Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields

2026-05-27

AI Summary

This work addresses the high computational cost of geometric mapping under spatially varying fields at high resolutions by proposing a resolution-agnostic neural surrogate model. The method operates without reliance on fixed grids or ground-truth solution labels, leveraging coordinate-augmented multi-resolution field encoding to predict mapping positions at arbitrary point sets. A geometry-aware unsupervised loss is formulated by integrating variational energy, diffusion equilibration, and quasiconformal theory. Experimental results demonstrate that the approach achieves efficient, accurate, and resolution-flexible geometric parameterization in both quasiconformal mapping and density equilibration tasks, significantly enhancing computational efficiency and generalization capability.

Abstract

Many imaging problems require computing spatial transformations induced by spatially varying intensity, feature, or density fields. Canonical examples include distortion correction, deformable image registration, atlas-based segmentation, and deformation-driven image analysis. These tasks can be formulated as geometric mapping problems in which the transformation is constrained to preserve local structure, control boundary behavior, or regulate angular distortion. Such formulations typically lead to variational models, diffusion processes, or elliptic partial differential equations. However, repeatedly solving high-resolution systems becomes computationally expensive when the underlying parameter fields vary across instances. In this work, we propose a resolution-free neural surrogate for geometric parameterization and mapping problems. Given a spatially varying parameter field $p:Ω\to\mathbb{R}^m$ and query locations $\{x_i\}_{i=1}^N\subsetΩ$, the model predicts mapped locations $\{u(x_i)\}_{i=1}^N$ on arbitrary structured or unstructured point sets. To avoid dependence on a fixed grid, we use a multi-resolution geometric encoding strategy that conditions the network on coordinate-augmented samples of the parameter field. The model is trained without labeled solution data by enforcing geometry-aware constraints derived from variational energies, diffusion-based density equalization, and quasi-conformal theory. Experimental results on quasi-conformal mapping and density-equalizing mapping problems are presented to demonstrate the effectiveness of our proposed method.

Problem

• geometric mapping
• spatially varying fields
• resolution-free
• neural surrogate
• computational efficiency

Innovation

• resolution-free
• neural surrogate
• geometric mapping
• spatially varying fields
• quasi-conformal