Ilias I. Giannakopoulos

Xinling Yu, José E. C. Serrallés, Ilias I. Giannakopoulos et al.

We propose Physics-Informed Fourier Networks for Electrical Properties (EP) Tomography (PIFON-EPT), a novel deep learning-based method for EP reconstruction using noisy and/or incomplete magnetic resonance (MR) measurements. Our approach leverages the Helmholtz equation to constrain two networks, responsible for the denoising and completion of the transmit fields, and the estimation of the object's EP, respectively. We embed a random Fourier features mapping into our networks to enable efficient learning of high-frequency details encoded in the transmit fields. We demonstrated the efficacy of PIFON-EPT through several simulated experiments at 3 and 7 tesla (T) MR imaging, and showed that our method can reconstruct physically consistent EP and transmit fields. Specifically, when only $20\%$ of the noisy measured fields were used as inputs, PIFON-EPT reconstructed the EP of a phantom with $\leq 5\%$ error, and denoised and completed the measurements with $\leq 1\%$ error. Additionally, we adapted PIFON-EPT to solve the generalized Helmholtz equation that accounts for gradients of EP between inhomogeneities. This yielded improved results at interfaces between different materials without explicit knowledge of boundary conditions. PIFON-EPT is the first method that can simultaneously reconstruct EP and transmit fields from incomplete noisy MR measurements, providing new opportunities for EPT research.

Ilias I. Giannakopoulos, Mikhail S. Litsarev, Athanasios G. Polimeridis

We present a method of memory footprint reduction for FFT-based, electromagnetic (EM) volume integral equation (VIE) formulations. The arising Green's function tensors have low multilinear rank, which allows Tucker decomposition to be employed for their compression, thereby greatly reducing the required memory storage for numerical simulations. Consequently, the compressed components are able to fit inside a graphical processing unit (GPU) on which highly parallelized computations can vastly accelerate the iterative solution of the arising linear system. In addition, the element-wise products throughout the iterative solver's process require additional flops, thus, we provide a variety of novel and efficient methods that maintain the linear complexity of the classic element-wise product with an additional multiplicative small constant. We demonstrate the utility of our approach via its application to VIE simulations for the Magnetic Resonance Imaging (MRI) of a human head. For these simulations we report an order of magnitude acceleration over standard techniques.

Ioannis P. Georgakis, Ilias I. Giannakopoulos, Mikhail S. Litsarev et al.

A stable volume integral equation (VIE) solver based on polarization/magnetization currents is presented, for the accurate and efficient computation of the electromagnetic scattering from highly inhomogeneous and high contrast objects.We employ the Galerkin Method of Moments to discretize the formulation with discontinuous piecewise linear basis functions on uniform voxelized grids, allowing for the acceleration of the associated matrix-vector products in an iterative solver, with the help of FFT. Numerical results illustrate the superior accuracy and more stable convergence properties of the proposed framework, when compared against standard low order (piecewise constant) discretization schemes and a more conventional VIE formulation based on electric flux densities. Finally, the developed solver is applied to analyze complex geometries, including realistic human body models, typically used in modeling the interactions between electromagnetic waves and biological tissue.

Xinling Yu, José E. C. Serrallés, Ilias I. Giannakopoulos et al.

Electrical properties (EP), namely permittivity and electric conductivity, dictate the interactions between electromagnetic waves and biological tissue. EP can be potential biomarkers for pathology characterization, such as cancer, and improve therapeutic modalities, such radiofrequency hyperthermia and ablation. MR-based electrical properties tomography (MR-EPT) uses MR measurements to reconstruct the EP maps. Using the homogeneous Helmholtz equation, EP can be directly computed through calculations of second order spatial derivatives of the measured magnetic transmit or receive fields $(B_{1}^{+}, B_{1}^{-})$. However, the numerical approximation of derivatives leads to noise amplifications in the measurements and thus erroneous reconstructions. Recently, a noise-robust supervised learning-based method (DL-EPT) was introduced for EP reconstruction. However, the pattern-matching nature of such network does not allow it to generalize for new samples since the network's training is done on a limited number of simulated data. In this work, we leverage recent developments on physics-informed deep learning to solve the Helmholtz equation for the EP reconstruction. We develop deep neural network (NN) algorithms that are constrained by the Helmholtz equation to effectively de-noise the $B_{1}^{+}$ measurements and reconstruct EP directly at an arbitrarily high spatial resolution without requiring any known $B_{1}^{+}$ and EP distribution pairs.

Ilias I. Giannakopoulos, Lokesh B Gautham Muthukumar, Yvonne W. Lui et al.

Parallel imaging techniques reduce magnetic resonance imaging (MRI) scan time but image quality degrades as the acceleration factor increases. In clinical practice, conservative acceleration factors are chosen because no mechanism exists to automatically assess the diagnostic quality of undersampled reconstructions. This work introduces a general framework for pixel-wise uncertainty quantification in parallel MRI reconstructions, enabling automatic identification of unreliable regions without access to any ground-truth reference image. Our method integrates conformal quantile regression with image reconstruction methods to estimate statistically rigorous pixel-wise uncertainty intervals. We trained and evaluated our model on Cartesian undersampled brain and knee data obtained from the fastMRI dataset using acceleration factors ranging from 2 to 10. An end-to-end Variational Network was used for image reconstruction. Quantitative experiments demonstrate strong agreement between predicted uncertainty maps and true reconstruction error. Using our method, the corresponding Pearson correlation coefficient was higher than 90% at acceleration levels at and above four-fold; whereas it dropped to less than 70% when the uncertainty was computed using a simpler a heuristic notion (magnitude of the residual). Qualitative examples further show the uncertainty maps based on quantile regression capture the magnitude and spatial distribution of reconstruction errors across acceleration factors, with regions of elevated uncertainty aligning with pathologies and artifacts. The proposed framework enables evaluation of reconstruction quality without access to fully-sampled ground-truth reference images. It represents a step toward adaptive MRI acquisition protocols that may be able to dynamically balance scan time and diagnostic reliability.