An RBF partition of unity method for geometry reconstruction and PDE solution in thin structures

TL;DR

The paper tackles the challenge of patient-specific diaphragm modeling by reconstructing a smooth, high-aspect-ratio geometry from medical images and solving PDEs on that geometry. It introduces an adaptive least-squares RBF-PUM with anisotropic patch scaling and cylindrical patches to generate an infinitely smooth level-set representation while avoiding spurious zero level sets. The approach couples geometry reconstruction and PDE solution within a single framework, enhanced by RBF-QR stability and geometry-quality metrics $F_H$ and $T_L$ to guide parameter choices. Numerical results in 2D and 3D demonstrate robust reconstruction from noisy data and accurate Poisson PDE solutions on the reconstructed domains, validating the method for biomechanical simulations and suggesting applicability to other thin-structure geometries in engineering and biology.

Abstract

The main respiratory muscle, the diaphragm, is an example of a thin structure. We aim to perform detailed numerical simulations of the muscle mechanics based on individual patient data. This requires a representation of the diaphragm geometry extracted from medical image data. We design an adaptive reconstruction method based on a least-squares radial basis function partition of unity method. The method is adapted to thin structures by subdividing the structure rather than the surrounding space, and by introducing an anisotropic scaling of local subproblems. The resulting representation is an infinitely smooth level set function, which is stabilized such that there are no spurious zero level sets. We show reconstruction results for 2D cross sections of the diaphragm geometry as well as for the full 3D geometry. We also show solutions to basic PDE test problems in the reconstructed geometries.

Figures (15)

• Figure 1: The initial point cloud data in 3D (left) and 2D (right), together with the initial normal data. The surface is divided into an inner and outer part whose normals are displayed in different colors.
• Figure 1: The adaptive cover generation with the initial patches (top left), the intermediate result (top right), and the final non-overlapping patches (bottom left). In each step, patches that that are marked for refinement (dashed lines) and the previous iteration (dotted lines) are indicated. The data points (dot markers) are shown for reference. Finally, the resulting patches after ensuring an overlap $\delta\approx0.15$ are shown (bottom right).
• Figure 1: Contour plots of the weight functions of two adjacent 2D patches.
• Figure 1: The local least-squares problem for one patch in the approximation of the 2D diaphragm data. The center points ($\times$), the data points (colored dot and arrow), and the additional points $X_{j,e}$ (colored dot) are shown. The color corresponds to the function value and the arrows are the normal directions used in conditions \ref{['eq:geomcond2']} and \ref{['eq:geomcond3']}.
• Figure 1: The best reconstruction for $n=21$ and $P=24$ with $F_H=0.016$ and $L_T=0.11$ is shown together with the spatial distribution of the quality measures to the left. The black curve is the zero level set. Negative losses are set to zero and correspond to dark blue. The quality measures are shown as a function of the number of patches $P$ to the right.

Theorems & Definitions (2)

• Remark 4.1
• Remark 8.1