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.

翻译：许多成像问题需要计算由空间变化的强度、特征或密度场诱导的空间变换。典型示例包括畸变校正、可变形图像配准、基于图谱分割以及变形驱动的图像分析。这些任务可建模为几何映射问题，其中变换需满足局部结构保持、边界行为控制或角度畸变约束。此类建模通常涉及变分模型、扩散过程或椭圆偏微分方程。然而，当底层参数场随实例变化时，反复求解高分辨率系统的计算成本会显著增加。本文提出一种无分辨率神经替代模型，用于几何参数化与映射问题。给定空间变化的参数场 $p:\Omega\to\mathbb{R}^m$ 和查询点集 $\{x_i\}_{i=1}^N\subset\Omega$，该模型可预测任意结构化或非结构化点集上的映射位置 $\{u(x_i)\}_{i=1}^N$。为避免依赖固定网格，我们采用多分辨率几何编码策略，通过坐标增强的参数场采样来调节网络。该模型无需标注解数据，通过由变分能量、基于扩散的密度均衡及拟共形理论导出的几何感知约束进行训练。实验展示了在拟共形映射和密度均衡映射问题上的有效性，验证了所提方法的性能。