cs.CVFeb 24, 2026

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

Authors: Chengshuai Yang

Organizations: NextGen PlatformAI C Corp, USA

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

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.

Figures & tables

ModalityCarrierDAG Primitives#NDeptheimge_{\mathrm{img}}
Full-validation modalities (forward model verified against reference implementation)
CASSI [ 22 ]PhotonM→W→Σ→DM\to W\to\Sigma\to D44<10−4<10^{-4}
CACTI [ 27 ]PhotonM→Σ→DM\to\Sigma\to D33<10−4<10^{-4}
SPC [ 28 ]PhotonM→Σ→DM\to\Sigma\to D33<10−4<10^{-4}
Lensless [ 29 ]PhotonC→DC\to D22<10−5<10^{-5}
Ptychography [ 30 ]PhotonM→P→DM\to P\to D334.2×10−44.2\times 10^{-4}
Table 1: Primitive decomposition of 31 imaging modalities. eimge_{\mathrm{img}} is the mean relative fidelity error over 20 test objects. All values satisfy eimg<0.01e_{\mathrm{img}}<0.01 .
Modalityeimge_{\mathrm{img}}#Nodes / DepthNew Prim.?
Held-out (existing primitives expected)
OCT3.8×10−43.8\times 10^{-4}4 / 3N
Photoacoustic7.1×10−47.1\times 10^{-4}3 / 3N
SIM2.5×10−42.5\times 10^{-4}3 / 3N
Phase-contrast X-ray1.2×10−31.2\times 10^{-3}4 / 4N
Electron ptycho5.6×10−45.6\times 10^{-4}3 / 3N
Table 2: Held-out closure test results. All modalities achieve eimg<0.01e_{\mathrm{img}}<0.01 with the frozen 9-primitive library; Compton’s failure ( eimg=0.34e_{\mathrm{img}}=0.34 ) motivated the addition of Scatter ( RR ), yielding the final 10-primitive library.
Figure 1: Basis-growth saturation. Number of distinct primitive types KK as a function of the number of modalities NN added to the registry. The curve saturates at K=11K=11 for N≥35N\geq 35 ; annotated points mark each primitive type introduction. Scatter ( RR ) enters at N=27N=27 (Compton imaging); Transform ( Λ\Lambda ) enters at N=35N=35 (beam hardening CT). Saturation is consistent with Theorem 24 : once all physics-stage families are covered, new modalities compose existing primitives.
ModalityDAG PrimitivesΛ\Lambda ?
Polychromatic CT (beam hard.)Π→Λexp⁡→ΣE→Λlog⁡→D\Pi\to\Lambda_{\exp}\to\Sigma_{E}\to\Lambda_{\log}\to DY
Phase-wrapped MRIM→F→S→Λwrap→DM\to F\to S\to\Lambda_{\mathrm{wrap}}\to DY
Nonlinear ultrasoundP→M→Λpoly→P→DP\to M\to\Lambda_{\mathrm{poly}}\to P\to DY
MPI (Langevin response)M→Λpoly→F→DM\to\Lambda_{\mathrm{poly}}\to F\to DY
DOT (strong mult. scatt.)M→(R∘P)N→Σ→DM\to(R\circ P)^{N}\to\Sigma\to DN
EIT (impedance tomo.)M→(P∘M)N→Σ→S→DM\to(P\circ M)^{N}\to\Sigma\to S\to DN
Table 3: DAG decompositions for nonlinear forward models. Λexp⁡\Lambda_{\exp} , Λlog⁡\Lambda_{\log} , Λwrap\Lambda_{\mathrm{wrap}} , and Λpoly\Lambda_{\mathrm{poly}} denote Transform with exponential, logarithmic, phase-wrapping, and polynomial families, respectively.
Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

PrimitiveRelative errorPass ( <10−6<10^{-6} )?
Propagate PP<10−14<10^{-14}Y
Modulate MM<10−15<10^{-15}Y
Project Π\Pi<10−12<10^{-12}Y
Encode FF<10−14<10^{-14}Y
Convolve CC<10−14<10^{-14}Y
Accumulate Σ\Sigma<10−15<10^{-15}Y
Appendix
Table S1: Adjoint consistency validation for all 11 canonical primitives.
ModalityPrimitive∥Hk∥\|H_{k}\|≤B\leq B ?
CASSIM(mmask)M(\mathbf{m}_{\mathrm{mask}})∥m∥∞≤1\|\mathbf{m}\|_{\infty}\leq 1 (binary mask)Y
W(α,a)W(\alpha,a)11 (unitary shift)Y
Σλ\Sigma_{\lambda}nλ≤3.9\sqrt{n_{\lambda}}\leq 3.9 ( nλ≤15n_{\lambda}\leq 15 )Y
D(g,η4)D(g,\eta_{4})g≤1g\leq 1 (normalized gain)Y
MRIM(scoil)M(\mathbf{s}_{\mathrm{coil}})∥s∥∞≤1\|\mathbf{s}\|_{\infty}\leq 1 (normalized)Y
F(k)F(\mathbf{k})11 (unitary DFT)Y
Appendix
Table S2: Per-primitive operator norms ∥Hk∥\|H_{k}\| for representative modalities. All satisfy ∥Hk∥≤B=4\|H_{k}\|\leq B=4 .
ModalityCarrierDAG Primitives#NDepth
Neutron imagingNeutronΠ→D\Pi\to D22
HolographyPhotonM→P→DM\to P\to D33
STEDPhotonC→DC\to D22
Light-fieldPhotonM→P→DM\to P\to D33
FPMPhotonM→P→DM\to P\to D33
Spectral CTX-rayΠ→S→D\Pi\to S\to D33
Appendix
Table S3: Template-validated modality decompositions (12 modalities completing the 31-modality set).
Q#QuestionYes →\toNo →\to
1Does HkH_{k} involve free-space carrier evolution (propagation through vacuum, air, tissue, etc.)?Propagation family →\to PP or CCGo to Q2
2Does HkH_{k} involve carrier–matter interaction?Go to Q2aGo to Q3
2aDoes the interaction change carrier direction or energy?Scattering family →\to RR (or R∘P∘RR\circ P\circ R for multiple scattering)Elastic interaction →\to MM
3Does HkH_{k} map spatial information to a measurement coordinate?Go to Q3aDetection–readout →\to {W,Σ,S,C,D}\{W,\Sigma,S,C,D\}
3aIs the mapping a line integral (projection geometry)?Π\PiFF (Fourier encoding)
Appendix
Table S4: Classification decision table: three questions determine the primitive assignment for each physics-stage factor.

Explore similar work

CardsList

  1. Eleven Primitives and Three Gates: The Universal Structure of Computational Imaging

    Mar 13, 2026Chengshuai Yang, Xin YuanComputational ImagingImage Reconstruction

    NextGen PlatformAI C Corp, USA. · School of Engineering, Westlake University, Hangzhou, China.

  2. BAM! Bayesian Anything Model: a foundation model for generative computational imaging

    Sep 30, 2026Alessio Spagnoletti, Charlesquin Kemajou Mbakam, Jonathan Spence +2Physics-Informed Generative ModelingPhysics-Guided Image Restoration

    Laboratoire MAP5, UMR 8145, Université Paris Cité, CNRS · Heriot-Watt University, School of Mathematical and Computer Sciences & Maxwell Institute for Mathematical Sciences

  3. Designing Any Imaging System from Natural Language: Agent-Constrained Composition over a Finite Primitive Basis

    Mar 26, 2026Chengshuai YangComputational Imaging

    NextGen PlatformAI C Corp, USA