所有成像模型都能用11个基础操作构成的图来精确逼近，统一了成像系统的设计范式。
The Finite Primitive Basis Theorem for Computational Imaging: Formal Foundations of the OperatorGraph Representation
- 将成像模型拆解为11个标准操作节点构成的有向无环图
- 在31个线性模态上误差小于0.01，最多5个节点、深度5
- 为物理世界模型框架提供数学基础，适合多模态成像研究者
计算成像前向模型（如编码孔径光谱相机、MRI扫描仪）传统上以专有代码实现。我们证明，在一个精确界定的算子类Cimg（涵盖临床、科研与工业成像，包括线性与非线性模态）中，任一前向模型H均可被ε近似表示为由11个规范原语构成的类型化有向无环图（DAG）：Propagate、Modulate、Project、Encode、Convolve、Accumulate、Detect、Sample、Disperse、Scatter和Transform。此即有限原语基定理。证明是构造性的：给出算法，对任意H ∈ Cimg，生成相对算子误差≤ε且图复杂度受控的DAG G。进一步证明该原语库最小：移除任一原语会导致至少一种模态无法被ε近似表示。对成像物理中非线性的系统分析表明，其结构仅分为两类：逐点标量函数（由Transform处理）与自洽迭代（可展开为现有线性原语）。在31个线性模态上的实证验证显示，平均误差eimg < 0.01，最多5个节点、深度5，并提供9个额外非线性模态的构造性分解。这些结果建立了物理世界模型（PWM）框架的数学基础。
原文摘要 · Abstract (English)
Computational imaging forward models, from coded aperture spectral cameras to MRI scanners, are traditionally implemented as monolithic, modality-specific codes. We prove that every forward model in a broad, precisely defined operator class Cimg (encompassing clinical, scientific, and industrial imaging modalities, both linear and nonlinear) admits an epsilon-approximate representation as a typed directed acyclic graph (DAG) whose nodes are drawn from a library of exactly 11 canonical primitives: Propagate, Modulate, Project, Encode, Convolve, Accumulate, Detect, Sample, Disperse, Scatter, and Transform. We call this the Finite Primitive Basis Theorem. The proof is constructive: we provide an algorithm that, given any H in Cimg, produces a DAG G with relative operator error at most epsilon and graph complexity within prescribed bounds. We further prove that the library is minimal: removing any single primitive causes at least one modality to lose its epsilon-approximate representation. A systematic analysis of nonlinearities in imaging physics shows they fall into two structural categories: pointwise scalar functions (handled by Transform) and self-consistent iterations (unrolled into existing linear primitives). Empirical validation on 31 linear modalities confirms eimg below 0.01 with at most 5 nodes and depth 5, and we provide constructive DAG decompositions for 9 additional nonlinear modalities. These results establish mathematical foundations for the Physics World Model (PWM) framework.
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。