Geometry-First Certification and Directional Fluence Efficiency in Limited-Angle Computed Tomography

Limited-angle tomographic acquisition induces directional anisotropy: certain feature orientations remain weakly observed regardless of downstream reconstruction method or additional measurements within the same angular support. We introduce a geometry-first framework in which limited-angle acquisition is formulated as a directional information balancing problem: directional detectability is determined directly by the second-moment geometry of the acquisition measure prior to reconstruction.

Under a Poisson–Radon edge-detection model, the Fisher information kernel for step-edge detection is cos²(φ−θ), whose directional variation is governed entirely by the second angular Fourier mode, reflecting the second Legendre polynomial P₂ structure associated with quadratic directional coupling. The directional coverage functional therefore admits an exact harmonic decomposition in which anisotropy is controlled entirely by the second angular moment c₂(J) of the acquisition measure, for the quadratic edge-detection model considered here. Within the stated model, this dependence is exact rather than asymptotic.

We prove that the minimum angular fluence certifying worst-case directional detectability at threshold κ in d dimensions is D* = dκ, achieved if and only if the induced directional sampling measure forms a tight frame on ; equivalently, its second-moment tensor is isotropic. The proof proceeds via an infinite-dimensional linear program over non-negative Borel measures whose dual certificate is the uniform orientation measure, establishing an information-theoretic lower bound prior to reconstruction and without reference to a specific reconstruction pipeline. For any geometry G, the fractional excess fluence is , computable in closed form, equivalently, the Frobenius distance of the unit-dose second-moment tensor from the isotropic matrix; in two dimensions this reduces to , with c₂ = 0 necessary and sufficient for fluence efficiency. Thus, within the stated model, directional fluence inefficiency is exactly geometric deviation from isotropic second-moment structure. Equivalently, worst-case directional certification reduces to balanced directional information geometry: optimal acquisition is precisely isotropic second-moment structure.

We introduce two pre-acquisition isotropy diagnostics: the isotropy defect  and directional conditioning ratio , related by  exactly, and extend the framework to task-weighted geometries, showing that optimal acquisition is isotropic relative to the task orientation distribution. Applied to standard limited-angle breast tomosynthesis geometries, the overhead formula reveals substantial model-predicted fluence penalties for contiguous arc designs, while symmetric multi-arc layouts satisfying C₂ = 0 achieve the isotropic lower bound at equal projection count. These results constitute a reconstruction-independent certification layer enabling pre-acquisition directional sufficiency evaluation, identification of geometrically redundant projections, and principled view allocation under constrained budgets and objective geometric audit of acquisition protocols without reconstruction or phantom data.

The optimality condition reflects a broader recurring phenomenon in systems with quadratic directional coupling: anisotropy is governed by deviation of the second directional moment from isotropy, and isotropic second-moment structure characterizes the optimum. In this sense, the tight-frame condition underlying optimal CT acquisition belongs to the same mathematical class as the magic-angle condition in NMR spectroscopy and spherical 2-design optimality.