorithmic Implementation, and Depth Bias Reduction

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.

math.NA↗

arXiv2025

Stable EEG Source Estimation for Standardized Kalman Filter using Change Rate Tracking

This article focuses on the measurement and evolution modeling of Standardized Kalman filtering for brain activity estimation using non-invasive electroencephalography data. Here, we propose new parameter tuning and a model that uses the rate of change in the brain activity distribution to improve the stability of otherwise accurate estimates. Namely, we propose a backward-differentiation-based measurement model for the change rate, which notably improves the filtering-parametrization-stability of the tracking. Simulated data and data from a real subject were used in experiments.

stat.AP↗

arXiv2025

Tracking EEG Thalamic and Cortical Focal Brain Activity using Standardized Kalman Filtering with Kinematics Modeling

Kalman filtering has proven to be effective for estimating brain activity using EEG recordings. In particular, the introduced post hoc standardization step of the algorithm, inspired by the sLORETA time-invariant method, reduces the depth bias and thus allows the estimation to appear at the correct depth from the electrode surface. In the current work, we propose first and second-order kinematic evolution models, where the state-space vector includes not only the dipolar source activity but also its velocity and acceleration. Compared to our previous study, this motion model yields smoother and more physically plausible estimates of brain activity even when the measurement noise is high, for both superficial and deep sources. In addition, we introduce a tunable power parameter that enhances the computational efficiency of the algorithm. Our simulation study, which involves thalamic and cortical activity in the somatosensory region, demonstrates that accurate estimation and tracking of both superficial and deep brain activity are feasible.

math.NA↗

arXiv2025

The Effect of Prior Parameters on Standardized Kalman Filter-Based EEG Source Localization

EEG Source localization is a critical tool in neuroscience, with applications ranging from epilepsy diagnosis to cognitive research. It involves solving an ill-posed inverse problem that lacks a unique solution unless constrained by prior knowledge. The Bayesian framework enables the incorporation of such knowledge, typically encoded through prior models. Various algorithms have been proposed for source localization, and they differ significantly in how prior knowledge is incorporated. Some approaches rely on anatomical or functional constraints, while others use statistical distributions or sampling-based techniques. In this landscape, the Standardized Kalman Filter (SKF) represents a dynamic Bayesian approach that integrates temporal modeling with a Gaussian prior structure. It addresses the depth bias, a common limitation in source localization, through a post-hoc standardization step that equalizes sensitivity across cortical depths and makes deep activity detection feasible. This study focuses on the development and optimization of Gaussian prior models within the SKF framework for simultaneous cortical and sub-cortical activity detection. Synthetic data similar to the P20 / N20 component of the somatosensory evoked potentials (SEP) was used to identify effective prior parameter configurations for reconstructing both deep and superficial sources under different noise levels. We also investigated the role of RTS smoothing in enhancing source separability. Our results indicate that raising the standardization exponent to 1.25, along with smoothing, significantly improves depth localization accuracy at low noise levels.

math.NA↗

arXiv2024

On Bias and Its Reduction via Standardization in Discretized Electromagnetic Source Localization Problems

In electromagnetic source localization problems stemming from linearized Poisson-type equation, the aim is to locate the sources within a domain that produce given measurements on the boundary. In this type of problem, biasing of the solution is one of the main causes of mislocalization. A technique called standardization was developed to reduce biasing. However, the lack of a mathematical foundation for this method can cause difficulties in its application and confusion regarding the reliability of solutions. Here, we give a rigorous and generalized treatment for the technique using the Bayesian framework to shed light on the technique's abilities and limitations. In addition, we take a look at the noise robustness of the method, which is widely reported in numerical studies. The paper starts by giving a gentle introduction to the problem and its bias and works its way toward standardization.

math.OC↗

arXiv2023

Pressure-Poisson Equation in Numerical Simulation of Cerebral Arterial Circulation and Its Effect on the Electrical Conductivity of the Brain

