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

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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.

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