发表机构
École de technologie supérieure（蒙特魁北克大学工程学院（École de technologie supérieure））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出CNN图像空间信息力学的统一数学框架，建立离散滤波器对称性与相对论能量-动量关系的对应，经3D图像验证可在多尺度图像中生成尺度不变莫尔斯临界点。
AI 中文摘要
本文提出了一种用于建模卷积神经网络（CNN）中信息传播的统一数学框架，旨在连接物理空间与信息空间的描述。针对广泛使用的非线性整流卷积操作，本文给出了离散滤波器对称性与相对论能量-动量关系之间的对应关系。具体而言，对称滤波器分量（例如求和Σ=[1,1]）的作用类似静能mc²，用于保留图像质心（例如各向同性扩散）；而反对称分量（例如梯度∇=[-1,1]）的作用类似动量项pc，通常会引发位移（例如振动或平移）。对于典型的小型离散滤波器，该位移由反对称分量与总滤波器能量的比值决定，类似相对论粒子的位移与洛伦兹变换的关系，其中β参数β=v/c=pc/E等于动量pc与总能量E的比值。重复滤波会产生高斯尺度空间和涌现的尺度不变特征。这些结构与经典热（扩散）方程具有拉普拉斯驱动的结构，通过标准数学对应关系，还与薛定谔方程以及弗里德曼方程的部分内容相关，同时伴随涌现的莫尔斯拓扑结构。在3D图像中的演示显示，在涵盖广泛物理尺度的图像中存在类斑点的尺度不变莫尔斯临界点，包括有机糖分子、无机硅晶体、磁共振图像（MRI）中的人类和灵长类大脑、星系以及宇宙微波背景（CMB）。
英文摘要
This paper introduces a unified mathematical framework for modeling information propagation through convolutional neural networks (CNNs), with the aim of connecting descriptions of physical space and information space. A correspondence is presented linking discrete filter symmetry and the relativistic energy--momentum relation under the widely used nonlinear rectified convolution operation. Specifically, symmetric filter components (e.g. the sum $Σ= [1,1]$) operate analogously to rest energy $mc^2$ in preserving the image centre of mass (e.g. isotropic diffusion), whereas antisymmetric components (e.g. the gradient $\nabla = [-1,1]$) operate analogously to the momentum term $pc$ in generally inducing a displacement (e.g. vibration or translation). For typical small discrete filters, this displacement is determined by the ratio of antisymmetric to total filter energy, analogously to how the displacement of a relativistic particle relates to a Lorentz transform with beta parameter $β= \frac{v}{c}=\frac{pc}{E}$ equal to the ratio of momentum $pc$ to total energy $E$. Repeated filtering leads to the Gaussian scale-space and emergent scale-invariant features. These constructions share a Laplacian-driven structure with the classical heat (diffusion) equation and, via standard mathematical correspondences, with the Schrödinger equation and aspects of the Friedmann equations, together with emergent Morse topological structure. Demonstrations in 3D images reveal blob-like, scale-invariant Morse critical points in images spanning a wide range of physical scales, including organic sugar molecules and inorganic silicon crystals, human and primate brains in magnetic resonance images (MRI), galaxies and the cosmic microwave background (CMB).
