Sharp deterministic stability for limited-angle reconstruction

Assume the Axiom of Countable Choice. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be any bounded open set with nonempty interior, and let $\mathrm{\Theta }\subset S^1$ be any nonempty proper open antipodally symmetric set whose complement contains a nonempty open arc. Write ${\theta }^{\perp }=(-{\theta }_2,{\theta }_1)$ and define $R_{\mathrm{\Theta }}f(\theta ,s)={\int }_{\mathbb{R}}f(s\theta +t{\theta }^{\perp })dt$ for $\theta \in \mathrm{\Theta }$, $s\in \mathbb{R}$. Equip $L^2(\mathrm{\Theta }\times \mathbb{R})$ with the norm $\mathrm{\parallel }g{\mathrm{\parallel }}_2=({\int }_{\mathrm{\Theta }}{\int }_{\mathbb{R}}\mathrm{\mid }g(\theta ,s){\mathrm{\mid }}^{2}dsd\theta {)}^{1\mathrm{/}2}$, where $ds$ is Lebesgue measure and $d\theta$ is arc-length measure, and equip $L^2({\mathbb{R}}^2)$ with Lebesgue measure. The line data of every compactly supported smooth input are regarded as elements of this data space. For $M>0$ define ${\mathcal{K}}_M(\mathrm{\Omega })=\{f\in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C}):\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}\le M\}$ with $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}^2=\mathrm{\parallel }f{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{\mathrm{\partial }}_{1}f{\mathrm{\parallel }}_2^2+\mathrm{\parallel }{\mathrm{\partial }}_{2}f{\mathrm{\parallel }}_2^2$. For deterministic noise radius $\delta >0$, define the worst-case reconstruction error ${\mathcal{E}}_{\mathrm{\Omega },\mathrm{\Theta },M}(\delta )={\inf }_{\mathrm{\Phi }:L^2(\mathrm{\Theta }\times \mathbb{R})\to L^2({\mathbb{R}}^2)}{\sup }_{\begin{array}{c}f\in {\mathcal{K}}_M(\mathrm{\Omega }),\eta \in L^2(\mathrm{\Theta }\times \mathbb{R})\\ \mathrm{\parallel }\eta {\mathrm{\parallel }}_2\le \delta \end{array}}\mathrm{\parallel }\mathrm{\Phi }(R_{\mathrm{\Theta }}f+\eta )-f{\mathrm{\parallel }}_2$. Determine the sharp asymptotic order of ${\mathcal{E}}_{\mathrm{\Omega },\mathrm{\Theta },M}(\delta )$ as $\delta \downarrow 0$, with matching upper and lower bounds and a deterministic reconstruction rule attaining the upper bound, for every such $\mathrm{\Omega }$ and $\mathrm{\Theta }$. In particular, determine the dependence of the sharp modulus on the geometry and aperture of the measured normal set.

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography, §§1.3.2 and 1.5.1

Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms, §§1.2–1.3

Daniel Hohmann and Hajo Holzmann, “Weighted angle Radon transform: Convergence rates and efficient estimation,” §3.1, Corollary 3.2

Daniel Hohmann and Hajo Holzmann, supplementary proof, Appendix A.1 (Theorem 3.1 and Lemma A.4)

Status as of 2026-10-19: the broad deterministic minimax question above remains unresolved in the checked source record. Salo proves the full-angle Fourier-slice and compact-support uniqueness baseline. Krishnan and Quinto prove compact-support uniqueness for limited-angle data while exhibiting nonzero Schwartz null functions; they also describe the visible covector condition. The local results on this page establish a no-Hölder lower-bound mechanism for each fixed compact support and blind arc, but none of the checked sources supplies matching deterministic minimax bounds for every bounded $\mathrm{\Omega }$, every such $\mathrm{\Theta }$, and this fixed $H^1$ ball. The analytic double-fibration result cited above is context only and is not a proof of this Euclidean noise question.

Related statistical result checked 2026-10-21: Hohmann and Holzmann, Corollary 3.2, proves the logarithmic minimax mean-squared risk $r_{\epsilon }({\mathrm{\Theta }}_i(\kappa ,L))\sim (\xi \mathrm{/}2{)}^{2\kappa }L\log (1\mathrm{/}\epsilon {)}^{-2\kappa }$ for $0<\eta <\pi \mathrm{/}2$, angle weight $\lambda ={\mathbf{1}}_{[-\eta ,\eta ]}$, and $\gamma >-1\mathrm{/}2$, in their Gaussian white-noise sequence model for the Radon SVD between weighted $L^2$ spaces on the unit disk and detector space and over their spectral ellipsoid classes ${\mathrm{\Theta }}_i(\kappa ,L)$. Its model and parameter classes do not settle the deterministic adversarial $L^2$-noise minimax error on the unweighted $H^1$ class for arbitrary $\mathrm{\Omega }$ and $\mathrm{\Theta }$ posed here.