发表机构
Faculty of Computer Science, Dresden University of Technology; Max Planck Institute of Molecular Cell Biology and Genetics; Center for Systems Biology Dresden; Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden/Leipzig; Cluster of Excellence Physics of Life, Dresden University of Technology; Chair for Mathematical Foundations of Artificial Intelligence, Ludwig-Maximilians-Universität München; Munich Center for Machine Learning (MCML); Department of Mathematical Modeling and Machine Learning, University of Zurich（德累斯顿工业大学计算机学院; 马克斯·普朗克分子细胞生物学与遗传学研究所; 德累斯顿系统生物学中心; 德累斯顿/莱可夫可扩展数据分析与人工智能中心; 德累斯顿工业大学“生命物理学”卓越集群; 慕尼黑大学人工智能数学基础讲席; 慕尼黑机器学习中心; 苏黎世大学数学建模与机器学习系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出PF-T数学框架，将其应用于SDF得到含边界与内部几何的连续表示，推导形态计量学量化形状相似性，在多类数据集上验证了方法的有效性。
AI 中文摘要
我们提出了一种基于从形状域映射函数到公共参考域的形状比较数学框架。该前推变换（Push-Forward Transform，PF-T）可实现形状的不变且鲁棒比较，同时保留内在几何信息。定量比较形状及其时间演化是图像分析中的基础挑战。有意义的形状比较需要对不改变形状本身的变换（如平移、旋转、反射、重参数化和均匀缩放）具有不变性，同时对内在几何变化保持敏感性。现有方法通常依赖敏感的参数化、地标对应或难以解释和复现的学习表示。我们表明，将前推变换（PF-T）应用于符号距离函数（Signed Distance Functions，SDFs）可产生连续表示，捕获边界和内部几何。我们推导了一种可解释的形态计量学，用于量化形状相似性并揭示骨骼拓扑和旋转对称性等特征。前推变换一致适用于二维和三维形状，可扩展到随时间演化的几何，并支持形状与形状上定义的其他标量场（如强度或分子信号）的联合分析。我们给出了数学公式，描述了高效算法，并在二维、三维和时间数据集上对该方法进行了基准测试。
英文摘要
We introduce a mathematical framework for shape comparison based on mapping functions from the shape domain to a common reference domain. This Push-Forward Transform enables invariant and robust comparison of shapes, preserving intrinsic geometric information. Quantitatively comparing shapes and their temporal evolution is a fundamental challenge in image analysis. Meaningful shape comparison requires representations that are invariant to transformations that do not alter shape itself, such as translation, rotation, reflection, re-parametrization, and uniform scaling, while remaining sensitive to intrinsic geometric variation. Existing approaches often rely on sensitive parameterizations, landmark correspondence, or learned representations that are difficult to interpret and reproduce. We show that the Push-Forward Transform (PF-T) applied to Signed Distance Functions (SDFs) yields a continuous representation that captures both boundary and interior geometry. We derive an interpretable morphometric that quantifies shape similarity and reveals features such as skeletal topology and rotational symmetries. The push-forward transform applies consistently to two- and three-dimensional shapes, extends to time-evolving geometries, and supports the joint analysis of shape and additional scalar fields defined over shapes, such as intensity or molecular signals. We present the mathematical formulation, describe an efficient algorithm, and benchmark the approach on 2D, 3D, and temporal data sets.