This study considers dynamic modelling of the cerebral arterial circulation and reconstructing an atlas for the electrical conductivity of the brain. While high-resolution 7-Tesla (T) Magnetic Resonance Imaging (MRI) data allow for reconstructing the cerebral arteries with a cross-sectional diameter larger than the voxel size, electrical conductivity cannot be directly inferred from MRI data. Brain models of electrophysiology typically associate each brain tissue compartment with a constant electrical conductivity, omitting any dynamic effects of cerebral blood circulation. Incorporating those effects poses the challenge of solving a system of incompressible Navier-Stokes equations in a realistic multi-compartment head model. We postulate that circulation in the distinguishable arteries can be estimated via the pressure-Poisson equation, which is coupled with Fick's law of diffusion for microcirculation. To establish a fluid exchange model between arteries and microarteries, a boundary condition derived from the Hagen-Poisseuille model is applied. The relationship between the estimated volumetric blood concentration and the electrical conductivity of the brain tissue is approximated through Archie's law for fluid flow in porous media. Through the formulation of the PPE and a set of boundary conditions based on the Hagen-Poisseuille model, we obtained an equivalent formulation of the incompressible Stokes equation. Thus, allowing effective blood pressure estimation in cerebral arteries segmented from open 7T MRI data. As a result of this research, we developed and built a useful modelling framework that accounts for the effects of dynamic blood flow on a novel MRI-based electrical conductivity atlas. The electrical conductivity perturbation obtained in numerical experiments has an appropriate overall match with previous studies on this subject.

math.AP↗

arXiv2023

The Effects of Peeling on Finite Element Method -based EEG Source Reconstruction

The problem of reconstructing brain activity from electric potential measurements performed on the surface of a human head is not an easy task: not just because the solution of the related inverse problem is fundamentally ill-posed (not unique), but because the methods utilized in constructing a synthetic forward solution themselves contain many inaccuracies. One of these is the fact that the usual method of modelling primary currents in the human head via dipoles brings about at least 2 modelling errors: one from the singularity introduced by the dipole, and one from placing such dipoles near conductivity discontinuities in the active brain layer boundaries. In this article we observe how the removal of possible source locations from the surfaces of active brain layers affects the localisation accuracy of two inverse methods, sLORETA and Dipole Scan, at different signal-to-noise ratios (SNR), when the H(div) source model is used. We also describe the finite element forward solver used to construct the synthetic EEG data, that was fed to the inverse methods as input, in addition to the meshes that were used as the domains of the forward and inverse solvers. Our results suggest that there is a slight general improvement in the localisation results, especially at lower noise levels. The applied inverse algorithm and brain compartment under observation also affect the accuracy.

math.NA↗

arXiv2023

Multi-compartment human head modeling: generating adaptive tetrahedral mesh with GPU acceleration

This paper introduces a highly adaptive and automated approach for generating Finite Element (FE) discretization for a given realistic multi-compartment human head model obtained through magnetic resonance imaging (MRI) dataset. We aim at obtaining accurate tetrahedral FE meshes for electroencephalographic source localization. We present recursive solid angle labeling for the surface segmentation of the model and then adapt it with a set of smoothing, inflation, and optimization routines to further enhance the quality of the FE mesh. The results show that our methodology can produce FE mesh with an accuracy greater than 1 millimeter, significant with respect to both their 3D structure discretization outcome and electroencephalographic source localization estimates. FE meshes can be achieved for the human head including complex deep brain structures. Our algorithm has been implemented using the open Matlab-based Zeffiro Interface toolbox with it effective time-effective parallel computing system.

math.AP↗

arXiv2022

Conditionally Exponential Prior in Focal Near- and Far-Field EEG Source Localization via Randomized Multiresolution Scanning (RAMUS)

In this paper, we focus on the inverse problem of reconstructing distributional brain activity with cortical and weakly detectable deep components in non-invasive Electroencephalography. In particular, we aim to generalize the previously extensively used conditionally Gaussian prior (CGP) formalism to achieve distributional reconstructions with higher focality. For this purpose, we introduce as a hierarchical prior, a general exponential distribution, refered to as conditionally exponential prior (CEP). The first-degree CEP corresponds to focality enforcing Laplace prior that suffers from strong depth bias making the deep activity unrecoverable. We sample over multiple resolution levels via RAMUS to reduce this bias as it is known to depend on the resolution of the source space. Moreover, we introduce a procedure based on the physiological a priori knowledge of the brain activity to obtain the shape and scale parameters of the gamma hyperprior that steer the CEP. The posterior estimates are calculated using iterative statistical methods, expectation maximization and iterative alternating sequential algorithm, which we show to be algorithmically similar and to have a close resemblance to the iterative reweighting methods. The performance of CEP is compared with the recent sampling-based dipole localization method Sequential semi-analytic Monte Carlo estimation (SESAME) in numerical experiments of simulated somatosensory evoked potentials related to the human median nerve stimulation. Our results suggest that a hybrid of the first-degree CEP and RAMUS can achieve an accuracy comparable to the second-degree case (CGP) while being more focal. Further, the proposed hybrid is shown to be robust to noise effects and compare well to the dipole reconstructions obtained with SESAME.

math.NA↗