# Rigorous Uncertainty Quantification for Helmholtz-based EPT: Application to Uncertainty-Guided Post-processing

Zhongzheng He, Julien Lamy, Alessandro Arduino, Luca Zilberti, Paulo Loureiro de Sousa

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Zhongzheng He, Julien Lamy, Alessandro Arduino, Luca Zilberti, Paulo Loureiro de Sousa. Rigorous Uncertainty Quantification for Helmholtz-based EPT: Application to Uncertainty-Guided Post-processing. 2026 ISMRM-ISMRT Annual Meeting and Exhibition, May 2026, Cape Town, South Africa. hal-05493748

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# Rigorous Uncertainty Quantification for Helmholtz-based EPT: Application to Uncertainty-Guided Post-processing

**Primary:** Contrast Mechanisms - Electromagnetic Tissue Properties ) **Secondary:** Acquisition & Reconstruction - Open source software, sequences, and reconstruction algorithms ) **Digital Poster** · 60 min | Novel Contrast Mechanisms II · Wednesday, May 13 at 05:00 PM ) **Keywords:** ELECTRICAL CONDUCTIVITY PERMITTIVITY UNCERTAINTY QUANTIFICATION EPT ULTRA-SHORT ECHO TIME (UTE)

**Zhongzheng He**<sup>1</sup>, **Julien Lamy**<sup>1</sup>, **Alessandro Arduino**<sup>2</sup>, **Luca Zilberti**<sup>2</sup>, **Paulo Loureiro de Sousa**<sup>1</sup>

<sup>1</sup>ICube, CNRS, Université de Strasbourg, Strasbourg, France

<sup>2</sup>Istituto Nazionale di Ricerca Metrologica, Turin, Italy

Presenting Author: Zhongzheng He (zhongzheng.he@unistra.fr)

## Impact

We provide a **ground-truth-free uncertainty metric** for Helmholtz-based EPT. The uncertainty maps can be thresholded to identify reliable voxels for analysis, while our **uncertainty-guided filter** further improves accuracy and reduces boundary artifacts. All code is openly available on GitHub.

## Synopsis

**Motivation:** Assessing the reliability of Electrical Properties Tomography is challenging in clinical settings where no ground truth is available.

**Goals:** To develop a framework that quantifies voxel-wise uncertainty from the measured data, and to use this metric to guide a post-processing filter for improved accuracy.

**Approach:** We use a bivariate error-propagation strategy accounting for the full covariance terms to generate uncertainty maps. A novel filter then improves reconstructions by selectively weighted-averaging only the lowest-uncertainty neighboring voxels.

**Results:** Our uncertainty metric strongly correlated ($r > 0.8$) with reconstruction error in simulations. The uncertainty-guided filter consistently achieved lower errors than the standard median filter in post-processing.

## Introduction

Helmholtz-based Electrical Properties Tomography (EPT) <sup>1</sup> is widely used in clinical research due to its conceptual simplicity. However, its reliance on the local homogeneity assumption (LHA) and the sensitivity of Laplacian estimation to noise remain major limitations <sup>2</sup>. While phantom studies with ground-truth vector network analyzer measurements have provided validation <sup>3</sup>, clinical translation is hindered by the inability to assess reconstruction reliability in vivo, where no reference ground truth exists. To address this challenge, we propose a rigorous framework for uncertainty quantification that incorporates the full covariance of complex-valued data. By extending the approach implemented in EPTlib <sup>4,5</sup>, which assumes uncorrelated real-imaginary parts in the input data, our method provides statistically robust voxel-wise uncertainty maps. Furthermore, we demonstrate the utility of these maps in guiding a novel post-processing filter, which significantly enhances EPT reconstruction accuracy and quality.

## Theory

The Helmholtz-based EPT family reconstructs tissue admittivity $\kappa = \sigma + j\omega\varepsilon_0\varepsilon_r$ from the measured magnetic field $B$:

$$ \kappa = \frac{\nabla^2 B}{j\mu_0\omega B} $$

In practice, the input data is locally modeled with a second-order polynomial <sup>6,7</sup>:

$$ B = c_0 + c_2x^2 + c_5y^2 + c_9z^2 + \dots $$

allowing admittivity to be computed directly from the fitted coefficients ($\hat{c}_i$):

$$ \kappa = \frac{2(\hat{c}_2 + \hat{c}_5 + \hat{c}_9)}{j\mu_0\omega\hat{c}_0} $$

This local fitting approach is applied to the various B-field representations that categorize EPT methods:

* **Standard Helmholtz-based(HB)** <sup>8</sup>: uses $B = |B_1^+|e^{j\phi_{tr}/2}$ with the transceive phase assumption (TPA) <sup>2</sup>.

