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

Chengshuai Yang

TL;DR

It is proved that every forward model in a broad, precisely defined operator class Cimg 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.

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.

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

TL;DR

It is proved that every forward model in a broad, precisely defined operator class Cimg 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.

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.
Paper Structure (85 sections, 8 theorems, 31 equations, 1 figure, 6 tables)

This paper contains 85 sections, 8 theorems, 31 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Forward model libraries and frameworks.
  4. Operator decomposition in imaging.
  5. Universal approximation.
  6. Computational graphs and domain-specific languages.
  7. Compressive sensing and measurement design.
  8. Preliminaries
  9. Imaging Forward Models
  10. Directed Acyclic Graphs
  11. DAG Composition
  12. The Primitive Library B
  13. Propagate P(d, lambda)
  14. Modulate M(m)
  15. Project Pi(theta)
  16. ...and 70 more sections

Key Result

Theorem 24

For every $H \in \mathcal{C}_{\mathrm{img}}$, there exists a well-formed typed DAG $G = (V, E, \tau)$ whose node types are drawn from $\mathcal{B}$ that is an $\varepsilon$-approximate representation of $H$.

Figures (1)

  • Figure 1: Basis-growth saturation. Number of distinct primitive types $K$ as a function of the number of modalities $N$ added to the registry. The curve saturates at $K = 11$ for $N \geq 35$; annotated points mark each primitive type introduction. Scatter ($R$) enters at $N = 27$ (Compton imaging); Transform ($\Lambda$) enters at $N = 35$ (beam hardening CT). Saturation is consistent with Theorem \ref{['thm:fpb']}: once all physics-stage families are covered, new modalities compose existing primitives.

Theorems & Definitions (40)

  • Definition 1: Imaging Forward Model
  • Definition 2: Typed DAG
  • Definition 3: Compose
  • Definition 4: Propagate
  • Definition 5: Modulate
  • Definition 6: Project
  • Definition 7: Encode
  • Definition 8: Convolve
  • Definition 9: Accumulate
  • Definition 10: Detect
  • ...and 30 more