计算成像的有限原始基定理：操作图表示法的正式基础
The Finite Primitive Basis Theorem for Computational Imaging: Formal Foundations of the OperatorGraph Representation
AI总结：
该研究提出有限原始基定理，证明所有成像前向模型可表示为由11个标准原语构成的DAG，并通过实验验证其在多种成像模态中的有效性。
摘要：
计算成像前向模型，从编码孔径光谱相机到MRI扫描仪，传统上都是作为单一的、特定模态的代码实现。我们证明了在广泛且精确定义的操作类Cimg中（涵盖临床、科学和工业成像模态，包括线性和非线性），每一个前向模型都可以以ε近似的方式表示为一个带类型的有向无环图（DAG），其节点来自恰好11个标准原语的库：传播、调制、投影、编码、卷积、累积、检测、采样、分散、散射和变换。我们称其为有限原始基定理。证明是构造性的：我们提供了一个算法，给定任何H在Cimg中，产生一个DAG G，其相对操作误差最多为ε，且图复杂度在规定的范围内。我们进一步证明该库是最小的：移除任何一个原语会导致至少一个模态失去其ε近似表示。对成像物理中非线性的系统性分析显示，它们分为两种结构类别：点wise标量函数（由Transform处理）和自一致迭代（展开为现有线性原语）。在31个线性模态上的实证验证确认eimg低于0.01，最多使用5个节点和深度5，我们还为9个额外的非线性模态提供了构造性的DAG分解。这些结果建立了物理世界模型（PWM）框架的数学基础。
英文摘要：
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.
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