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

arXiv: https://arxiv.org/abs/2602.20550 · HTML: https://mlsanity.com/paper/2602.20550/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 54 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 5 of 5)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Coded aperture compressive temporal imaging](https://mlsanity.com/paper/1302.2575/) (2013)
- [DeepInverse: A Python package for solving imaging inverse problems with deep learning](https://mlsanity.com/paper/2505.20160/) (2025)

## Built on by (most cited first, 1 of 1)

- [Eleven Primitives and Three Gates: The Universal Structure of Computational Imaging](https://mlsanity.com/paper/2603.13521/) (2026)

## Abstract (arXiv)

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.