* **Image-based(IB)** <sup>9</sup>: uses $B = \sqrt{B_1^+ B_1^-}$ (from complex UTE/ZTE signals).

* **Phase-based(PB)** <sup>10</sup>: uses the transceive phase $\phi_{tr}$ for conductivity $\sigma = \frac{\nabla^2\phi_{tr}}{2\mu_0\omega}$.

## Methods

Admittivity uncertainty is quantified by propagating the coefficient covariance through a first-order Taylor expansion <sup>11</sup>:

$$ \text{Cov}([\Re(\kappa), \Im(\kappa)]^T) \approx J_{\kappa} \text{Cov}(c_{\text{biv}}) J_{\kappa}^T, $$
where the Jacobian matrix $J_κ \in \mathbb{R}^{2 \times 20}$ maps the real–imaginary coefficient vector $c_{biv} = [\Re(c), \Im(c)] \in \mathbb{R}^{20}$ to the real–imaginary components of $κ$:

$$J_κ = \begin{bmatrix} \Re(\frac{\partial κ}{\partial c}) & -\Im(\frac{\partial κ}{\partial c}) \\ \Im(\frac{\partial κ}{\partial c}) & \Re(\frac{\partial κ}{\partial c}) \end{bmatrix},$$

with non-zero derivatives

$$\frac{\partial \kappa}{\partial c_0} = -\frac{2(\hat{c}_2 + \hat{c}_5 + \hat{c}_9)}{j\mu_0\omega\hat{c}_0^2}, \quad \frac{\partial \kappa}{\partial c_{2,5,9}} = \frac{2}{j\mu_0\omega\hat{c}_0}.$$

Voxel-wise uncertainties follow as

$$u_\sigma = \sqrt{[\mathrm{Cov}]_{11}}, \quad u_{\varepsilon_r} = \frac{1}{\omega\varepsilon_0} \sqrt{[\mathrm{Cov}]_{22}}.$$

The coefficient covariance is

$$\mathrm{Cov}(c_{\mathrm{biv}}) = \Sigma_{\varepsilon}^{(2)} \otimes (F^T F)^{-1},$$

where $F$ is the design matrix and, assuming bivariate Gaussian noise $ε$, the residual ($r$) based covariance is

$$\Sigma_{\varepsilon}^{(2)} = \frac{1}{n-10} \begin{bmatrix} \sum \Re(r_i)^2 & \sum \Re(r_i) \Im(r_i) \\ \sum \Re(r_i) \Im(r_i) & \sum \Im(r_i)^2 \end{bmatrix}.$$

For PB, both data and coefficients are real-valued, reducing $\Sigma_{\varepsilon}^{(2)}$ to $\sigma_{\varepsilon}^2$ and

$$\mathrm{Cov}(\hat{c}_{\phi}) = \sigma_{\varepsilon}^2 (F^T F)^{-1}, \quad u_{\sigma} = \sqrt{J_{\sigma} \mathrm{Cov}(\hat{c}_{\phi}) J_{\sigma}^T},$$

with

$$J_{\sigma} = [0, 0, \frac{1}{\mu_0 \omega}, 0, 0, \frac{1}{\mu_0 \omega}, 0, 0, \frac{1}{\mu_0 \omega}].$$

Moreover, non-biophysical estimates are **penalized**. If an estimate's 1-sigma confidence interval $[κ - u_κ, κ + u_κ]$ lies entirely outside the plausible biophysical range (e.g., [0, 2.5] S/m for conductivity and [1,100] for relative permittivity), its uncertainty is increased to match the estimate's distance from the nearest boundary of that range.

These final uncertainty maps then guide post-processing¹² by averaging only the **first-quartile lowest-uncertainty voxels**:

$$\bar{\kappa} = \frac{\sum_i w_i \kappa_i}{\sum_i w_i}, \quad w_i = \frac{1}{u_i^2},$$

yielding minimum-variance reconstructions.

