数学推导常见分布的傅里叶变换，助力MRI信号建模
A Selection of Distributions and Their Fourier Transforms with Applications in Magnetic Resonance Imaging
- 从拓扑空间出发严格定义分布与傅里叶变换
- 推导高斯函数傅里叶变换，用常微分方程法
- 适用于有泛函分析基础的医学成像研究者
本文系统介绍信号处理中常见的一类分布及其傅里叶变换，特别聚焦于磁共振成像（MRI）中的应用。不同于多数侧重信号处理的MRI教材，本文采用更严格的数学方法，明确所关注的拓扑空间，并精确阐述分布及其傅里叶变换的定义方式。关键结果包括泊松求和公式及通过常微分方程（ODE）推导高斯函数的傅里叶变换。尽管读者需具备泛函分析与分布理论基础，本文力求自包含。
原文摘要 · Abstract (English)
This note presents a rigorous introduction to a selection of distributions along with their Fourier transforms, which are commonly encountered in signal processing and, in particular, magnetic resonance imaging (MRI). In contrast to many textbooks on the principles of MRI, which place more emphasis on the signal processing aspect, this note will take a more mathematical approach. In particular, we will make explicit the underlying topological space of interest and clarify the exact sense in which these distributions and their Fourier transforms are defined. Key results presented in this note involve the Poisson summation formula and the Fourier transform of a Gaussian function via an ordinary differential equation (ODE) argument, etc. Although the readers are expected to have prior exposure to functional analysis and distribution theory, this note is intended to be self-contained.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。