Stable Reconstruction of Poisson Cylinder Directions

Definitions and mathematical statement

A stationary Poisson cylinder process is a Poisson process of cylinders in a Euclidean space whose law is invariant under translations. The supplied record does not specify the ambient dimension, the dimension of the cylinder axes, or the distribution of cylinder cross-sections.

For standard spherical cosine-transform notation, let $d\geq 2$, let $S^{d-1}=\{u\in{\mathbb{R}}^d:\|u\|=1\}$, and let $\mu$ be a finite Borel measure on $S^{d-1}$. Its spherical cosine transform is \[ (\mathcal C\mu)(u)=\int_{S^{d-1}}|\langle u,v\rangle|\,\mu({\,\mathrm{d}} v),\qquad u\in S^{d-1}. \] If the cylinder directions are unoriented lines, their directional distribution can be represented by an even probability measure $\mu$, meaning $\mu(A)=\mu(-A)$ for every Borel set $A\subseteq S^{d-1}$. This representation is conditional on the intended axis model; that model is not specified in the record. The notation above does not impose a dimension or a range of dimensions on the source problem.

Source question. Given finitely sampled intersection intensities $g$ of a stationary Poisson cylinder process, can its directional distribution be estimated stably through a continuous inversion of the spherical cosine transform?

In the conditional spherical model above, finite directional samples would have the form \[ \left(u_i,g(u_i)\right)_{i=1}^{m},\qquad m\in{\mathbb{N}},\quad u_i\in S^{d-1}. \] The requested task is to reconstruct an estimate of the directional distribution from such finite observations using cosine-transform inversion. The source does not give the equation relating $g$ to $\mathcal C\mu$, including any process-intensity or cross-section factors. It also does not specify sampling assumptions, an observation-error model, admissible directional distributions, topologies for continuity, or a stability or convergence criterion. Consequently, this is an unresolved formulation of an estimation problem, not a fully specified existence assertion for a continuous inverse or estimator. In particular, whether "continuous inversion" concerns complete transform data or a reconstruction procedure acting on finite samples remains unresolved.

Short English statement

The problem asks whether finitely many intersection-intensity observations of a stationary Poisson cylinder process allow stable estimation of its directional distribution through spherical cosine-transform inversion. The geometric model, observation equation, sampling assumptions, and meaning of stability require clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30001459$, title Stable Reconstruction of Poisson Cylinder Directions, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2010. No individual proposer is supplied.
• Supplied report citation: Mathematics and Algorithms in Tomography (2010), Oberwolfach Reports, pp. 1017--1099, DOI 10.4171/owr/2010/18. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2010/18. No precise problem page or excerpt is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that known cosine-transform inversion results and special numerical methods do not resolve the general estimation problem described here. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note reports only that the intensity symbol was placed in a mathematical delimiter without changing the question. This metadata does not establish mathematical correctness or independent source verification by this draft.