* **In-silico data** from a ADEPT ¹³ brain model were simulated (Sim4Life, 128MHz) to generate 1~mm isotropic $|B_1^+|, |B_1^-|$, and $\phi_{tr}$ fields. PB, IB, and HB reconstructions were performed across four SNRs ($\infty$, 90, 60, 30). An SNR-adapted second-order-polynomial fitting cube-shaped kernel ($3^3, 9^3, 11^3$ and $15^3$ voxels) and a fixed $21^3$ post-processing kernel were used, both adapted by the segmentation.

* **In vivo brain data** were acquired on a 3T Siemens Vida scanner using a birdcage transmit coil and a 20-channel receive array. A UTE acquisition ¹⁴ (TE/TR=50~μs/3.6~ms, 1.2~mm isotropic), with white matter SNR of ~30, is used for PB and IB reconstructions. An MPRAGE image for anatomical reference was registered to the UTE space. Kernel sizes were fixed $15^3$ for second-order-polynomial fitting and $21^3$ for post-processing.

## Results

The proposed uncertainty metric demonstrated a strong correlation ($r>0.8$) with absolute reconstruction error in all simulations (Figure 1A). Across all SNR levels and EPT methods, the uncertainty-guided post-processing consistently achieved the lowest normalized mean absolute error (NMAE) compared to raw and median-filtered reconstructions (Figures 1B-C). Visually, our method showed superior artifact reduction and anatomical preservation at complex boundaries in both simulations (Figures 2 and 3 ▼) and in vivo experiments (Figure 4). The in vivo histograms (Figure 5) confirm the proposed method yielded the narrowest distributions, indicating reduced variance.
# Conclusion

We have developed and validated a rigorous framework for uncertainty quantification in Helmholtz-based EPT. The method generates statistically-grounded uncertainty maps that reliably predict voxel-wise reconstruction error assuming complex bivariate Gaussian noise. We further demonstrated the practical utility of these maps by introducing a novel uncertainty-guided filtering technique. The filter consistently outperformed median filtering, yielding EPs maps with lower error and enhanced anatomical fidelity. This work provides a comprehensive strategy for improving the overall reliability and accuracy of Helmholtz-based EPT reconstructions.

# Acknowledgements

The study was funded by ANR-21-CE19-0040 (ELECTRA project). We thank Thomas Benkert (PhD, Application Development, Siemens Healthcare, Erlangen, Germany) for providing the UTE sequence. All code is openly available on GitHub: https://github.com/zhongzheng-he/uqEPT.

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**Figure 1:** Simulation results. (A) Pearson's correlation coefficient between uncertainty and absolute error (all p-values < 0.01). (B) Normalized mean absolute error (NMAE) bar plots for conductivity ($\sigma$) reconstruction methods. (C) NMAE bar plots for relative permittivity ($\epsilon_r$) reconstruction methods. NMAE was calculated within the ROI without any tissue boundary erosion.

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**Figure 2:** Simulation results with SNR = $\infty$. The normalized mean absolute errors (NMAE) were calculated within the ROI without any tissue boundary erosion. Here, the noiseless uncertainty reflects principally the EPT systematic errors (e.g., errors from LHA, TPA, and/or phase-based approximation). Pearson's correlation coefficient ($r$) between the uncertainty and the absolute error (AE) for Phase-based (PB), Image-based (IB) and Helmholtz-based (HB) reconstruction is also displayed.
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**Figure 3:** Simulation results with SNR = 30. The normalized mean absolute errors (NMAE) were calculated within the ROI without any tissue boundary erosion. Pearson's correlation coefficient ($r$) between the uncertainty and the absolute error (AE) for Phase-based (PB), Image-based (IB) and Helmholtz-based (HB) reconstruction is also displayed.

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**Figure 4:** In vivo EPs maps reconstructed using phase-based (PB) and image-based (IB) methods, alongside their corresponding uncertainty maps. The segmentation map obtained by SynthSeq$^{\&\{15\}}$ was used as the reference image for the anatomical kernel.
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<caption>**Figure 5:** Histograms of EPs in white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) without any tissue boundary erosion. Red vertical lines indicate ex vivo reference values$\{16-18\}$, while green vertical lines show histogram peak values from the proposed method.</caption>