PINGS-X: Physics-Informed Normalized Gaussian Splatting with Axes Alignment for Efficient Super-Resolution of 4D Flow MRI

TL;DR

PINGS-X addresses the challenge of efficiently obtaining high-spatiotemporal-resolution velocity fields in 4D Flow MRI by replacing slow implicit neural representations with an explicit, axis-aligned Gaussian splatting framework. It introduces Normalized Gaussian Splatting (NGS) to guarantee stable, convergent predictions, uses axes-aligned Gaussians to reduce dimensional complexity, and employs Gaussian merging to maintain scalability. The method couples an explicit, physics-informed loss with a streamlined optimization pipeline, achieving faster training times and higher SR accuracy than PINN- and Gaussian-based baselines on both synthetic CFD and real 4D Flow MRI data. This work bridges explicit rendering techniques and physics-informed learning, offering a practical path to patient-specific, high-fidelity flow reconstructions without prohibitive per-scan retraining.

Abstract

4D flow magnetic resonance imaging (MRI) is a reliable, non-invasive approach for estimating blood flow velocities, vital for cardiovascular diagnostics. Unlike conventional MRI focused on anatomical structures, 4D flow MRI requires high spatiotemporal resolution for early detection of critical conditions such as stenosis or aneurysms. However, achieving such resolution typically results in prolonged scan times, creating a trade-off between acquisition speed and prediction accuracy. Recent studies have leveraged physics-informed neural networks (PINNs) for super-resolution of MRI data, but their practical applicability is limited as the prohibitively slow training process must be performed for each patient. To overcome this limitation, we propose PINGS-X, a novel framework modeling high-resolution flow velocities using axes-aligned spatiotemporal Gaussian representations. Inspired by the effectiveness of 3D Gaussian splatting (3DGS) in novel view synthesis, PINGS-X extends this concept through several non-trivial novel innovations: (i) normalized Gaussian splatting with a formal convergence guarantee, (ii) axes-aligned Gaussians that simplify training for high-dimensional data while preserving accuracy and the convergence guarantee, and (iii) a Gaussian merging procedure to prevent degenerate solutions and boost computational efficiency. Experimental results on computational fluid dynamics (CFD) and real 4D flow MRI datasets demonstrate that PINGS-X substantially reduces training time while achieving superior super-resolution accuracy. Our code and datasets are available at https://github.com/SpatialAILab/PINGS-X.

Figures (21)

• Figure 1: Super-resolution of a carotid artery velocity field. From a low-resolution (LR) input, PINGS-X optimizes a set of 4D spatiotemporal Gaussians (visualized here in 3D at a single time frame). This explicit representation allows us to reconstruct a high-resolution (HR) output that faithfully recovers the complex flow patterns of the ground truth.
• Figure 2: Illustration of the architectural parallel between neural rendering and physics-informed learning. Our work is motivated by the shared use of slow implicit representations (NeRFs and PINNs) and aims to transfer the efficiency of the explicit 3DGS 3dgs framework to our domain.
• Figure 3: Our PINGS-X framework. For training, we initialize (axes-aligned) spatiotemporal Gaussians from low-resolution data and iteratively optimize them. In each step, we (1) predict values using Normalized Gaussian Splatting (NGS), (2) compute a combined data ($L_{data}$) and physics ($L_{PDE}$) loss, and (3) update the Gaussians parameters. The Gaussian density is adjusted via splitting, cloning, and our proposed merging. At inference, we use the trained Gaussians for prediction via NGS.
• Figure 4: A 1D visualization comparing unnormalized and normalized sums of Gaussians ($\sigma=12$). (a) Unnormalized sum: with only two Gaussians ($x=\{5, 25\}$, $v=\{1, 3\}$), the prediction (dotted red) decays to zero in the middle. Adding a third Gaussian at $x=15$, $v=1.3$ (orange) to fill this gap results in an oscillatory profile (solid blue). (b) Normalized sum: we observe a smooth prediction curve between the two Gaussians. Adding the third Gaussian adjusts the curve while maintaining a stable, monotonic result.
• Figure 5: Absolute error maps for the Rosenbrock function (yellow indicates higher error). Our PINGS/PINGS-X converges as the number of Gaussians (in parentheses) increases, while (unnormalized) PIGS does not.

Theorems & Definitions (2)

• Theorem 1
• Corollary 1