Roberto G. Beraldo’s research while affiliated with Centro Universitário Fundação Santo André and other places

In this book, written in Portuguese, we discuss what ill-posed problems are and how the regularization method is used to solve them. In the form of questions and answers, we reflect on the origins and future of regularization, relating the similarities and differences of its meaning in different areas, including inverse problems, statistics, machine learning, and deep learning.

Electrical Impedance Tomography (EIT) is an imaging modality that allows the visualization of internal resistivities of a region of interest from electrical measurements external to the same region. In this work, we reconstruct 3D static images using two regularization terms, an anatomical atlas with $\ell _1$-norm and a total variation (TV) term. We chose the iteratively reweighted least squares (IRLS) algorithm to approximate the $\ell _1$-norms by quadratic terms and the Gauss-Newton algorithm to perform the optimization of the resulting functional. Together with the anatomical atlas, using a traditional $\ell _2$-norm and a high-pass filter as the regularizer tends to reconstruct the target on the mesh elements near the region boundary. In comparison, the reconstructed targets with the proposed method are better located, especially when reconstructing multiple targets, in addition to detecting a higher resistivity variation with the same number of iterations.

Time-Difference Electrical Impedance Tomography (TDEIT) is an imaging technique to visualize resistivity changes over time in a region of interest. Regularization is necessary because TDEIT is an ill-posed problem. In this work, we use Regularization by Denoising (RED) with four different denoisers to reconstruct brain images in a simplified 2D head model. We compared the RED results to two traditional reconstruction methods, generalized Tikhonov regularization and total variation regularization. In both the noiseless and the noisy scenarios, we achieved the best results using RED with non-local means as the denoiser in relation to figures of merit such as ringing, resolution and shape deformation.

Electrical impedance tomography (EIT) is a medical imaging modality that has the potential to benefit diagnosing, monitoring, and understanding several pathological conditions. However, some regions of the body, such as the brain, are more challenging to reconstruct, demanding improvements before the technique can be used in clinical practice. In this study, we implemented and evaluated an algorithm for 2D static EIT image reconstruction based on the Deep Image Prior (DIP) method. The method was tested in measurements calculated from a computational human head model, where we included a region representing the occurrence of a stroke. The results showed that the DIP-based algorithm had some advantages compared to a more classical method, such as robustness to noise and independence of an initial solution. Therefore, this method could be better suited for real-life EIT image reconstructions.

Electrical impedance tomography (EIT) is a technique that can be used to estimate resistivity distribution from the inside of a domain based on surface measurements. This could be useful, for example, in the diagnosis of cerebral strokes. However, a method to acquire EIT images of the head with enough quality to achieve this task is still needed. In this work, an automated method for the calculation of a statistical atlas of the human head is presented to be used as prior information for the ill-posed inverse problem associated to EIT. Fifty magnetic resonance images of healthy subjects were used for this purpose. Numerical simulations using a realistic head model with hemorrhagic and ischemic stroke were used to evaluate its effect. The results show that, when the atlas was used, there was a decrease in the root mean square error of the images obtained. Also, some artifacts observed in the image generated without the use of the atlas were eliminated or diminished. These findings hint to the possibility of using a statistical atlas of the head to improve the quality of EIT images.KeywordsAnatomical atlasElectrical impedance tomographyImage processingStroke

Continuous monitoring of brain hemodynamics is important to quickly detect changes in healthy cerebral blood flow, helping physician decision-making in the treatment of the patient. Resistivity changes in the brain happen as a result of the pulsatile characteristic of the blood in the arteries or pathological conditions such as ischemia. We developed a dynamic model of cerebral circulation capable of portraying variations in resistivities in arteries within a cardiac cycle. From the hypothesis that the resistivity changes in the brain can be detected by Electrical Impedance Tomography (EIT), we included this model as prior information in time-difference image reconstruction algorithm. With this prior information, image reconstruction of the brain with pre-existing ischemia was possible, showing that EIT is a potential technique for brain hemodynamic monitoring.KeywordsBlood flow modelElectrical impedance tomographyDifference imagingStroke

Objective. The objective of this work is to develop a 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications. Approach. The atlas was constructed based on 3D magnetic resonance images (MRI) of 107 human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. T1w+T2w MRI images were used to segment the main structures of the head while angiography MRI was used to segment the main arteries. The proposed atlas also comprises a time-varying model of arterial brain circulation, based on the solution of the Navier–Stokes equation in the main arteries and their vascular territories. Main results. High-resolution, multi-frequency and time-varying anatomical atlases of resistivity, conductivity and relative permittivity were created and evaluated using a forward problem solver for EIT. The atlas was successfully used to simulate electrical impedance tomography measurements indicating the necessity of signal-to-noise between 100 and 125 dB to identify vascular changes due to the cardiac cycle, corroborating previous studies. The source code of the atlas and solver are freely available to download. Significance. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically. Ideally, the model should be made with patient-specific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability. Anatomical atlases are important tools for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g. machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers.