Coupled Reconstruction of 2D Blood Flow and Vessel Geometry from Noisy Images via Physics-Informed Neural Networks and Quasi-Conformal Mapping

Abstract

Blood flow imaging provides important information for hemodynamic behavior within the vascular system and plays an essential role in medical diagnosis and treatment planning. However, obtaining high-quality flow images remains a significant challenge. In this work, we address the problem of denoising flow images that may suffer from artifacts due to short acquisition times or device-induced errors. We formulate this task as an optimization problem, where the objective is to minimize the discrepancy between the modeled velocity field, constrained to satisfy the Navier-Stokes equations, and the observed noisy velocity data. To solve this problem, we decompose it into two subproblems: a fluid subproblem and a geometry subproblem. The fluid subproblem leverages a Physics-Informed Neural Network to reconstruct the velocity field from noisy observations, assuming a fixed domain. The geometry subproblem aims to infer the underlying flow region by optimizing a quasi-conformal mapping that deforms a reference domain. These two subproblems are solved in an alternating Gauss-Seidel fashion, iteratively refining both the velocity field and the domain. Upon convergence, the framework yields a high-quality reconstruction of the flow image. We validate the proposed method through experiments on synthetic flow data in a converging channel geometry under varying levels of Gaussian noise, and on real-like flow data in an aortic geometry with signal-dependent noise. The results demonstrate the effectiveness and robustness of the approach. Additionally, ablation studies are conducted to assess the influence of key hyperparameters.

Figures (15)

• Figure 1: Overview of the iterative pipeline: starting from an initial mask defining the flow domain, points are sampled and input to a PINN for flow prediction. The predicted flow is resampled over the image domain, followed by quasi-conformal mapping to align with the noisy image. The mask is updated via this mapping, and the process repeats until convergence.
• Figure 1: Qualitative comparison of flow and region reconstruction. Red contours show ground truth; yellow contours indicate predicted domains. Top two rows: low noise group; bottom two: high noise group. Each pair shows $x$- (top) and $y$-direction (bottom) velocity. Yellow contour in "Noisy" column marks the initialization.
• Figure 2: Illustration of the steps used to divide the entire boundary into wall, outlet, and inlet boundaries. Red indicates the full boundary or the unclassified segments; yellow denotes wall boundaries; green represents outlets; and pink indicates inlets.
• Figure 2: Discrepancies in velocity fields among the noisy input (gray), reconstructed results (red for positive, blue for negative), and ground truth (green “+”). The upper indicates the velocity in $x$-axis while the lower for $y$-axis. (a) Low noise; (b) High noise.
• Figure 3: Architecture of the PINN for fluid flow prediction, featuring two separate networks dedicated to velocity and pressure estimation.

Theorems & Definitions (9)

• Definition 1: Quasi-conformal map
• Definition 2
• Theorem 3
• Remark 1
• Proof 1
• Lemma 1
• Proof 2
• Lemma 2
• Proof 3