The Finite Primitive Basis Theorem for Computational Imaging: Formal Foundations of the OperatorGraph Representation

TL;DR

This paper proves that all models in a broad class of computational imaging can be approximated by a finite, minimal set of primitive operations represented as directed acyclic graphs, with constructive algorithms and empirical validation.

Contribution

It introduces the Finite Primitive Basis Theorem, showing all imaging models in class Cimg can be represented with a minimal set of primitives as DAGs, with constructive algorithms and validation.

Findings

All models in Cimg can be approximated with primitive DAGs within epsilon error.

The primitive library is minimal; removing any primitive breaks the approximation.

Empirical validation confirms low error and small complexity for multiple modalities.

Abstract

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.